A Globally Optimal Portfolio for m-Sparse Sharpe Ratio Maximization
Authors: Yizun Lin
Zhao-Rong Lai
Cheng Li
Venue: NeurIPS 2024 poster (2024)
Keywords: Sharpe ratio
$\ell_0$ constraint
proximal gradient algorithm
global optimality
Primary area: optimization
TL;DR: We exploit the Kurdyka-Lojasiewicz property to develop an efficient proximal gradient algorithm that leads to a portfolio which achieves the globally optimal m-sparse Sharpe ratio.

Abstract: The Sharpe ratio is an important and widely-used risk-adjusted return in financial engineering. In modern portfolio management, one may require an m-sparse (no more than m active assets) portfolio to save managerial and financial costs. However, few existing methods can optimize the Sharpe ratio with the m-sparse constraint, due to the nonconvexity and the complexity of this constraint. We propose to convert the m-sparse fractional optimization problem into an equivalent m-sparse quadratic programming problem. The semi-algebraic property of the resulting objective function allows us to exploit the Kurdyka-Lojasiewicz property to develop an efficient Proximal Gradient Algorithm (PGA) that leads to a portfolio which achieves the globally optimal m-sparse Sharpe ratio under certain conditions. The convergence rates of PGA are also provided. To the best of our knowledge, this is the first proposal that achieves a globally optimal m-sparse Sharpe ratio with a theoretically-sound guarantee.

A Globally Optimal Portfolio for m-Sparse Sharpe Ratio Maximization
Yizun Lin'
Zhao-Rong Lai' 'Department of Mathematics
Cheng Li'
College of Information Science and Technology Jinan University, Guangzhou, China {linyizun, (aiiizuu,,aizzrrrrrrr laizhr]@jnu edu. cn 1i 1ichengdstu2020. jnu edu, cn
Abstract
The Sharpe ratio is important and widely-used risk-adjusted return in financial engineering. In modern portfolio management, one may require an m-sparse (no more than active assets) portfolio However, few existing methods can optimize the Sharpe ratio with the m-sparse constraint, due to the nonconvexity and the complexity of this constraint. We propose to convert the m-sparse fractional optimization problem into an equivalent m-sparse quadratic programming problem. The semi-algebraic property of the resulting objective function allows us to exploit the develop an efficient Proximal Gradient Algorithm (PGA) that leads portfolio which achieves the globally optimal m-sparse Sharpe ratio under certain conditions. The convergence rates of PGA are also provided. To the best of our knowledge, this is the first proposal that achieves globally optimal m-sparse Sharpe ratio with a theoretically-soutt guarantee.
managerial and financial costs.
property
Introduction
The Sharpe ratio (SR) [33] is an important and widely-used performance metric in finance. Suppose an investing strategy is represented by a portfolio w E RN of N assets from a financial market. RN and E Nx denote the expected return vector (in excess of the risk-free rate) and its covariance matrix for the assets, respectively. can be seen that fu and Vwe Ew represent the expected return and its standard deviation (i.e., risk) for the portfolio w. The original definition of SR is given as the follow quotient between return and risk:
wc /
So(w)
(1.1)
Ew
Ever since the proposal of SR, how to maximize it becomes an attractive research topic. Ordinary portfolio optimization methods based on either the mean- variance approach [5, 10] or the exponential growth rate approach [22, 24J can reduce the portfolio risk and increase the portfolio return to some the other hand, direct SR optimization methods are
extent [23], and hence improve the SR. also proposed. Hung et al. [18], Yu and Xu [35] consider the SR as differentiable function of the portfolio, which can be solved via the augmented Lagrangian method. Pang [29] converts the SR maximization under the self-financing and long-only constraints into linear complementarity problem, which can be solved via the Parametric Linear Complementarity Technique (PLCT) and the
principle pivoting algorithm [12]. Note that PLCT requires μu; > 0 for at least some asset i in order to be feasible.
"Corresonneence to: Zhao-Rong Lai
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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In modern portfolio management, it is pidely-recognized that the number of selected assets should be restricted to a manageable size, in order keep simplicity and save time and financial costs. Managerial strategies provide approach to achieve this objective, such as the revenue driven resource allocation [11], the endowment model [15], and selling stocks after market crashes [27]. However, the managerial approaches still require intensive administration and abundant experience in management and finance. Hence researchers turn to sparsity models for solutions via the com- putational approaches. In [10], Sparse and Stable Markowitz Portfolio (SSMP) is proposed by imposing & &rregulaarization on the portfolio. Ao et al. [2] further propose mean-variance model with an & constraint based maximum-Shaan Shar----aaaaattiii estimation. In [24], the exponential growth rate (EGR) criterion [1, 13, 19, 23] is exploited to develop a Short-term Sparse Portfolio Optimization (SSPO). Furthermore, Short-term Sparse Portfolio Optimization with fg-regularization (SSPO-Lu) is developed in [28]. In [25], a nonlinear shrinkage of the covariance matrix is proposed to obtain an appropriate size of free parameters. Motivated by this strategy, Lai et al. [22] characterize sparse structure for covariance estimation to construct a portfolio via the machine learning approach.
The é reggllarization and the & constraint cannot control the exact number of selected assets. One has to tune the sparsity parameter to roughly adjust this number. On the other hand, suppose we want to select more than m active assets out of N assets to construct a portfolio, then this can be exactly represented by the m-sparse (or Eu) constraint || /wllo & m, where the to norm || llo denotes the number of nonzero components of vector. Although many sparsity models are established for the Markowitz portfolio, few existing methods can optimize the SR (1.1) with the lo constraint, due to the nonconvexity and the complexity of this constraint. In addition to the Eo constraint, other realistic constraints should also be imposed to ensure feasibility. For example, the self-financing constraint represents full re-investment and no external loans; the long-only constraint represents no short position. If all these constraints are imposed, the whole model becomes even much more difficult solve.
To overcome these difficulties, we observe that this optimization is essentially an m-spprre fractional optimization that can be transformed into an equivalent m-sparse quadratic programming. Then (KL) property
the resulting objective function is semi-algebraic, so that the [3] can be exploited to develop an efficient Proximal Gradient Algorithm (PGA) [30]. It converges to a portfolio which achieves the globally optimal m-sparse SR under certain conditions. To the best of our knowledge, this is the first proposal that achieves globally optimal m-sparse SR with theoretically-sound guarantee. Our main contributions can be summarized as follows.
1) We propose to directly maximize the SR with the to constraint, the self-financing constraint and the long-only constraint on the portfolio. This model aims to obtain a feasible and realistic portfolio that optimizes the SR with exact sparsity.
2) SR maximization is essentially a fractional optimization. We convert this m-sparse fractional optimization problem into equivalent m-sparse quadratic programming problem, which reduces the difficulty of solving it.
3) We observe that the resulting objective function is semi-algebraic,, thus exploit the KL property to develop an efficient PGA that leads to globally or at least a locally optimal solution of the m-sparse SR maximization model. The convergence rates of PGA are also provided.
Besides the above contributions, our approach also has several advantages: (i) It can be extended to a wide range of optimization problems with semi-algebraic objective functions and constraints. (ii) The actual sparsity is robust to the choice of m. (ii) It needs very little parameter tuning. (iv) It does not require any external algorithms commercial optimizers.
2 Related Works and Existing Problems
There are some existing works that indirectly or directly optimize the SR to some extent via the computational approach. We introduce some examples and then analyze some unsolved problems.
2.1 Ordinary Portfolio Optimization
An intuitive approach is to directly optimize the portfolio, so that the expected return is maximized and/or the risk is minimized. These methods can be categorized into the mean-variane approach and the exponential growth rate approach [23]. Let R E RT'N be the sample asset return matrix with T' trading times and N assets, and 1,, denotes the vector of ones. Brodie et al. [10] propose

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to impose & -ergglaarizatinn on the mean-variance model, forming a Sparse and Stable Markowitz Portfolio (SSMP):
L +
i = argmin WERN
R Rw - pprl2 1- + =1120111 s.t. w p = p. w T IN = 1, (2.1)
where á = + R 1r denotes the column vector of sample mean returns, and T 3 0 is the regular- ization parameter. ||-J|a and -| denote the f--norm and the finnrmm, respectively. The quadratic form 7 R2w - plrll actually computes the mean squared error between the sample portfolio return r:() (r-(t) is the t-th of R) and the given return level p. Equations w f and w 1v 1 are the expected return constraint and the self- financing constraint, respectively. Model (2.1) can be approximately solved by surrogate model [10] and the Least Absolute Shrinkage and Selection Operator (Lasso) [34]. Goto and Xu [17] also exploit the Lasso to solve a mean-variance model through sparse hedging restrictions.
Ao et al. [2] propose a Beniimum-Sharpe-rane estimated and sparse regression (MAXER) to approach
mean-variance efficiency. Assume there are sufficient observations T > (N + 2). MAXER first
computes the maximum chauuum--haar-rttiii estimated regression response f. as follows:
; u 0 (T N 2)0.
1+0
fo
(2.2)
T
where j and Σ denote the sample mean and the sample covariance, respectively, is the risk constraint parameter. Then it adopts the Lasso to obtain the portfolio:
+ = argmin WERN T Rw falTlle s.t.
|laa| $
(2.3)
Instead of the mean-variance approach, our method takes an essentially different objective that directly maximizes the SR in (1.1). Besides, our method does not require T > (N + 2).
Based on the exponential growth rate criterion [1, 13, 19, 23), Lai al. [24] propose to minimize kind negative potential return w with é Zy-regularization but without any risk term, forming a Short-term Sparse Portfolio Optimization (SSPO) model
i argmiin {w WERN
4p + +]]l]])) s.t.
w IN 1.
(2.4)
develops an unconstrained augmented Lagrangian with the existence of saddle point that can be solved by the alternating direction method of multipliers (ADMM), Luo et al. [28] further propose the SSPO-& model
i = argmin
{ w s0 + r 1 = w E ERNL w > On and w T "N = 1},
(2.5)
wEA
where A is the N-dimensional simplex. This simplex constraint E A is the combination of the long-only and the self-financing constraints. Under this constraint, the lg-regularization problem
(2.5) has closed-form solution imain : - ii & minjENN 9 + }) with tolerance > where NN {1,
N}. NN,,,
On the other hand, Lai al. [22] propose rank-one covariance estimator based on the operator space decomposition, in order to capture the
risk structure in the financial market:
x V,EUJ, D =2 - Gi -1(( (r(2) (N(T -
0,, Eno = =
where x R + 17xN denotes the price relative matrix and ViEUJ is its singular value decompo- sition (SVD). 8, and u, are the largest eigenvalue and its eigenvector, respectively.
Although the above portfolio optimization methods may partly improve the SR, they may not be competitive to direct SR optimization. Hence direct SR optimization methods should still be developed and investigated.

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2.2 Sharpe Ratio Optimization
Pang [29] proposes to optimize the following SR model:
So(w)) s.t. w A, Cw & d, WEBN
max
(2.6)
where So (w)) is defined by (1.1), C E R'XN and d E R' form a linear constraint for w. It can be transformed into the following equivalent parametric linear complementarity problem:
u p Ew + (C - 1yd
T yy e ON, w e ON,
t1 = - (C - d ) > O2, y > Or,
(2.7)
u w = a - 0
Problem (2.7) can be efficiently solved by the principle pivoting algorithm [12], but it requires μ4 > 0 for least some asset in order to be feasible. Moreover, if we aim to construct an m-sparse w. this approach becomes invalid. In Section 3, we will convert the m-sparse SR optimization into an equivalent m-sparse quadratic programming, and the latter is still nonconvex optimization. We further elaborate proximal gradient algorithm to obtain a globally or locally optimal SR.
Another viable approach is consider the SR as a function of the portfolio and directly optimize it under some realistic constraints. Hung et al. [18] propose the following IPSRM-D model to optimize the SR:
w μ + 11 w Uw 3 {(w)
max WEA
+ K22 T (IN - un)
(2.8)
w Dw
where U E N and D E RNNN are upside and downside risk matrices, respectively, w" T (1N w) is diversification term. K₁ and K2 are hyperparameters that control the strength of upside risk and diversification, respectively. Interested readers can further refer to [35] for some practical estimators for μ. U and D.
However, ((w) in model (2.8) is different from the original SR Sow (1.1) in several significant parts. First, (w) uses second-order moments w" T Uww and w Dw as risk metrics, but So(w) uses the first-order moment Vw T Ew instead. In general, a first-order moment is more appropriate because the expected return w fa should remain in the same order of magnitude as Vw" Ew. Second, the numerator of Sy(w) does not contain any risk term, while the numerator of (ww) contains w Uw. This may change the meaning of SR as an equilibrium point in the efficient frontier based on the CAPM theory [32]. These facts may affect the performance of SR optimization.
Another problem is the lack of effective solving algorithms that could really maximize the SR under constraints. A conventional way is to adopt gradient methods, since -(w) is a differentiable function when * Ov. Hung et al. [18]. Yu and Xu [35] propose to adopt the augmented Lagrangian method to optimize (2.8). Though they do not specify which form of Lagrangian models used, we give the following one without loss of generality:
L(w, = (ww) +
2 ( [I IN - 1)) + A w
(2.9)
where [ww] 1N 1)2 with hyperparameter e & 0 is regularization term for the self-financing constraint, and À E TR is the dual variable with respect to (w.r.t)) us for the long-only constraint w > ON. R. denotes the set of all N-dimensional nonnegative vectors. The update scheme is
w (R+ (k+1) (k) + 71 V u (k)
(8) (())),
(2.
()(+1)= = (1) 7/ V L (k++1) (8)),
where 77, 0 are update step sizes. Note that y is nonconvex function w.r.t. w, and the augmented Lagrangian method is a surrogate method that approximates model (2.8). Hence (2.10) does not necessarily lead to the maximum SR. Worse still, due to the augmented term ((w 1 1)3.
(2.10) may not even decrease the objective function P. Moreover, the Lagrangian (x) X (
increased by the (k) updates but decreased by the (A) updates, hence (2.10) cannot guarantee convergence to point (w", X*) without thorough investigation of the update scheme. To summarize, the augmented Lagrangian method and most existing gradient methods cannot guarantee global or local optimality for model (2.8).

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3 PGA for m-Sparse Sharpe Ratio Maximization
In this section, we formulate the maximization problem of SR as nonconvex fractional optimization under constraints. Instead of directly solving the proposed model, we develop an efficient proximal gradient algorithm to solve a simpler surrogate model (subtraction form) that equivalent to the original constrained fractional optimization model.
3.1 m-Sparse Sharpe Ratio Maximization Model
In order to retain the risk premium meaning of SR in finance, directly maximize the original SR in (1.1) instead of a variant like (2.8). In the perspective statistical estimation, suppose we have a sample asset return (in excess of the risk-free rate) matrix R E RTxN with T trading times and N assets. Then the original SR (1.1) )) can be represented by
R1,Rw
Sww) =
(3.1)
Rw 1 Raw) Rw)l1113 + c//w011
where cl| e||=|] is a regularization term for a positive definite Q. defined in (3.2). The parameter é be arbitrarily-small positive parameter, whose effect the risk term can be negligible. To simplify the notation, we define
p: p: -
Q
R
1 T
IIxT
and Q. :.===QQ+>
(3.2)
VT
Then the maximization of SR under the m-sparse, long-only and self-financing constraints is given by
S(w) p w
max WEA
(3.3)
Vw Q.w
where the simplex A is defined in (2.5). Note that minimizing S(w)) under the constraint ||w]|o & m essentially quite different from minimizing the fo-regularization version -S(w) > with some positive т. In general, the latter is easier because it incorporates the lg norm into the objective function and enlarges the feasible set by dropping the constraint ||w]oo & m. We simply call (3.3) the m-Sparse Sharpe Ratio Maximization (mSSRM) model. In fact, to solve the mSSRM model, suffices solve the following simpler constrained minimization model
1 T Q.v-p-u {2"
min 020 |
(3.4)
To see this, establish the relation between the solutions of these models in the following theorem, whose proof is provided in Appendix A.1. We define the constraint sets in model (3.3) and (3.4) by
Ω = fw E A ||wllo & m} and Ω = {u E I > ON and ||ollo & m].
(3.5)
respectively. is obvious that ft S
Theorem Suppose that there exists some i E Q2 such that p w > O. If an optimal solution of model (3.3), then _ é solution of model (3.4), then 5 b is an optimal solution of model (3.3).
Q.5" is an optimal solution of model (3.4), Conversely, if 6 is optimal a
Defining the indicator function by
if u E Q; (v) = {000 otherwise,
0,
(3.6)
we can rewrite model (3.4) as the following two-term unconstrained minimization model:
min V (f(v) mn()),, where f(v) = _ TQv pTv.
1
Q
(3.7)
We then turn to solving model (3.7) instead of the mSSRM model in (3.3).
5

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3.2 Proximal Gradient Algorithm
To develop a proximal gradient algorithm for solving model (3.7). we recall the notion of proximity operator, and then establish the relation between the solution of model (3.7) and the proximity characterization (3.8) in Theorem 2. For a proper function v IR" + R. its proximity operator æ1 (u)).. We remark that for function that nonconvex, prox, (ax) may not be unique. Throughout this paper, the formula prox y (x:) represents h E prox, & (a)). For 8² E R. and 8 > O, we denote by B(v"; 6) the neighborhood of v² with radius 8. If there exists some 8 > 0 such that f(e²) < f(v) holds for all Q2 B 8), then say that v" is a locally optimal solution of model (3.7).
æ Rn is defined by prox, (x) ;= argmin {
Theorem 2 Let function io and f be defined by (3.6) and (3.7), respectively. If v is a globally optimal solution of model (3.7), then for any a E 0. 1Q.12
e² prox (v" - -VV(((()))
(3.8)
Conversely, we have the following two statements:
If 2 1 and (3.8) holds, then v² is globally optimal solution of model (3.7).
(ii) For any > 0, if (3.8) holds, then o² is locally optimal solution of model (3.7).
The proof of Theorem 2 is provided in Appendix A.2. Based on this theorem, the Proximal Gradient Algorithm (PGA) for solving model (3.7) can be given by the following iterative scheme:
e (k++)) prox, (v(*) f((())) where E N. a > O, (0)) Q.
(3.9)
We then compute the closed form of prox, For vector E R.N, we denote by my and Jos the number of positive components and the index set of positive components of v. If m. >> m, then we denote by Jm-pos an index set the m-largest positive components v. Specifically, by letting Nen be an rearrangement of {uilsENy such that Un > V/ 2 > Vix then Jmnpos == {ji: J2,......... Jm Throughout this paper, for a given vector a E R. we shall always compute prox,, (v) according to the following proposition. Its proof is given in Appendix A.3.
Proposition 3 Let in be defined by (3.6), v E RN,, and define the index set J by
Ju Impooss
if my > m, if m < m.
J
pos:
if J";
Then the vector h given by h2
10 if E No satisfies that h E prox, (v)-
3.3 Convergence Analysis of PGA
In this subsection, we delve into the convergence analysis of PGA. We aim to demonstrate that PGA possesses the capability to converge to globally optimal solution of model (3.4). The limit point obtained by PGA can also yield globally optimal solution of the original model (3.3), under certain conditions. We also demonstrate the convergence rates of PGA.
Firstly, we introduce a proposition that illustrates the convergence and monotonic decreasing behavior of the objective function values for the iterative sequence, as well the vanishing gap between consecutive iterates. The proof of this proposition is provided in Appendix A.4.
Proposition 4 a (. ). then for arbitrary initial vector (0)) RRN the sequence (olh))res generated by PGA satisfies the following properties:
Let function in and f be defined by (3.6) and (3.7), respectively. and let F: := f + io-
(k) E Ωt for all K E N:
i
F(u(++1)) (allo)))) -x)) & ((u())) for all N, where := (1 1QQel2) >
limg-+20 (R)( exists;
(iv)
limg-+20 (++++) v(0) 0.

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Though we have established the convergence of {F(+()) kEN and the vanishing gap between consecutive iterates, efforss are necessary to rigorously confirm the convergence of the iterative sequence {e(k) JAEN. We demonstrate the convergence of {o(*) JAEN local minimizer of function F and the corresponding convergence rates in the following theorem. In order to maintain consistency with the original SR maximization model (as outlined Theorem 1), we further define sequence (w(k) JANN based {u(k)) kEN and conduct analysis of the convergence rate of (w(() J.NN To prove the theorem, we need to introduce the notions of subdifferential, semi-algebraic function and Bardyka-E.jjsiewiin property, along with several technical lemmas. Detailed proofs and relevant content can be found in Appendix A.S.
Theorem Suppose that there exists some w E Q such that p i > O. For arbitrary initial vector (1()) E RR,, let (v!k) teEN be generated by PGA, and let (w(l)) JREN be defined by
(8)
( #0;
w (k)
T Ix
ON:
otherwise.
Ifac E
(0,
14.112 then the following statements hold:
1
(i) (o(h)) TKEN converge to a locally optimal solution w" of model (3.4) with convergence rates [b(k) -- v²12 = O(1 Vk) and (f(w(*)) - f(v" 0
(ii) The limit point v* of (v(1)) JREN satisfies that v > ON and v* + #N.
'¹ (iii) (20(") converge w² = (o*) 11x with convergence rates ((
0(1/VA)
and ())) S(w")] O(1V/k), where S is defined in (3.3).
In the remainder of this section, we always let v² E Ω be the locally optimal solution of model (3.7) that sequence (x((k) keN converges to. We recall that my and J denote the number of positive components and the index set of positive components of v", respectively, Suppose that there exists some v E Ω such that in > O. Then item (ii) in Theorem 5 together with 8² Ω yields that 1 & my- < m. In fact, v² is also the globally optimal solution of the convex model
min wen 2º²² Q. p- vj. where (v E R p>0n and ey=0 for all E NN NN (3.10) QUu..upTo
Certainly, Ω + Ø due to the condition m, > 1. According to the definition of Ω, it is straightforward to observe that v' E Ω. Furthermore, 8 is a closed convex set and Ω C Q. To analyze the relation between o" and the original m-sparse Sharpe ratio maximization model (3.3), define
Ω {u A/ 02 0 for all j NN J
(3.11)
where A is given by (2.5). It is easy to see that Ω C 21, where 22 defined in (3.5) is the constraint set of model (3.3). We then have the following theorem, whose proof is provided in Appendix A.6.
Theorem Suppose that there exists some E â such that p i > 0, where fi defined in (3.10),
TT
and let wn
n" Io Then the following stutements hold:
v² is the unique globally optimal solution of model (3.10).
(ii) w" is a globally optimal solution of model max S(wr).
west, (iii) If mm- = m, then wn is locally optimal solution of model (3.3).
Item (iii) in Theorem 6 demonstrates that the limit point of the sequence obtained by PGA can yield a locally optimal solution of model (3.3). In fact, according to item (i) in Theorem 2, we have the following theorem that provides sufficient conditions for obtaining a globally optimal solution of model (3.3), whose proof is provided in Appendix A.7.
Theorem 7 Suppose that there exists some i E Si such that p i > O, and let w" (w)) If one of the following two conditions holds:
(i) m <. m;
7

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(it) my- = m and V.f(e")> -E- -e- min E supp(v")) }} for all i Nx Ny\supp("")
then w" is a globally optimal solution of model (3.3).
Combining Theorem 7 and item (tii in Theorem 6, see that the proposed method can obtain a globally optimal solution of model (3.3) when My < m. Even if this condition does not hold, we can obtain least a locally optimal solution. To test validation of PGA's global optimality, we conduct set of simulation experiments, whose details are presented in Appendix A.8. The codes for the simulation experiments are accessible via the link: https: /tt::/gitttt com//inyizun2024// fttp//oooeeeoooeeeeeeeeee for MhloorSimull Simmlatiinn.nnn...
MSSRM/ htt///teeemmmaa/ee http//maann/nneeeeeeeeeeee
We call the existence of in E Ω such that p > in Theorem 6 the Existence of Positive Expected Return (EPER) condition. Although the EPER condition is required to guarantee the convergence of wn to a locally optimal solution of the original model (3.3), the proposed method is still of high practical significance in the case that the EPER condition does not hold. From the proofs of item (i) in Theorem 5 and item (i) in Theorem we see that even if the EPER condition does not hold, the sequence generated by PGA still converges to a locally optimal solution v* of model (3.4), which also the globally optimal solution of model (3.10). In these two models, the objective function 1 Q. v p' (subtraction form) w.r.t. v represents risk minus expected return, whose minimization gives smaller risk and less loss in revenue, even the expected return is not positive. For the case that the expected return is not positive, compared with the failure of Sharpe ratio of fractional form, the globally or locally optimal solution of the subtraction form seems to have more realistic significance. We recall from item (ii) in Theorem 5 that v" may be equal to Ox if the EPER condition does not hold. In this case, we shall set w" Ov and keep all the wealth the risk-free asset to avoid loss in revenue. To close this section, we summarize the whole m-sparse Sharpe ratio maximization method, which we abbreviate to MSSRM-PGA, in Appendix A.9.
4 Experimental results
Extensive experiments with real-world financial data sets are conducted to evaluate the performance
of the proposed mSSRM-PGA. Moreover, we also consider one baseline method: 1/N [14], as well as 9 state-of-the-an art methods: IPSRM-D [18], PLCT [29], SSMP [10], MAXER [2], SSPO [24]. SPOLC [22], S1, S2 and S3 [28], as competitors in the experiments. We use yeal-world monthly benchmark data sets: FF25, FF25EU, FF32, FF49, FF100 and FF FF100MEINY to compare different methods. These data sets are collected from the baseline and commonly used Kenneth R. French's Real- world Data Library². Details regarding these competitors and data sets are given in Appendix A0 As for mSSRM-PGA, we examine three levels of sparsity m = 10, m = 15, m = 20 and set = 10-3 The setting of other parameters are presented in Appendix A.9. The codes of mSSRM-PGA are accessible via the link: https http::////hhbbb http://yy..u/22///////////////// Experiments in -apttttss___app..____
4.1 Results for Sharpe ratios
We adopt the moving windoww trading framework in [23] imitate real-world portfolio management. For each method, we use the asset returns {T or the price relatives (x()) := r(t) + 1N lT= in the time window t = [1 T] to update the portfolio (h(T+1) for the next trading time. On the (T + 1)-th time, we compute the portfolio return by 1) = x) ( - and then turn to the next round where the time window moves to t = [2 (T + 1)] and new portfolio w is computed, This procedure is repeated till the last trading time J, which yields a return sequence (fr)) 1 This sequence can be used to compute the test SR:
Ei 1/
SR- =
V EZ
(E A()) w/(( (19/1))))((---
The I/N strategy does not involve the time window size T. For all other methods, we examine two conventional settings for the time window size in the finance industry [2, 17]: T 60 and T 120.
Phttp://mbor tuck, dartmouth. wwwrtrtttt..euupggg/////ttt////e www..r//encchattee aalittttllabbraaaaaaaaaaaaaaaala .html
8

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Table shows the (monthly) SRs of the 11 compared methods. Because MAXER requires T > (N + 2), it is unavailable on FF100 and FF10OMEINV when T 60. is worth noting that the trivial strategy I/N outperforms most competitors in most situations. The reason is that 1/N diversifies the risk over all the assets, which also an effective risk control approach [14]. However, mSSRM-PGA outperforms all the competitors including 1/N on all the 6 data sets when T 60 and on 5 data sets when T = 120. For example, its SR is more than 70% higher than that of I/N on FF25EU whether = 60 or T 120. Hence mSSRM-PGA achieves competitive SRs with sparse portfolios, which saves much managerial cost.
Table 1: Sharpe ratios of different portfolio optimization methods on 6 benchmark data sets.
Stralegy
FF25
FF25EU
FF32
H149
FF100
FFIOUMEINV FF25
FF25EU
FF32
FF-49
1F100 FFTOMEINN
60
= 120
IN
0.2276
0.1574
0,2057
0.2087
0.2151
0.2276
0.1574
0.2055
0.2087
0.2151
SPOLC
0,1452
0.0315
0.1734 0.0752 0,0562
0,1009
0.1545
0,0350
0.1830
0.1291 0.0988
0.1218
SSPO
0.1544
0,0411
0.1181 0.0588 0.0425
0.0872
0.1789
0.0719
0.1557
0.0601 0.0529
0.1109
S1
0.1497
0.0369
0.1169 0.0559 0.0327
0.0879
0,1789
0.0736
0.1525
0.0648 0.0467
0,0999
S2
0.1382
0.0633
0.1225 0.0573 0.0456
0,1034
0.1578
0.0725
0,1438 0.0605 0.0602
0.1203
S3
0.1428
0.0607
0.0469
0,1100
0,1609
0.0617
0.1215
SSMP
0.1934
0.1596
0.1535 0658 0.0883
0,1448
0,1920
0.0849
0,1512 0.1581 0.0573
0,1495
MAXER
0.1825
0.2229
0.1625
1581 N/A
NIA
0.19
0.2379
0.1465
0.1433 0.1351
0.1479
IPSRM-D
0.2239
0,1994
0.1952
0.1436 0,1766
0,1662
0,2439
0.2358
0.2240 0.1410 0,2012
0.1712
PLCT
0,2475
0.2708
0,2600 0,2119 0.2270
0.2220
0.2468
0.2796
0,2577 0.2025 0,2369
0,2279
MSSRM-PGAAI=10))
0.2481
0.2712
0.2612 0.2151 0.2290
0.2217
0,2472
0.279
0.2592 0.2041 0.2391
0.2271
MSSRM- (m=15)
15)
0.2481
0.2708
0.2615
0.2135 0.2289
0,2232
0.2474
0.2796
0,2592
0.2040 0.2381
0.2293
MSSRM-PGA(m=220)
0.2481
0.2708
0.2615 0.2134 0.2285
0.2234
0.2474
0.2796
0.2592
0.2041 0.2384
0.2292
4.2 Results for Cumulative Wealths
Ordinary investors are also concerned about how much they gain when using an investing strategy.
Without loss of generality, we can set the initial wealth for an investing strategy as S(o) = 1, then the
final cumulative wealth can be conveniently computed by S = II (f(o). + 1). The results of final cumulative wealths shown in Table 2. The two competitors 1/N and PLCT perform well in general. Nevertheless, MSSRM-PGA achieves the best final cumulative wealths on out of the 6 data sets. Besides, it outperforms each competitor on at least 5 out of the 6 data sets. For example, MSSRM-PGA is about 20% higher than the second best competitor PLCT on FF49 when T = 60 and m = 10. On the data sets where mSSRM-PGA is not the best method, it is still the second best method. These results indicate that mSSRM-PGA is an effective strategy for pursuing return gain in practical perspective.
Table 2: Cumulative wealths of different portfolio optimization methods on 6 benchmark data sets.
Strategy
FF23
FF25EU
FF32
F1-99
FF110
FFTOMMEINY HF23 FF2SEU
FF32
FF-49
FF100 FF FFIOMMEINY
120
I/N
355.98
13.05
424.42
235.48
364.87
428.70
355.98
13.05
424.42
235.48
364.87 87
428.70
SPOLC
57.53
0.96
169.58
5.44
2.39
14.05
70.46
1.03
259. 74
100.49
16.03
36.20
SSPO
129.35
1.22
30.20
1.33
0.89
8.98
286.51
2.67
130.21
1.61
1.74
25.62
$1
100.76
1.08
29.47
1.09
0.54
9.25
265.82
2.78
121.47
2.23
1.27
15.57
66.24
2.17
39.27
1.39
1.15
20.45
130.31
2.73
93.13
1.89
2.67
43.66
S3
70.88
2.01
36,88
1,28
1.20
23.73
129,90
2.61
89.51
1.92
2.62
38.70
SSMP
248.67
13.47
158.98
186.79
10.09
154.27
237.45
3.25
149.65
143.18
2.26
222.35
MAXER
173.39
47.56
200.03
142.31
N/A
N/A
216.94
55.71
117.42
98.85
79.82
188.54
IPSRM-D
398.55
37.25
243.83
69.57
240.12
146.40
567.76
77.79
507 47
50.04
457 86
188. 34
PLCT
126.04
918.62
238.27
471.4
354.70
608.65
148.19
854,83
157.50
552.41
399.48
MSSRM-PGA (m=10)
615.34
126.02
991.89
285.02
527.099
375.75
640.89
147.17
928.19
188.38
635.66
421.97
mSSRM-PGA (m=15) 614.71
125.19
996.32
262.54 522.28
383.44
643.44
147.17
927.21
172.95 597.67
435.01
MSSRM-PGA (m=20) 614.70
125.19
996.23
262.06 515.50
384.65
643.44
147.17
927.16
173.27
603.05
433.15
4.3 Results for Transaction Costs
Cumulative wealth with transaction cost can also be tested to see how the transaction cost influences the performance of different methods. We adopt the proportional transaction cost model [8, 26, 21]
z TT (()
(1 V
N
i0o - w ].
S"=
II
2
(r, w (00-1),x((-11))) + ((-1) ((-1)
2
where ((1 the evolved portfolio weight of the i-th asset at the end of the (1 - 1)-th period, and v is the bidirectional transaction cost rate. When the cost rate of buying is the same as that

---

of selling, updating the evolved portfolio () as the next portfolio w()) yields proportional
transaction cost of &/2 Dim1 ((e)) 1(.-)), Figure 2 in Appendix A.10 shows the final cumulative wealths of different methods as V varies from to 0.5% with T = 60. mSSRM-PGA outperforms all other competitors on FF25, FF25EU and FF32 for all v E [0, 0.5%], and on FF100 for & 0.45%. MSSRM-PGA the second best method on FF100MEINY, following 1/N. This is because 1/N naturally keeps small trading volume. Note that a manager for mutual fund with sufficient trades and capital is able to negotiate for a small enough v. Thus mSSRM-PGR PGA is applicable to scenarios with certain level of transaction cost.
4.4 Sparsity for mSSRM-PGA
In this subsection, we examine the sparsity of the portfolios {1)()) generated by mSSRM-PGA. The sparsity can be measured by the cardinality of the support set of (e)); ( For each data set and each setting of m. the mean and the standard deviation (STD) of {|supp( (1)) (1) are computed
to provide a general description, shown in Table It indicates that mSSRM-PGA further increases sparsity compared with the preseted sparsity level m. Moreover, mSSRM-PGA keeps stable sparsity w.r.t. the change of m. For example, the average sparsity for mSSRM-PGA is about 4.9 when 7 = 60 (or 4.4 when T 120) on FF25EU, for all the settings assets increases, MISSRM-PGA gets more advantageous in sparsity. For example, the average sparsity for MSSRM-PGA is about 8 ~ 11 on FF100 and FFI FFIOOMEINV, compared with N = 100. indicates that mSSRM -PAA selects only 8% ~ 11% of the assets in the whole asset pool, while the widely-used I/N strategy has to maintain the whole asset pool. Therefore, mSSRM-PGA can save much managerial cost by reducing the proportion of selected assets, while keeping a competitive performance in SR optimization.
10, 15, 20. As the total number of
Table 3: Sparsity of the portfolios generated by mSSRM-PGIE PGA: supp((t()) ())
m
FF25 FF25EU FF32
FF49 FFI00 FFIOMMEINV FF25
FF25EU
FF32
FF49
FF100
EFIOOMEINY
T' 60
T
120
Mean 6.3511
4.842
6,4159
8.1214 8.0097
7.9175
7.1359
4,4560
7.8255
8,4790
8.3706
8.8722
10
STD 2,4164
2.1763
2.3654
1.9918 2.2473
2.3915
2.2221
1.4648
2.4464
1,9080
2.3343
2.0117
Mcan 6,4746
4.9352
7,4286
9.0000 8.9709
9.0825
7.1637
4.4430
7.4919
9.6003
9,8754
10.5906
STD 3.0451
2.3573
3,0590
3.0713 3.3964
3.5906
2.2995
1.4462
2.2567
3.1731 3.6169
3.4845
Mean 6.4763
4.9352
7,4692
9.0421 9.1974
9.2994
7.1637
4.4430
7,4919
9.6828
10.0437
10.7621
STD 3.0462 2.3573 3.1734
3,1620 3.8949
4.0125
2.2995
1.4462
2.2567
3.3349
3.9160
3.7460
5 Concluding Remarks
The Sharpe ratio (SR) is very important measurement for the performance of returns attributable to risk in finance. On the other hand, modern portfolio management usually restricts the number of selected assets relatively small size, order to save managerial and financial costs. The m-sparse (fo) constraint is an exact constraint for a sparse portfolio, but it is nonconvex and complex. Thus few existing methods can optimize the SR with the m-sparse constraint. In this study, we convert the m-sparse fractional optimization problem into an equivalent m-sparse quadratic programming problem. Then we develop an efficient, masy-to-implemet and mathematically sound proximal gradient algorithm to solve this nonconvex problem. We theoretically prove that this algorithm yields a portfolio that achieves the globally optimal m-sparse Sharpe ratio under certain conditions.
We conduct extensive experiments on real-world monthly benchmark data sets built on the Kenneth R. French's 's widely-used public data library. The numerical results demonstrate that the proposed MSSRM-PGA improves the SR, compared with 9 state-of-the-art portfolio optimization methods including SPOLC, SSPO. SI. S2 S3, SSMP, MAXER. IPSRM-D. PLCT and one baseline method I/N. For another evaluating metric cumulative wealth, MSSRM-PGA outperforms each competitor on at least 5 out of the 6 data sets. Besides, mSSRM- PGA can withstand a considerable level of transaction cost rate. Sparsity experiments indicate that MSSRM-PGA M-A successfully generates portfolios with stable sparsity, and its advantage increases as the size of the whole asset pool increases. In summary, the proposed MSSRM-PGA is promising approach in managing portfolios or other financial issues, which is worth further investigations. limitation of this research lies in its inability to directly apply to fractional optimization models featuring nondifferentiable numerator and denominator. Future work will strive to broaden the theoretical and methodological foundations, ultimately enabling its application to broader spectrum of fractional optimization models in machine learning.
10

---

Acknowledgements
The authors thank the anonymous reviewers for their constructive comments and valuable suggestions in improving this paper. This work was supported in part by National Natural Science Founda- tion of China under Grants 12401120 and 62176103, in part by Guangdong Basic and Applied Basic Research Foundation under Grants 2021 10541 and 2023B 151512006666 in part by the Science and Technology Planning Project of Guangdong under Grant 2023A0505330013 and in part by the Science and Technology Planning Project of Guangzhou under Grants 2024A0413940, 2024A0419896, 202206030007, Nansha District: 2023ZD001 and Development District: 2023GH01.
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12

---

A Appendix
A.1 Proof of Theorem
To prove Theorem 1, we need the following lemma.
Lemma 8 Suppose that there exists some vo E Pi such that p i > 0. If 6 is optimal solution of model (3.4), then 6 + On and p T
Q. > O.
Proof. Since é is an optimal solution of model (3.4), we know that ê E Qt Let w == p
w. The
Q.
facts E Q and p T > 0 imply that w E Ω. Then it follows that
1 (p D)²
2 w Q p b - <
< 0.
2 w Q.
Hence p > f Q > 0 and ú + ON. Now by letting
= 8 Q.0 o, then and
(p) 0)2
8 T Q. p 0 2 Q.u p-u=
(A.1)
2 0 Q.o
Multiplying both sides of (A.1) by 26 Q. ú yields
p i 5 Q.0)² <0,
which implies that ò 6 Q. 0.
Proof of Theorem 1. Let be optimal solution of model (3.3). Then 2 and
p è O
p
i
> 0.
V 1
Q.
Defining ö = p'ú é, we see that e E Ω and
Q.o
(p 6)2
(p
(p 6)
2 1
Q. T.
p 2 ( Q.092
Q.
20 Q. O.
(A.2)
QQ.
For any u E Ω such that Tuu pT u < O, we have TTu u > 0, u + ON and à = 1x Ω2
Then the fact 0 is optimal solution of model (3.3) implies that
(p 0)
(p u)
u)
(A.3)
Ô QQ.
ū Q.u
u Q.u
Note that
1 (p 2²
1
2 u Q.u pm +
(u' Qau Quu-p"u) u=))²> 0,
2ul Q.u
2u Q.u
which combined with (A.3) and (A.2) yields
1 u Q.u p a u Q.u
1 Tp)
1 (p 6)
1 -
Q.o
T
2
2
Q.ū 2
Therefore, an optimal solution of model (3.4).
Conversely, let be an optimal solution of model (3.4). It follows from Lemma 8 that e / 0 and
p T Q. > O. Thus ê
_ For any u Ω such that p Jo 0, we let
p Q.
Q.
Then u E Ω and
1 0)
1 Q.Q.u p pT== Q.o poo
1 ( 1ppvv) v)²
(A.4)
0 Q.
2
2
20 Q.
Now we let ō b
$
Then ē E Ω. Inequality (A.4) yields that
T
P o
vv v
Vū Q.u
V
Q.
Q.v
Q
Note that 2 C. Ω. Therefore, is an optimal solution of model (3.3).
13

---

A.2 Proof of Theorem 2
To prove Theorem we first investigate the properties of function f in Proposition and then recall two well-known results as Lemmas 10 and 11. Let vb be a function from R" to |-x,, -oo]. Then db is proper if -oo & ((R") and {o E IR" (x)) < }o0 * Ø. Let Φ R" - R be a proper function. We say that & is convex if for any a; y E R and any À E (0, (Ax (1 A)y) <
(rb()) + (1 - A))(y) If there exists 3 > 0 such that vi - lll is convex, then & is said to be 8-strongly convex.
Proposition 9 Let f: R. + R be defined in (3.7). Then the following hold:
f is e-strongly convex on :.;
(ii) Vf ||Q Lippshiit continuous on R
Proof. Let f(v) == f(v) - = Qu p_v, v. v E RN.. Since the Hessian matrix
Q D'Qof of f is positive semidefinite, we know that f is convex EN (see Proposition B.4 of [7]). Thus item (i) holds. The gradient of f given by Vf(()) = Q.u p. For all I, E R
[Vf(z) Vf(y)ll2 1|Q.112|= yll2. which implies item (ii).
Lemma 10 (Proposition A.24 of (71) Let function vb R"
+
I be difserentiable with L-
Lipschitz continuous gradient, where L >> Then
L xx1
(v)) (bxx) & (Vo(a), m) +
2
holds for all x. y. E IR..
Lemma (Exercise 17.5 of [6]) Let vp IR" + be differentiable and B > 0. Then si is B- strongly convex if and only if
( - (x))) > (Vv(a)). - x)) + x
holds for all æ, y.y. 1."
Proof of Theorem 2. We first show that (3.8) holds when v* is globally optimal solution of model (3.7). By the definition of proximity operator, (3.8) is equivalent to
1 I|u - +aVf(=")1
x² argmin (u) + 2
WERN
that is,
1 in (u) lu v"" aVV( > in(²)) + a (b=)), for all u E R
1
According to the definition of 'q in (3.6) and the fact v² E Ω, the above inequality can be simply rewritten as
1
(V((v)), u v*) +
u
0. for all u E Q.
(A.5)
2a
To prove (3.8), it suffices to show that (A.5) holds. From Proposition 9, we know that f is e-strongly convex and Vf is ||Q::l-Lipschic continuous on R.N. Then Lemma 10 yields that
f(u) f(0') (VI(v"), v²) ||Q
2 | "² v*1 for all E Ω.
(A.6)
Since o" is globally optimal solution of model (3.7), f(u) ʃ(x*) > for all E Ω, which together with (A.6) and the fact E (0, | implies (A.5). This proves that (3.8) holds.
Conversely, if a > ! and (3.8) holds, then we have (A.5). Recall that f is c-strongly convex. It follows from Lemma 11 that
f(u) f(v*)= (Vf(x"), wa) + 2
1
14

---

which together with the fact a >> and (A.5) implies that f(u) - f(())(())))))))))){))) > 0 for all u Q. Thus the assertion item (a) holds.
We then prove item (ii). The fact (3.8) holds implies (A.5). For d >> 0, we define
) v²) 0). Ωg = {u E B(v"; (Vf(v")
Note that when u tends to v², the quadratic term L u o" 1 is of higher order infinitesimal than the linear term (Vf((") u ----% v-)l. There must be some 5 > 0 such that
2a
(Vf(v"), 1- u->) au
for all a E B00":5)(.2..
(A.7)
2a
We then show that
(V((v"), 0²)=0, for all (B(0":8)\(ss) no
(A.8)
Otherwise, there exists some (B(o);6)10(s nit such that (Vf(v"), -0) 0. It follows from (A.7) that
(Vf(v"), ú 10--0 0²) + 1
v 111 <
2a
which contradicts (A.5). Hence (A.8) holds. This together with the definition of Ωg yields that
(Vf(="), u- 0*) 0, for all E B(o"; 5)) n
(A.9)
Recall that f is convex and differentiable on R,, According to (A.9) and the first order condition for convexity (Proposition B.3 of [7]).
f(u) f(0") > (Vf(v"), v²) > 0, for all B(o"; n
which implies that vr is a locally optimal solution of model (3.7).
A.3 Proof of Proposition 3
Proof. By the definitions of to and its proximity operator, have
prox sn (v)
argmin ||u v/22.
uef
To prove that h E prox, (v), it is equivalent to show that
|| - 011 < Iu 1 for all E Q.
(A.10)
For any u E Ω2 there exists an index set Ju E NN with m elements such that u, = 0 for all j E N Ju. Let Jegg be the index set of negative components in and Ju (NN (Ju U. Since > ON, ||u w112 > Ejer uP. Let J/ Nv Then Jhg C JJ and | - v v122 = Σje uz.
If mm > m, then J = j
m-ppos We are easy to see from the definition of J m.pos that
Σ
Σ
of
Σ VENN In (
3 0.
JEJ
JEJ
JENN (J.UJJ)
If m. & m, then _ J and Ejery 03 -EjEY, 13 ESECSNNLJUIR - 03 0) 20. Now we conclude from the above two cases that
2
1/ Σ 0320.
jE
JEJ
that is, (A.10) holds. This completes the proof.
15

---

A.4 Proof of Proposition 4
Proof. Item (i) follows from (3.9) and the definition of prox directly. Then we have that to (0((())))))))))))))())
0 for all k E N. To prove item (ii), it suffices to show that
f(blk+1) + all
-0(1)1 1/2 & f(o(k)), for all & N.
(A.11)
Note that 1Qc112 0, since (0.112.11 (12 Let
v(() p(u) 2 u (x)) + a (()(((()))) ||/ + un(u), UER"
1
(A.12)
Then (3.9) implies that (++1))))) - )((((k)) that is.
(V v((+1) -((1))
||o(kt)) - for all & E N,
(A.13)
2a
It follows from Lemma 10 that
f(v(++1) f(((()) < (V)(v)))) (((++)) - ((())) 1Qel2 | (A++1) (1
(A.14)
Combining (A.13) and (A.14) yields (A.11). Thus item (ii) holds. Now that F is monotonically
decreasing, according to the monotone convergence theorem, to prove item (iii). it suffices to
show that function F is bounded below Ω. Solving VS(v") 0 gives v² =Q-1p Since I is convex and differentiable on R attains the minimum value at Q₂ p on IR Hence
f(v)= > (Q. = 1/pp" Qp'p for all v E IR,, which implies that F(v) > "/pp" Q_ for all Ω. Therefore, item (iif) holds. Now taking the limit on both sides of the inequality in item (ii) yields item (iv) immediately. This completes the proof.
A.5 Proof of Theorem 5
In order prove Theorem 5, necessary to review several definitions and establish several preliminary results. First, We recall the notions of subdifferentials and critical point. The lower limit of function &b x and the domain of sb are defined by
lim inf (by) lim yー
inf ((:())
v'())
(A.15)
8-50
and
dom := = (xc E R (xx) +00):
respectively. We say that lower semicontinuous at E R if (xx) lim inf (uu). If lower semicontinuous at every R²,, then v is lower semicontinuous on R" [31].
Definition (Subdifferentials and critical point) Let vp R R be a proper lower semicontinu- ous function.
i For each x E dom vb, the Fréchet subdifferential of ip at x, written by òu((x), is the set of all vectors R." which satisfy
(
(xxx))
(u, x) > 0.
lim inf
11
x
vfn
When x & dom vb, set drb(x)= =
(ii) The limiting- subdifferential, simply the subdifferential ob at a dom th, written by (w)),, , is defined through the the following closure process
db((z) {u E I Fx + x, vx(x²²) +(x) and nt e drb(x²)) u as k to0).
We call an element aup((z)) subgradient of 1b at a. We say that a critical point of vb if
On E (v)
16

---

We also recall the following known results about subdifferential from Theorem 8.6, Exercise 8.8 (c) and Theorem 10.1 of [31], respectively.
Fact 12 For a dom 1b, (u)(ar)) C (x)(
Fact 13 Let 2/1 E - E and 222 IR" + T. be two proper lower semicontinuous functions and E RR, If thi is differentiable on neighborhood of x and 1/2 is finite at æ, then
O + (2) (x) + 0b2(2)).
Fact 14 (Fermat's rule) If E IR" is a local minimizer of v, then On E @v((.).
We shall use Theorem 2.9 of [4], which is recalled as Proposition 15, to prove the convergence of the PGA. For this purpose, recall the notions of Kurdyka-F.ojasievice (KI.) property and KI. function.
Definition (KL property) Let tb R - be a proper semicontinuous function. We say that ti satisfies the KL property at & E dom Oi there exist n E (0, +oo), neighborhood U of a and continuous concave function y (0.n) > [0, +o0 such that
(0) 0:
(ii) >p is continuously differentiable on (0, n) with yp' 0;
(iii) (((()) (+))) (v)((() > for any U Un R" v()) < ((x)
(()) +1).
Definition 3 (KL function) We call a proper lower semicontinuous function th R R KL. function if ib satisfies the KL property at all points in dom db.
Proposition 15 Let th IR" + R be proper lower semicontinuous furiicooniuooussuuniiooon Consider a sequence (x( JEEN Re satisfying the following conditions:
ii) There exists a > 0 such that
(k++ + a (o -
for all le e N.
(ii There exist and y (++1) (n(((o((tttt such that
| (k+))) & bllxx((+1)) (k)))
ll2, for all h E N.
ii There exist subsequence (x() JEN,, and x E R such that
lim (k,) = xx and lim (xx())))) = (bxx)). +00
db satisfies the KL property x,, then
lim (k) x² and O, E dú((*).
We then focus on verifying that the sequence (M) KEN generated by PGA satisfies all the conditions in Proposition 15. The satisfaction of item (i) has been shown in Proposition 4. We next consider the satisfaction of item (ii) Proposition 15.
Proposition 16 Let (k( JREN be generated by PGA. Then there exist (1+1) aF(v(++1) and b > 0 such that
(1 < (++)( - v(4) l|2, for k N.
(A.16)
Proof. Let
q( (k+1) = 1 (v((k) ((++))) + - KeN. a
17

---

and function be defined by (A.12). We first prove that ((*+1) E (F(x()+)) It follows from (3.9)
and Fact 14 that O E @p(v((+t)), which together with Fact 13 yields that
a n bl++1) a f( (l for all lo e N.
Note that Io onn- The above inclusion relation can be rewritten as
(v(R) v(++1)) -V(+))))((((())) E din(v((t1)), for all k E N.
(A.17)
+ Now combining (A.17) and the fact (F - yields that q ( FF(((t+1)), N.
We next prove that (A.16) holds. Since f is Q.-Lipschitz continuous,
11a (x+1)) 2 < | (k+1)) (1)) + V [v (S(D())) & blll)(+t)) - (x(()) 12,
where b = ( ||Q I12). This completes the proof.
We then consider the satisfaction of item (iit) in Proposition 15, To this end, we need the following two lemmas.
Lemma Let (o(k)))en be generated by PGA. If E (0, I.a... then (o kkEN bounded.
Proof. We let y || o 2|,, and denote by Amax (Q.), Amin Amin(Q.) (Q.) the maximum and the minimum eigenvalues of Q., respectivell....... Since Q. is symmetric positive definite and < [IQ. Amax (Q.)
we have 7 1- o- Amin (Q.) E (0, 1). From Proposition 3, we are easy see that [prox (v)lla (v))la S ||l||2 for all RV,, which together with (3.9) yields that
[1+0++1100 E (bv(M) f(o ))) = II(I aQc)olk) + +apl12 & <1100 112 + a 2l1pl12
for all & E N. The above inequality implies that
1 1- 1-
||o(+tD)|2 & 1| + allp|2 Σr=
122+aalppll a|ll||2
1 1-7
for all & E N. Therefore, (b(k) JREN is bounded, since y (0,
Lemma 18 Let (u(K) be generated by PGA. vf is an accumulation point of (o(kk))eth then v" E Ω
Proof. Since ot is an accumulation point of (u Jee, there exists a subsequence (u(k)) JEN, of (u()) such that lim (Aa)) v" Note that the set Q is closed and (Aa))) E S2 for all e N+. j_
Hence v" E Q, which completes the proof.
Proposition 19 Let (o((k) keN be generated by PGA and F := f ig- If (0, 14.11a then
there exist a subsequence (x) JEN, of JoeN and o² E Q such that
lim (kk)) = and lim F(w(t))) F(u"). 10000
Proof. It follows from Lemma 17 that {u!k) JReN is bounded. So there exists a subsequence (v(t)) hien_ of (x(*) JeEN converges to some v= E R.N It follows from Lemma 18 that o² E Ω. By the continuity of ff on RN, we have lim f(((ks)) f(v"). We also know that (n()") (n()())) 0 for all k N. Therefore, lim (()()))) F(v"), which completes the proof. D
To employ Proposition 15 for the convergence of PGA. We still need to show that F satisfies the KL property at x². To this end, we recall the notions of semi-algebrai sets and functions, and recall known result in [4, 9] that establishes the relation between semi-algebraie property and KI. property as Lemma 20.
18

---

Definition 4 (Semi-algebraic sets and functions) A subset S C I² is called real semi- -mmi-algebrai if can be represented by
S f ( = 0, (,)) < 0}, ==1==1
(A.18)
where Pij and dij are real polynomial functions for i No. N.. for some 8, N+. A function R - R. is called semi- aoni-algebraic if its graph {(z, ((x)) x dom / is a semi- algebrai subset of R"-1
Lemma 20 Let vo IR + R be a proper lower semicontinuous function. 1/ is semi-algebraic, then satisfies the Kl. property any point of dom Oib == {u [R. (vp( # 23).
Proposition 21 Let F ;= f+ ti where function ig and are defined by (3.6) and (3.7), respectively, Then dom 0F Zi and F is semi-algebraic.
Proof. We first prove that dom OF S. Note that dom F Q2 is closed, which together with the definition of limiting-subdifferenn
implies that dom OF C Ω. For any E Ω, it is easy to verify that ON E (v). Then we see from Fact 12 that Oλ E @v)(v). Now by Fact 13, we have f((v) (v), which means that (F)) + Ø, that is, dom DF. Hence 5 dom OF. This proves dom OF = S2.
We then prove that F is semi-algebraic. From the definition of semi- semi-algebraic function, to see that the sum of semi-al alli-algebraic functions is still eemi-algebraic. It is obvious that function f is semi-algebraic. To prove that F semi-algebraic, it suffices show that in is semi-algebraic. The graph of to is given by
easy
gra 40 {xER' AECENN-. RNI FI.N > ON, | Ilo & and EN+1 = =0}
(A.19)
N where #1.N == (I1, T2 ................................ xN) T Note that there are K == (N-m) combinations to choose an index set with (N m) elements out of the set (1,2...., .). We denote these index sets with size
(N m) by J1, J2...., JR, and let J, J, (N 1), Nx. Then the graph of to in (A. can be represented by
gra to
= j=1
U
(
R 2, 00
( LigJz
n fNEEE
(x B < 00) ]
VEJ,
which implies that 4n is semi-algebraic function. This completes the proof.
We show in the following proposition that the objective function : f + ig satisfies the KL property at any accumulation point of sequence (x(*) JAEN-
Proposition 22 Let (v(k) JeeN be generated by PGA. of is an accumulation point of (b()) JREN, then F satisfies the KL property at v".
Proof. Since o² is an accumulation point of (x) JAEN, it follows from Lemma 18 that v' E Q. By Proposition 21, know that semi-algebraic and v² dom OF. Thus the desired result follows from Lemma 20 immediately.
We then show the continuity of the proximity operator prox,, in the following proposition, which is also required to prove the convergence of PGA.
Proposition 23 Let (x(1)) JEEN C RN be sequence converges some x" Ω and let (k)
prox, (x())) for k N and h" prox,, be given according to Proposition 3.
Then lim h (k) h*
80000
Proof. Since xx E Ω2 max & m. We first consider the case ma- = 0, that is, 2. < for all j E Nv
Then ht = Oy. For all E > 0, we let 61 = VN Then there exists Ki e N such that (0))) < 8 for
19

---

all j E Ny and k > K1. By the definition of h (k),, we know that 0 < ngg < 81 for all j E Ny and
k >
K1. Hence
h (8) h- h (4)
la < VN = for all & > K1,
which implies lim
(k)) h"
We then consider the case 0 < mmx & m. In this case, the set ii E NN 23 > 0} is nonempty. For
all > O, we let 62 min { F post
where
VN
min--oo
min {z5] 23 >> 03.
JEHN
There exists K2 E N such that for all h: 3 K2, ||x ) x²112 112 & 62. which indicates that
2 > 23 - 62 >
(k)
2
min.pos
> 0, for j E J
(A.20)
and
(k)
23 + 82 E 82 <
min--oss for j E NN (J Jraa
(A.21)
By the fact mæ- & m and the definitions of (k) and h", can conclude from (A.20) and (A.21) that for all K2.
11th x. h2 = zJ., for j E Joee
(R)
and
0 & h
(1)
< 62, n3 = 0, for j E NN J
pos-
Thus [[(((k] - 121l2 & VNo2 < E, which yields lim (b) = ht, This completes the proof. k-000
We are now a position to utilize Proposition and Theorem to prove Theorem 5.
Proof of Theorem 5. We first prove item (i). According to Propositions 4, 16, 19 and 22, the convergence of (x(*) to a critical point v' E Ω of F := f + in follows from Proposition 15 immediately. By item (ii) Theorem 2. prove that is locally optimal solution of model (3.7) (or model (3.4)), suffices to show that (3.8) holds. Since lim (kk) v², the Lipschitz continuity of f yields that
lim (b()) - aVf(((()) = -V
Now by letting k + do on both side of (3.9) and employing Proposition 23, we obtain (3.8), which proves the convergence of (x(*) JeEN locally optimal solution v" of model (3.4).
We then prove the convergence rates of (o(* JAEN- Let (v) == || - (*) + a (v))))
RVV. It obvious that 4& 2-strongly convex, since the Hessian matrix of function 4y
is positive semidefinite. To prove the convergence rate, we first show that there exists K E N such that
"f(""), v²)= > O and (o v² - (*(+1))) > 0
(A.22)
for all k > K. From the proof of Theorem in Appendix A.2, we see that (A.9) holds. It has been (see item (i) in Proposition 4). Then there exists Ki such that (kk) B(u; 5) n for k: > Ki, which together with (A.9) implies that the first inequality in (A.22) holds for all k K1. According to the definition of db and (3.9), see that
shown that lim (kk) vn and (A) E for all
1
e (k++) argmin NERN 2
(k)
a
f(u (1) (n(v)= argmin (v).
Ven
By a procedure similar to the first paragraph of the proof of Theorem 2, we can establish that
(*+1) prox,g (b((++1))
V (v(
Note that & is also strongly convex. Using proof analogous to that of the first inequality in (A.22), we can establish the existence of K2 N such that the second inequality (A.22) holds for all k > K2. By setting K max K1, K2). we conclude that (A.22) holds for all k > K.
20

---

It follows from Lemma 11 that
2 (()) > Lx (v(+t))) + (V (v( v² - r((tt1) + 1100 - 012-1))
(A.23)
which together with the second inequality in (A.22) implies that
Φx((") > (x((((+))) + 1|02 (1
that is,
||0" - (8) aVV > ||b((+))
- (h)) + (VS(((())) + 1100 - (1
(A.24)
for all 2 K. For simplicity of notation, we define (R) = w(++1] wb(,, & E N. Expanding (A.24) and dividing the resulting inequality by 2a yields
(V)(v(()), (8)) + 2αx
2 (k)
1
_ -1 2a
1
<
- 1| - 01+ ( -
2a
(A.25)
for all k > K. Recall that (A.14) in the proof of Proposition 4 holds. Since (0, liQ: (A.14)
implies that
f(b((+1)) f(b(k)) ()
_ || 12: for all & e N.
1
(A.26)
2a
Combining (A.26) and (A.25), obtain that
((b(bt1) - (1) +
1 _ ||00 ((+11)
v0" - 1 (VS(v()) --- ((k)), (A.27)
2a
2a
for all k 3 K. It follows from the first order condition for convexity (Proposition B.3 of [7]) that
f(v*) > s(x(()) + (VS(1(()), v² v(()),
which together with (A.27) yields
1
f(u((+1)) + - 11100 ok+ < f(v)) +
2a -100 200)
20
that is,
f(b((+1)) f(()) &
1 (lo(())
-0 - ||o(k+1) - for all & > K.
o'1l2).
2x
We see from (A.11) that (f(r)())
is monotonically decreasing. Then for all j N+.
K K-ý-1
( -
<
Σ
((((ttt))-((0)
i=
K
K-j-1
(1|(00) -00112 - -1 (116t1)-00+11
2a
i=K
1
(lo(k) - ||((K())) - v²1l3)
2a
Hence
[(((Kt)))- - s(0*) < _ (k) - -0"113-
1
(A.28)
2nx
Note that [kx( - 001 [1 constant. Let k: K + and == _l(o(k) - 0013. Then (A.28)
implies that
1
((n(k)) - ((0))) C
2j & C
for > K.
T
Thus
1 - =0 (
(A.29)
21

---

Combining Lemma 11 and the first inequality in (A.22), we obtain that
(( ʃ(0)) > _ _1lo(() 0= for all k > K.
2
This together with (A.29) yields that || 0"112 1000 = O (+) Vk
We next prove item (it). The fact o* 3 O,y follows from (3.8) and Proposition 3 directly. Assume, to reach contradiction, that v" ON Item (i) in this theorem shows that v² is a locally optimal solution of model (3.7). Then there exists > such that
f(0) > f(0²) = s(xx) 0, for all v vEΩn
(A.30)
We recall from the assumption of this theorem that there exists w E Ω such that p ~ > 0. Let Wa = ant. where a > Then ti, Q and p T w. 0. Note that when a tends to 0 the quadratic term ti a, is of higher order infinitesimal than the linear term p W.. There exists some sufficient small a > 0 such that w. B(w";8) and
f(w,, w Q. pT.. < <0,
which contradicts (A.30). Therefore, 8* + ON-
Lastly, we prove item (ii)) Since o" > ON and 0² + ON. There exit 6o > 0 and K' E N such that
(u*)T 1N > Eo and (x)) [T 1N > so for all k > K', and hence w(a) (w(lr) ly for all & e K'. Note that en and {x((k kEN are both bounded. There exist C₁ > and C2 > such that
(kk
( 1|2
& C1 and CIVN (w+) IN
C2.
(v())
1N (00) IN
Then for all k > K"
(R)
(k)
|
(w)) 1N
((o(k))TIN TIN (w+) 1N
-((0()) - =.'
((o(A)) (ovi))TIN| (2o)TTNN
CIvNN - 02112.
and hence
12
20112
(k)
+ (u=TTI 1N
w
(
1N
2
< C1 (
- +
20"
(x+) IN
< (2)
v" 0"112-
This implies that (k) w= 2|2 0 ( (1z) Similarly, we can prove that there exists C33 > 0 such
that
(k) - (((w)()) & C3llw 12,, for all k > K',
This completes the proof. which implies that (k(w())) S(w*)) vi
A.6 Proof of Theorem
To prove Theorem 6, we recall Proposition 11.4 of [6] as the following lemma.
Lemma 24 Let 1b I - Th. be be proper and convex. Then every local minimizer of vb is global minimizer.
22

---

Proof of Theorem 6. We first prove item (t). Let in2 be defined by
if E Q; (v) = ( otherwise.
0,
Then model (3.10) is equilvalent to min F(v)) where F == Of course, 'n is proper. The wERR
convexity S implies that in is convex (see Example 8.3 of [6]). Recall that f is strictly convex, and hence is proper and strictly convex. Since v² is locally optimal solution of model (3.7) and Ô c 22 there exists d > 0 such that
f(u) > f(v"), for all u E B(v";5)
(A.31)
The fact v" E St gives '1(0") O. Then (A.31) implies that F(u) > F(v") for all E B(u";)). that is, v" a local minimizer of F. Now it follows from Lemma that v² also global minimizer of F. The strict convexity of F implies the uniqueness of its global minimizer.
We next prove item (ii). From item (i) in this theorem and item (ii) in Theorem 5, we see that 0 e 9 is globally optimal solution of model (3.10) and v* * Ox. Then are able prove that
p pTo" o* = (w") Qu0" > O. We omit this proof here since it is very similar to the proof of Lemma
Now have v² [.)]
p'v² Q..."
V² For any a E St such that p > 0, we let
p'v U. Then v Q.
5t and p TTu O. Hence
(p 2-
1 TuQuu < 2"
1 vp u)
(v)) Q 2 "Q0'
-p
Q.u p p=u
20 Q.v
which implies that
p 0
p
for all E á
v (v)) Qu
VuTQ.a
Q.
Since o² E Ω and va * ON by the definition of Q4 see that wt E Ω4 Note that 22 Ω. For all w E Qq
p w
pdu v
p w
w Q
Q
V(w*) Q.
which implics that w is globally optimal solution of model max S(w). wei;
Lastly, we prove item (iii). Note that w5- > 0 for all j E Jpm Let w min-pos = min {wz }.
jE
6 = 3 and let w be any vector in B(w";)) n Ω2. Then wy > 26 > 0 for j E Jp and |uy] < for Nv' Wpit Since w e 9 and my- = m, we conclude that wy 0 for j Nx Jps which implies that w E Ω2 It has been shown that wn is an optimal solution of model max S(w).
weit; Therefore, S(w) S(w") for all w E B(w"; n Q2 that is, w" is a locally optimal solution of model (3.3). This completes the proof.
A.7 Proof of Theorem
Proof. According to Theorem and item (1) in Theorem 2, to prove the desired result, it suffices to show that
1 (
prox i0 0²
(A.32)
c
From the proof of Theorem 5, we know that (3.8) holds. According the computation of prox, in Proposition 3, to guarantee the validity of (3.8), we have v.f(v*) = 0 for all i E supp(v"), Otherwise, there exists some in supp(v*) such that vi A of aV which together with Proposition 3 implies that v" * prox, (v* aaVs(o" )), contradiction (3.8).
Suppose that mp- < m. Then we have (z)("") > for all i E Ny (upp(("") Otherwise, there exists some i E Ny supp("")" such that vi a f(v") > 0. Note that m < m. The operation
23

---

of , will preserve the positive value "i a, f. f(v") instead of truncating it as 0, which violates (3.8). In this case, we have [V.f(v*) ii for i E supp(v") and is V.f(e*) & 0 for E Ny (supp(""), which imply that (A.32) holds.
Suppose that item (it) holds. Let 5 = m E supp(v")} since (v') > - and = O, we have of - (V)''")) < 6. Note that my = m. The
0. For i E Nx Ny\sspp(e"
operation of prox, makes uf - vv"" = of for i E supp(v") and 2i Vif('") = 0 for i Ny' (suppp"") that is, (A.32) holds. This completes the proof.
A.8 Validation of PGA's Global Optimality Through Simulation Experiments
To test the validation of PGA's global optimality, we conduct set of simulation experiments by considering model (3.7), where Q. := Q Q ee. The iterative scheme of PGA for solving this model is given by (3.9) with
2
In the simulation experiments, we set E E 1 by Σg := 0.5 and randomly generate matrix Q E R050 10 from the multivariate normal distribution, with mean vector O10 and covariance matrix Σ. We set p as a random vector with components that are randomly generated numbers in the range [-10, 10), and casually set e = 0.001 and the sparsity m = 3.
The direct exhaustive approach enumerates all possible support set configurations, totaling C% 120 cases. In each case, we solve a 3-dimension quadratic programming problem. By comparing the optimal solutions corresponding to these 120 cases, we obtain the exact globally optimal solution of model (3.7). After that, we can evaluate the optimality of PGA's convergence.
For each experiment, we performed 500 iterations of PGA. To ensure the robustness of our findings, we used three different initializations: ON 1, 1N/N and 1v. We repeated the experiments 10* times for each initialization, with different Q and p in each run. We found that in over 7,200 of the 10 trials, for any of the three initializations, both the normalized error of the iterative sequence (l) - 0/// ||2 and the normalized error of the function value (f(v(*) - f(v)((()))))))))r))) f(=5)lllf((")) were smaller than 10-10, Here o" denotes the globally optimal solution, and (k) represents the iterative sequence the k-th iteration of PGA. We show the plots of ||e(()) vz ||22/||0"11 Il2 and (f()(() f()))l/lf I/Il((" [f= in the following Figure 1, and show in Table 4 these two normalized
errors obtained at 500 iterations of PGA in ten simulation experiments.
0.8
0.5
0.4
0.6
0.3
0.4
0.2
0.1
0
0
100
200
300
400
500
0
100
200
300
400
500
Iteration number
Iteration number
Figure 1: Simulation results of PGA for model (3.7), Left: normalized error of the iterative sequence versus number of iterations. Right: normalized error of function value versus number of iterations.
Table 4: The normalized errors obtained at 500 iterations of PGA in 10 simulation experiments.
d
1
2
4
5
6
8
9
10
2.45F-16
1.62- 16
3.03F.
16
6.23F-01
1.14-116
7.68F-16
7.41F-16
9,55F-117
1.50F.
2.52F-16 16
0.00
1.48E-16
1.68E-16
16
1.17E-01
0.00
1.40E-16
4.73E-16
1.36E-16
0.00
0.00
From the simulation experiments, we conclude that the proposed PGA has high probability (over 72%) of directly converging a globally optimal solution of model (3.7). This finding consistent with the sufficient conditions for global optimality in Theorem 7.
24

---

A.9 Solving Algorithm: mSSRM-PGA
Algorithm A1 mSSRM-PGA
Input:
Given the sample asset return matrix R E R and the positive parameter c.
Preparation: Let p = 7 R 1T. Q vT-I R
+ [TxxRR and Q. - Q
Q + +el Compute
the largest eigenvalue of Qc and a A:
0.999
Initialization: Set w p, tol = 10-5, Maxlter = 10 and k = 0.
repeat
1. 2 (k++)) prox
(u(*) (Q. p))
k k
(0(0) piik 1122
until
< tol or l > Marlter.
|o(k-1112
(4) *ON
3. w w(N)
(
else
4. w" = ON
Output: The portfolio w".
A.10
Additional Experimental Results
The 1/N strategy rebalances to the equally- portfolio on each trading time. S1. S2 and S3 are different versions of SSPO-Lo in (2.5), among which SI is deterministic but S2 and S3 are randomized. The hyperparameters of these competitors are set according to the original papers.
FF25 contains 25 portfolios developed by BE/ME (book equity to market equity) and investment in the US market. FF25EU contains 25 portfolios developed by ME and prior return in the European market. FF32 contains 32 portfolios developed by BE/ME and investment in the US market. FF49 contains 49 industry portfolios in the US market. FF100 contains 100 portfolios developed by ME and BE/ME, while FF100MEINV contains 100 portfolios developed by ME and investment, both in the US market. The information of these data sets are given in Table 5.
Table 5: Information of 6 real-world monthly benchmark data sets.
Data Set
Region
Time
Months
Assets
FF25
US
Jul/1971 ~ May/2023 /2023
623
25
FF25EU
EU
Nov/1990 May/2020
391
25
FF32
US
Jul May/ May/2023 /0223
623
32
FF49
US
Jul/1971
Moy/2023
623
FF100
US
Jul/1971 ~ May/ Ma2/2023 /2023
623
100
FF100MEINV
US
Jud/1971 ~ May/2023 2023
623
100
There is relaxation approach based the semi-definite programming (SDP Relaxation, [20]) that intends to address nearly the same mSSRM model (3.3) of this paper, except for relaxing the cardinality constraint and the long-only constraint. Therefore, this method fails to control cardinality exactly and a simplex projection [16] should be implemented to ensure feasibility. Its experimental results also provided in Table 6, which not good those of mSSRM-PGA.
Table 6: Final cumulative wealths (CW) and Sharpe Ratios (SR) SDP Relaxation and mSSRM-PGA on 6 data sets (T' 60).
Data Set
HF25
FF2SEU
HRI:NCH32
FF49
1F100
FF100MHINV
Strategy
CW
SR
CW
SR
CW
SR
CW
SR
CW
SR
CW
SR
SDP Relaxation
323.76
0.2340
14,25
0.1674
290.24
0.2224
280.46
0.2151
0.51
0.0218
194.09
0.1528
mSSRM-PGA (m=10)
615.34
0.2481
126.02
0.2712
991.89
0.2612
285.02
0.2151
527.09
0.2290
375.75
0.2217
mSSRM-PGA (m=15)
614.71
0.2481
125.19
0.2708
996.32
0.2615
262.54
0.2135
522.28
0.2289
383.44
0.2232
mSSRM-PGA (m=
614.70
0.2481
125.19
0.2708
996.23
0.2615
262.06
0.2134 515.50
0.2285
384.65
0.2234
As for practical issues, Table shows the running times for different methods with T 60, where MSSRM-PGA achieves competitive computational efficiency. Figure 2 shows the final cumulative wealths of different methods as the transaction cost rate varies from to 0.5% with 60, which indicates that mSSRM-PGA can withstand considerable levels of transaction cost rates.
25

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Table The average running times (in seconds) different portfolio optimization models for one
period on 6 data sets.
Data Set
SPOLC
SSPO
52
SSMP
MAXER
IPSRM-D
PLCT
SDP Relaxation
MSSRM-PGA
FF25
0.0263
0.0234
5.72F 05
5.631.005
05
8.46E 05
0.0122
0.0525
0,00009
0.0020
1.01155
0,0052
FF25EU
0.0222
0.0239
1.70E-05
2.89E-05
2.59E-05
0.0316
0.0015
0.8178
0.0059
FRENCH32
0.0239
0.0250
2.93E-05
2.81E-05
3.08E-05
0.0075
0.0392
0.0012
0.0021
1.2862
0.0075
FF49
0.0252
0.0458
2.94E-05
3,51E-05
2.82E-05
0.0083
0.0270
0.0029
0.0034
12,3780
0.0114
FF100
0.0306
0.0854
5.38E-05
4.48E.05
4.27E:05
0.0451
0.0458
0.0132
0.0052
24.5852
0,0713
FFIOOMEINV
0.0296
0. 0.0864
5.08E-05
4.81E-05
4,53E-05
0.0145
0.0152
0.0144
0.0059
23,9911
0.0696
700
FESANID
120
33PD
PLCP
600
53
SEP solnaton
MAXER
SPOLC
1-PSRM-
Cunulativ Weath
500
Cumulative Wealth
322
3380
FLEF
80
400
5509
60
300
20
0.4
Transaction Cost Rate x(%)
Transaction Cost Rate x15))
(a) FF25
(b) FF25EU
1000
MAKER
SPOLC
350
SSPO
PLCP
#04/mw001
300
Cumulative Wealth
700
IPERN
Cumulatik Wealth
-55HP
SSSSIN FGA/TH20)
35PO
600
53
SCP eceRon
YSSAN FOARY
500
200
150
200
50
100
00
0.1
0.2
0.3
0.4
0.5
0.1
0.2
0.3
0.4
0.5
Trensaction Cost w1%)
Transaction Cost
(c) FF32
(d) FF49
550
700
MAKER
500
ISSRI
600
450
reaesdon
Cumulative Wealh
Cumulative Wealth
500
51
55HP
HSSAM 7G4/m=20)
NAXER
SPOLC
SSPO
PLCP
200
150
S5MP
200
100
50
01
Transaction Cost Rate v/%]
Transaction Cost Rate o(56)
(c) FF100
FF100MEINY
Figure 2: Final cumulative wealths of portfolio optimization methods w.r.t. transaction cost rate on 6 benchmark data sets.
26

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NeurIPS Paper Checklist
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Answer: [Yes]
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Does the paper discuss the limitations of the work performed by the authors?
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Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
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27

---

Does the paper describe safeguards that have been put in place for responsible release of data models that have high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
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28


Reviews:
[{"review_id": "M6CyXuMkb6", "reviewer": "Reviewer_iiAu", "summary": "This paper studies Sharpe ratio optimization with sparsity constraints. The paper transforms the original m-sparse fractional optimization problem into an m-sparse quadratic programming problem and develops a proximal-gradient algorithm to solve it. Numerical experiments show that the proposed method improves the Sharpe ratio compared with existing methods.", "questions": "See the question in the weakness section.", "limitations": "The authors have addressed the limitations.", "rating": 6, "confidence": 1, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "I am not familiar with this area. This paper first represents the Sharpe ratio by substituting the mean and variance with unbiased estimates, and then rewrites the problem in a quadratic objective. From a purely mathematical perspective, this entire procedure is quite standard. What’s interesting seems to be the convergence guarantee and the experiment performance of the proposed algorithm. I have some questions about them but if they turn out justified, I think this work brings fair contribution.", "weaknesses": "While the paper transforms the original problem to (3.4), the $\\ell_0$ constraint is non-convex. In (3.7), the objective function is still non-convex and it seems to contradict with standard optimization theory that you can guarantee convergence to global optimum in this non-convex objective. Indeed, this is a combinatorial optimization problem and has computation lower-bounds. In sum, I think the theoretical results need further justification on why they do not contradict with existing hardness results.\n\nPOST REBUTTAL: the author justified their results in rebuttal.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "In fact, our method guarantees a locally optimal solution to the non-convex optimization in a general case, which is consistent with standard optimization theory. Only under certain conditions (Theorem 2 $(i)$), the locally optimal solution become a globally optimal solution.\n\nAdditionally, based on Theorem 2 $(i)$, we have now further proven more intuitive sufficient conditions for global optimum, during the process of rebuttal. For a detailed proof, please refer to **1. Sufficient conditions for PGA's convergence to a global optimum** in the **'Author Rebuttal for Global Response'**. \n\nSpecifically, if one of the following two conditions holds for the limit point $v^*$ of the sequence $\\\\{v^k\\\\}$ generated by PGA:\n\n(1) $\\\\|v^*\\\\|_0<m$,\n\n(2) $\\\\|v^*\\\\|_0=m$ and $\\nabla_i f(v^*)>-\\epsilon\\cdot\\min\\\\{v_i^*|i\\in{\\rm supp}(v^*)\\\\}$ for all $i\\in\\mathbb{N}\\backslash{\\rm supp}(v^*)$,\n\nthen $v^*$ is a globally optimal solution of of model (3.7).\n\nWe would like to add that the non-convex fractional model with the non-convex $\\ell_0$-constraint presents a highly complex doubly non-convex optimization problem. Despite this complexity, our new result successfully provides theoretically verifiable sufficient conditions for convergence to the global optimum, which is a significant achievement. These rigorous theoretical results have meaningful implications for sparse multi-objective optimization in machine learning.\n\nTo further demonstrate that our proposed method has a high probability of converging to the global optimum of the model, we conducted additional simulation experiments. For more details of the simulation experiments, please refer to **3. Validation of PGA's global optimality through simulation experiments** in the **'Author Rebuttal for Global Response'**. In our simulations with 10,000 random data sets for each of the three different initializations, over 72% of the experiments for each initialization showed that both the normalized error of the iterative sequence $\\\\|v^k-v^*\\\\|_2/\\\\|v^*\\\\|_2$ and the normalized error of the function value $|f(v^k)-f(v^*)|/|f(v^*)|$ obtained after 500 iterations of PGA was less than $10^{-10}$ from the global optimum. We show the plots of $\\\\|v^k-v^*\\\\|_2/\\\\|v^*\\\\|_2$ and $|f(v^k)-f(v^*)|/|f(v^*)|$ in Response Figure 1, and show in Response Table 1 these two normalized errors obtained after 500 iterations of PGA in ten simulation experiments (see the attached PDF file).", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "C0sizY1Xbw", "reviewer": "Reviewer_EtEi", "summary": "This paper studies the optimization of an m-sparse portfolio, which has an additional sparsity constraint on the portfolio compared to traditional portfolio optimization. \nInstead of the mean-variance approach, this work proposed to directly optimize the fractional objective which is the Sharpe ratio. The paper shows that the m-sparse SR optimization can be converted into an equivalent m-sparse quadratic programming, which is non-convex. It then proposed to use a proximal gradient algorithm to obtain a locally optimal solution.", "questions": "- Directly optimizing the fractional objective can be very unstable with real-world noisy financial data. How does the proposed PGA algorithm compared to the traditional optimization in terms of robustness?\n- What are the computational time and costs for the different baselines?", "limitations": "Yes.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- this paper studies an important and practical problem of sparse portfolio optimization, and it is presented with a good clarity.\n- The paper proposed to directly optimize the fraction - Sharpe ratio, which can be turned into a nonconvex fractional optimization under constraints. Such an idea to alternatively formulate the problem is natural and very interpretable. \n- It is shown theoretically that the proposed proximal gradient algorithm is guaranteed to converge to a locally optimal Sharpe. Experimental results further show that PGA outperforms other baselines in terms of the achieved Sharpe ratios.", "weaknesses": "- Although the proposed PGA algorithm is guaranteed to converge to a locally-optimal solution, there is a lack of analysis and results on how suboptimal the converged solution might be in the worst case.\n- Moreover this is a lack of discussions on the convergence rate for the proposed PGA algorithm.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Answer for Weakness 1**\n\nIn fact, under certain conditions, our method can directly converge to a globally optimal solution. Based on Theorem 2 $(i)$ in the original manuscript, we have now further proven more intuitive sufficient conditions for convergence to global optimum. For a detailed proof, please refer to **1. Sufficient conditions for PGA's convergence to a global optimum** in the **'Author Rebuttal for Global Response'**. \n\nSpecifically, if one of the following two conditions holds for the limit point $v^*$ of the sequence $\\\\{v^k\\\\}$ generated by PGA:\n\n(1) $\\\\|v^*\\\\|_0<m$,\n\n(2) $\\\\|v^*\\\\|_0=m$ and $\\nabla_i f(v^*)>-\\epsilon\\cdot\\min\\\\{v_i^*|i\\in{\\rm supp}(v^*)\\\\}$ for all $i\\in\\mathbb{N}\\backslash{\\rm supp}(v^*)$,\n\nthen $v^*$ is a globally optimal solution of of model (3.7).\n\nWe would like to add that the non-convex fractional model with the non-convex $\\ell_0$-constraint presents a highly complex doubly non-convex optimization problem. Despite this complexity, our new result successfully provides theoretically verifiable sufficient conditions for convergence to the global optimum, which is a significant achievement. These rigorous theoretical results have meaningful implications for sparse multi-objective optimization in machine learning.\n\nTo further demonstrate that our proposed method can indeed converge to the global optimum of the model, we conducted additional simulation experiments. For more details of the simulation experiments, please refer to **3. Validation of PGA's global optimality through simulation experiments** in the **'Author Rebuttal for Global Response'**. In our simulations with 10,000 random data sets for each of the three different initializations, over 72% of the experiments for each initialization showed that both the normalized error of the iterative sequence (NEIS) $\\\\|v^k-v^*\\\\|_2/\\\\|v^*\\\\|_2$ and the normalized error of the function value (NEFV) $|f(v^k)-f(v^*)|/|f(v^*)|$ obtained after 500 iterations of PGA were less than $10^{-10}$. We show the plots of NEIS and NEFV in Response Figure 1, and show in Response Table 1 these two normalized errors obtained after 500 iterations of PGA in ten simulation experiments (see the attached PDF file).\n\nIn 10,000 experiments, we averaged the normalized errors for cases that did not converge to the global optimum ($>10^{-5}$). The average NEIS values for the three different initializations were 0.5132, 0.5550, and 0.6123, while the average NEFV values were 0.0889, 0.1080, and 0.1334, respectively. This indicates that even when not converging to the global optimum, the local optima achieved by our algorithm exhibit good overall performance.\n\n\n**Answer for Weakness 2:**\n\nThanks for the insightful comments regarding convergence rate. We have conducted a rigorous theoretical analysis to conclusively demonstrate that the convergence rate of PGA in terms of the function value is $f(v^k)-f(v^*)=O(1/k)$. Since this proof is relatively lengthy, we did not include it here. In the upcoming revised version of the manuscript, we will incorporate the comprehensive proof to substantiate this finding.\n\n**Answer for Question 1:**\n\nWe have carefully transformed the fractional model (3.3) into an equivalent quadratic model (3.4) in subtractive form. Hence we just need to solve the quadratic model (3.4), which is more robust than directly solving the fractional model (3.3) with respect to the data noise.\n\n\n**Answer for Question 2:**\n\nThe time complexity of S1, S2 and S3 is $O(TN+Nlog(N))$, and the time complexity of PLCT is $O(TN+N^2)$, where $T$ and $N$ denote the window size and the number of assets, respectively. On the other hand, the time complexity of SSMP, SPOLC, SSPO, IPSRM-D, MAXER and mSSRM-PGA is $O(TN\\epsilon^{-1})$, where $\\epsilon$ denotes the convergence tolerance in the objective function value. Response Table 2 in the attached PDF shows the running times for these methods, which are consistent with their time complexities. Therefore, mSSRM-PGA is competitive in time complexity, and achieves good investing performance.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "6FPCfCY9nM", "reviewer": "Reviewer_4h7E", "summary": "In summary, this paper studies Sharpe ratio optimization in portfolio management and contributes the achievement of sparse distribution iterates converging to a local optimum by converting the fractional optimization problem into a quadratic programming.", "questions": "Major Questions:\n- Why are self-financing and long-only constraints needed?\n- What does Section 2.1 accomplish? It is too superficial with no heed towards the roles of the introduced models and parameters in their respective optimization scenarios, akin to a laundry list of past works. How are they related to SR maximization? It appears each method aims for something different.\n- What does \"guarantee suboptimality\" mean?\n- Considering the definition of $Q_\\epsilon$, when is (i) possible?\n- What is the convergence rate?\n\nMinor Questions:\n- Page 1 Line 33: How is \"market crashes\" from [25] a strategy?\n\nSuggestions:\n- Please correct the reference numbering.\n- Page 6 Line 240: Grammar error.", "limitations": "The authors addressed the limitations. For improvement, they are suggested to update their limitations with lack of convergence rate and global optimality.", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "Originality:\nThe task of optimizing SR with sparse distributions is somewhat new. \nThe work incorporates quadratic programming and its well-known techniques into SR optimization in a novel way.\n\nQuality:\nThe submission is technically sound.\nThe existing claims are generally supported by theory and experiments.\nThe methods are appropriate.\n\nClarity:\nThe writing of the submission is mostly clearly, well organized and adequately informs the reader.\n\nSignificance:\nThe work advances the state of the art in a demonstrable way with its theory.\nOther researchers and practitioners are likely to use the ideas, possibly build on them, considering the experimental results.\nWith a somewhat unique theoretical approach, it provides unique conclusions about existing optimization targets in the form of sparse optimization.", "weaknesses": "Originality:\nIt is not clear how this work differs from previous contributions beyond the fact that they did not study this setting. Due to this, it is also possible that the related work is not adequately cited. More explanations are needed in that regard.\n\nQuality:\nThis is more of a work in progress. The authors are at times not careful and possibly honest about evaluating their work. Their achievement is only to a local optimum and the feasibility of the scenario for global optimality is questionable. They also do not provide convergence rates, which is important in performance analysis.\n\nSignificance:\nThe importance of the results is a bit questionable due to the lack of specific comparisons with the literature. Hence, it is also not demonstrated that the submission addresses a difficult task in a better way than the previous works.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Answer for Weakness 1 (Originality):**\n\nThe crucial contribution of our work is maximizing SR under two constraint simultaneously: the m-sparse (cardinality) constraint and the simplex constraint. While there are indeed a bunch of works elaborating SR maximization under various constraints, few of them consider the former two constraints, especially the m-sparse constraint. This setting makes sense because the m-sparse constraint can accurately control the size of selected assets, and the simplex constraint ensures feasibility of the portfolio in practical issues.\n\nTo further improve the completeness and contribution of our work, we have added three main components: 1. sufficient conditions for PGA's convergence to a global optimum; 2. analysis of the convergence rate of PGA; 3. validation of PGA's global optimality through simulation experiments. Please refer to **'Author Rebuttal for Global Response'** for more details.\n\n**Answer for Weakness 2 (Quality):**\n\nOur work is already a complete one that elaborates the best theoretical results on the addressed problem. We have already provided the loosest conditions under which local and even global optimality is guaranteed. Based on the existing results in the original manuscript, we have now proven more intuitive sufficient conditions for the convergence of PGA to a globally optimal solution of model (3.7). For a detailed proof, please refer to **1. Sufficient conditions for PGA's convergence to a global optimum** in the **'Author Rebuttal for Global Response'**. We also conducted a set of simulation experiments to show that the proposed PGA has a high probability (over 72%) of directly converging to a globally optimal solution. For more details of the simulation experiments, please refer to **3. Validation of PGA's global optimality through simulation experiments** in the **'Author Rebuttal for Global Response'**.\n\nBesides, we have conducted a rigorous theoretical analysis to conclusively demonstrate that the convergence rate of PGA in terms of the function value is $f(v^k)-f(v^*)=O(1/k)$. Since this proof is relatively lengthy, we did not include it here. In the upcoming revised version of the manuscript, we will incorporate the comprehensive proof to substantiate this finding.\n\nWe would like to add that the non-convex fractional model with the non-convex $\\ell_0$-constraint presents a highly complex doubly non-convex optimization problem. Despite this complexity, our new result successfully provides theoretically verifiable sufficient conditions for convergence to the global optimum, which is a significant achievement. These rigorous theoretical results have meaningful implications for sparse multi-objective optimization in machine learning.\n\n**Answer for Weakness 3 (Significance):**\n\nThis is because few approaches are proposed to solve the proposed problem (3.3). We have to compare our method with other ones that solve simpler models like (2.6) and (2.8). For another example, [Kim et al., 2016] develops a semidefinite programming (SDP) relaxation of (3.3). As suggested by Reviewer pxYT, this method fails to control cardinality exactly and actually solves another simplified optimization model. We add this method as comparison and the results in Response Table 3 (see the attached PDF file) show that it is not so good as ours.\n\n\n**Answer for Major Questions:**\n\n(1) The self-financing constraint restricts the position that an investor can use to ensure feasibility. For example, suppose there are two assets A and B. An investor can eligibly invest 40\\% and 60\\% of the whole position on A and B, respectively. The self-financing constraint is (40\\% + 60\\%)=1 in this case. But if without a self-financing constraint, the investor may invest 50\\% and 70\\% on A and B, respectively. Then the whole position becomes (50\\%+70\\%)>1, which is infeasible. The long-only constraint means that an investor can only buy assets instead of short-selling, which is a conventional setting in portfolio optimization.\n\n(2) These are ordinary portfolio optimization models. Some of them are also competitors in the experiments. Since SR maximization originates from ordinary portfolio optimization, we provide a subsection for such background knowledge to the readers.\n\n(3) It means \"guarantee the convergence to a locally optimal solution\".\n\n(4) Thank you for mentioning the validity of the conditions in item $(i)$ of Theorem 2. Based on this question, we have proven more intuitive sufficient conditions for the convergence of PGA to a globally optimal solution of model (3.7). For a detailed proof, please refer to **1. Sufficient conditions for PGA's convergence to a global optimum** in the **'Author Rebuttal for Global Response'**. \n\nSpecifically, if one of the following two conditions holds for the limit point $v^*$ of the sequence $\\\\{v^k\\\\}$ generated by PGA:\n\n(i) $\\\\|v^*\\\\|_0<m$,\n\n(ii) $\\\\|v^*\\\\|_0=m$ and $\\nabla_i f(v^*)>-\\epsilon\\cdot\\min\\\\{v_i^*|i\\in{\\rm supp}(v^*)\\\\}$ for all $i\\in\\mathbb{N}\\backslash{\\rm supp}(v^*)$,\n\nthen the conditions in item $(i)$ of Theorem 2 holds, that is, $v^*$ is a globally optimal solution of of model (3.7).\n\n(5) We have conducted a rigorous theoretical analysis to conclusively demonstrate that the convergence rate of PGA in terms of the function value is $f(v^k)-f(v^*)=O(1/k)$. Since this proof is relatively lengthy, we did not include it here. In the upcoming revised version of the manuscript, we will incorporate the comprehensive proof to substantiate this finding.\n\n\n**Answer for Minor Questions:**\n\nIt is a shorting (selling) based strategy that exploits market crash events.\n\n\n**Answer for Suggestions:**\n\nWe appreciate the reviewer pointing out the issues with citation numbers and grammatical errors in the manuscript. We will thoroughly review the entire manuscript and make the necessary corrections.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "8iv2DyOS5h", "reviewer": "Reviewer_pxYT", "summary": "This study addresses the problem of Sharpe ratio maximization under a cardinality constraint, referred to in the paper as an m-sparse constraint. Adding a cardinality constraint typically makes optimization problems NP-hard. Existing studies usually approach this problem using heuristic methods or relaxations. The former can be time-consuming and do not guarantee optimality, while the latter struggles to control cardinality accurately. This research proposes transforming the m-sparse fractional optimization problem into an equivalent m-sparse quadratic programming problem, which ensures convergence to a locally optimal solution and allows for more precise control of cardinality. However, as will be discussed in the question section, the study essentially transforms the problem into a mean-variance optimization form. This raises the question of why not simply perform m-sparse mean-variance optimization. Since Sharpe ratio maximization is just one of many Pareto optimal points in mean-variance optimization, addressing the more general problem could be more advantageous. Also, as this paper deals with practical issue (i.e., cardinality constraint), the authors should discuss more about practical implementation of the proposed method (e.g., computational time, adding more practical constraints).", "questions": "The authors emphasize several times in bold that directly maximizing the Sharpe ratio is a key point of this study. However, as shown in equations (3.4) and (3.7), the problem is eventually transformed into a mean-variance optimization form, with Theorem 1 demonstrating how this can be converted back to a solution for the Sharpe ratio maximization problem. This raises the question of why the study is framed as a Sharpe ratio maximization problem. Why not simply present it as m-sparse mean-variance optimization, which is more general? After all, Sharpe ratio maximization is just one of many Pareto optimal points in mean-variance optimization. Is this framing due to overlapping aspects with previous research? Further clarification on this point would be helpful.", "limitations": "The authors mention the inability to directly apply fractional optimization as a limitation. However, I believe that more emphasis should be placed on practical implementation. Since this study defines and solves a problem based on practical needs, more attention should be given to practical implementation rather than purely mathematical formulation. Ensuring that the proposed method can be effectively implemented in real-world scenarios is crucial for its overall utility.", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The proposed method appears to be theoretically sound and effectively controls cardinality.\n- The approach has been tested on various datasets, and the experimental results indicate strong performance.\n- The writing structure is well-organized and the content is presented in a clear and readable manner.", "weaknesses": "- Ultimately, this research focuses on a portfolio optimization problem with a cardinality constraint driven by practical needs, making practical implementation the most critical aspect. However, there is a lack of discussion or analysis on this aspect.\n- Practical implementation would require consideration of various additional constraints (e.g., lower or upper bound of portfolio weights, turnover constraint). It would be beneficial to demonstrate whether the proposed model can accommodate a range of additional constraints.\n- There is a need for a comparative analysis of computation time to evaluate the efficiency of the proposed method.\n- The following paper addresses the exact same problem of cardinality constrained Sharpe ratio maximization. While their approach, based on relaxation, has the advantage of transforming the problem into a convex optimization issue, it fails to control cardinality exactly. \n\n[1] Kim, M. J., Lee, Y., Kim, J. H., & Kim, W. C. (2016). Sparse tangent portfolio selection via semi-definite relaxation. Operations Research Letters, 44(4), 540-543.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Answer for Weakness 1:**\n\nIn modern portfolio management, it is widely-recognized that the number of selected assets should be restricted to a manageable size, in order to keep simplicity and save time and financial costs. Managerial strategies provide an approach to achieve this objective. However, the managerial approaches still require intensive administration and abundant experience in management and finance. Hence researchers turn to sparsity models for solutions via the computational approaches.\n\n**Answer for Weakness 2:**\n\nLower or upper bound of portfolio weights, as well as the turnover constraint, can also be deployed in our method, as long as they are convex and have closed-form proximal mapping. One can replace the simplex constraint by the afore-mentioned constraints. The theory and the algorithm throughout the paper still hold.\n\n**Answer for Weakness 3:**\n\nWe have conducted a rigorous theoretical analysis to conclusively demonstrate that the convergence rate of PGA in terms of the function value is $f(v^k)-f(v^*)=O(1/k)$. Since the proof is relatively lengthy, we did not include it here. In the upcoming revised version of the manuscript, we will incorporate the comprehensive proof to substantiate this finding. This convergence rate result implies that the computational complexity of the proposed PGA is $O(TN\\varepsilon^{-1})$, where $T$ and $N$ denote the window size and the number of assets, respectively, $\\varepsilon$ is the convergence tolerance in the objective function value.\n\nResponse Table 2 in the attached PDF shows that the running time of mSSRM-PGA is competitive to those of the competitors. Hence mSSRM-PGA has high computational efficiency besides good investing performance.\n\n**Answer for Weakness 4:**\n\nThis method fails to control cardinality exactly and actually solves another simplified optimization model. We add this method as comparison and the results in Response Table 3 show that it is not so good as ours.\n\n**Answer for Questions:**\n\nPlease note that (3.4) is an equivalent optimization model to (3.3), but other general m-sparse mean-variance optimization models may not be. Solving a general m-sparse mean-variance optimization model may lead to a solution far from the locally optimal ones of (3.3). Therefore, we have carefully deduced (3.4) from (3.3) while preserving equivalence, which is a novel contribution.\n\nAdditionally, we have now provided two important theoretical results beyond the original manuscript. First, we established more intuitive sufficient conditions for the proposed PGA algorithm to converge to a global optimal solution. Second, we proved that the convergence rate of PGA in terms of the function value is $f(v^k)-f(v^*)=O(1/k)$. For more details of these two results, please refer to Point 1 and Point 2 in the **'Author Rebuttal for Global Response'**. We believe that these new results significantly improve the completeness and contribution of our paper.\n\nWe would like to add that the non-convex fractional model with the non-convex $\\ell_0$-constraint presents a highly complex doubly non-convex optimization problem. Despite this complexity, our new result successfully provides theoretically verifiable sufficient conditions for convergence to the global optimum, which is a significant achievement. These rigorous theoretical results have meaningful implications for sparse multi-objective optimization in machine learning.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We greatly appreciate the reviewers' professional feedback, which has significantly improved this paper. Based on their suggestions, we have added three main components: 1. sufficient conditions for PGA's convergence to a global optimum; 2. analysis of the convergence rate of PGA; 3. validation of PGA's global optimality through simulation experiments. We believe that these components significantly improve the completeness and contribution of our paper.

Next, we provide a detailed description of these three aspects.

**1. Sufficient conditions for PGA's convergence to a global optimum**

We have already proven in Theorem 5 of the original manuscript that the sequence $\\{v^k\\}$ generated by PGA converges to a locally optimal solution of model (3.7).

If one of the following two conditions holds:

$(i)$ $\\|v^*\\|_0<m$,

$(ii)$ $\\|v^*\\|_0=m$ and $\nabla_i f(v^*)>-\epsilon\cdot\min\\{v_i^*|i\in{\rm supp}(v^*)\\}$ for all $i\in\mathbb{N}_N\backslash{\rm supp}(v^*)$,

then $v^*$ is a globally optimal solution of model (3.7).

${\boldsymbol{Proof.}}$ According to item $(i)$ of Theorem 2, to prove the desired result, it suffices to show that

$$
v^*={\rm prox}_{\iota{\tiny\Omega}}\Big(v^*-\frac{1}{\epsilon}\nabla f(v^*)\Big).\ \ \ \ (Eq\ 1)
$$

From the proof of Theorem 5 in Supplementary A.5, we know that

$$
v^*={\rm prox}_{\iota{\tiny\Omega}}(v^*-\alpha\nabla f(v^*))\ \ \ \ (Eq\ 2)
$$

holds. According to the computation of ${\rm prox}_{\iota{\tiny\Omega}}$ in Proposition 3, to guarantee the validity of $(Eq\ 2)$,  we have $\nabla_i f(v^*) = 0$ for all $i\in{\rm supp}(v^*)$. Otherwise, there exists some $i_0\in{\rm supp}(v^*)$ such that $v _ {i{\tiny0}}^*\neq v _ {i{\tiny0}}^*-\alpha\nabla _ {i{\tiny0}} f(v^*)$, which together with Proposition 3 implies that $v^*\neq{\rm prox} _ {\iota{\tiny\Omega}}(v^*-\alpha\nabla f(v^*))$, a contradiction to $(Eq\ 2)$.

Suppose that $\\|v^*\\|_0<m$. Then we have $\nabla_i f(v^*)\geq 0$ for all $i\notin{\rm supp}(v^*)$. Otherwise, there exists some $i_1\in\mathbb{N}\backslash{\rm supp}(v^*)$ such that $v_{i{\tiny1}}^*-\alpha\nabla_{i{\tiny1}}f(v^*)>0$. Note that $\\|v^*\\|_0<m$. The operation of ${\rm prox} _ {\iota{\tiny\Omega}}$ will preserve the positive value $v _ {i{\tiny1}}^* - \alpha\nabla _ {i{\tiny1}}f(v^*)$ instead of truncating it as 0, which violates $(Eq\ 2)$. In this case, we now have $v_i^*-\frac{1}{\epsilon}\nabla_i f(v^*)=v_i^*$ for $i\in{\rm supp}(v^*)$ and $v_i-\frac{1}{\epsilon}\nabla_i f(v^*)\leq0$ for $i\in\mathbb{N}\backslash{\rm supp}(v^*)$, which imply that $(Eq\ 1)$ holds.

Suppose that item $(ii)$ holds. Let $\delta:=\min\\{v_i^*|i\in{\rm{supp}}(\boldsymbol{v}^*)\\}>0$. For $i\in\mathbb{N}\backslash{\rm{supp}}(v^*)$, since $\frac{1}{\epsilon}\nabla_i f(v^*)>-\delta$ and $v_i=0$, we have $v_{i}^*-\frac{1}{\epsilon}\nabla _ i f(v^*)<\delta$. Note that $\|v^*\| _ 0=m$. The operation of ${\rm{prox}} _ {\iota{\tiny\Omega}}$ makes $v_{i}^*-\frac{1}{\epsilon}\nabla _ i f(v^*)=v_i^*$ for $i\in{\rm{supp}}(v^*)$ and $v_{i}^*-\frac{1}{\epsilon}\nabla _ i f(v^*)=0$ for $i\in\mathbb{N}\backslash{\rm{supp}}(v^*)$, that is, $(Eq\ 1)$ holds. This completes the proof.

**2. Analysis of the convergence rate of PGA**

We have conducted a rigorous theoretical analysis to demonstrate that the convergence rate of PGA in terms of the function value is $f(v^k)-f(v^*)=O(1/k)$. Since this proof is relatively lengthy, we did not include it here. In the upcoming revised version of the manuscript, we will incorporate the comprehensive proof to substantiate this finding.

**3. Validation of PGA's global optimality through simulation experiments**

To test validation of PGA's global optimality, we conducted a set simulation experiments by considering the following model:

$$
\min_{v\in\mathbb{R}^N}\left\\{\frac{1}{2}v^\top Q_{\epsilon}v-p^\top v+\iota_{\Omega}(v)\right\\},\ \ \ \ (Model\ 1)
$$

where $Q_{\epsilon}:=Q^\top Q+\epsilon I$ and $\Omega:=\\{v\in\mathbb{R}^N | \ v>0 _ N\ and\ \\|v\\| _ 0\leq m\\}$. The iterative scheme of PGA for solving this model is given by

$$
v^{k+1}={\rm prox} _ {\iota{\tiny\Omega}}\big(v^k-\beta Q_{\epsilon}v^k+\beta p\big),
$$
where $\beta$ is set as $\frac{0.99}{\\|Q\\|_2}$.

In the simulation experiments, we set $\Sigma\in\mathbb{R}^{10\times 10}$ by $\Sigma_{ij}: = 0.5^{|i-j|}$, and use the matlab function 'mvnrnd' to randomly generate a matrix $Q\in\mathbb{R}^{50\times10}$ from the multivariate normal distribution, with mean vector $0_{10}$ and covariance matrix $\Sigma$. We set $p$ as a random vector with components that are randomly generated numbers in the range $[-10,10]$, and casually set $\epsilon=0.001$, $m=3$.

The direct exhaustive approach enumerates all possible support set configurations, totaling $C_{10}^{3}=120$ cases. In each case, we solve a 3-dimension quadratic programming problem. By comparing the optimal solutions corresponding to these 120 cases, we obtain the exact globally optimal solution of $(Model\ 1)$. After that, we can evaluate the optimality of PGA's convergence.

For each experiment, we performed 500 iterations of PGA. To ensure the robustness of our findings, we used three different initializations: $0_N$, $1_N/N$ and $1_N$. We repeated the experiments $10^4$ times for each initialization, with different $Q$ and $p$ in each run. We found that in over 7,200 of the $10^4$ trials, for any of the three initializations, both the normalized error of the iterative sequence $\\|v^k-v^*\\|_2/\\|v^*\\|_2$ and the normalized error of the function value $|f(v^k)-f(v^*)|/|f(v^*)|$ were smaller than $10^{-10}$. Here $v^*$ denotes the globally optimal solution, and $v^k$ represents the iterative sequence at the $k$-th iteration of PGA.

From the simulation experiments, we conclude that the proposed PGA has a high probability (over 72\%) of directly converging to a globally optimal solution of $(Model\ 1)$. This finding is consistent with our newly proven sufficient conditions for global optimality.

Meta-review: I would like to thank both the authors and reviewers for the good discussion. The reviewers agree that the paper is interesting and all recommend acceptance although there is no single strong supporter. Having looked at the paper and read the reviews, I recommend acceptance (but really borderline acceptance), however I urge the authors to seriously take the reviewers feedback into account when preparing the camera-ready version.
Decision: Accept (poster)