LEARNING CONTEXT-AWARE ADAPTIVE SOLVERS TO ACCELERATE QUADRATIC PROGRAMMING
Authors: Haewon Jung
Junyoung Park
Jinkyoo Park
Venue: Submitted to ICLR 2023 (2023)
Keywords: quadratic optimization
convex optimization
reinforcement learning for optimization
graph neural network
contextual learning

Abstract: Quadratic programming (QP) is an important sub-field of mathematical optimization. The alternating direction method of multipliers (ADMM) is a successful method to solve QP. Even though ADMM shows promising results in solving various types of QP, its convergence speed is known to be highly dependent on the step-size parameter $\rho$. Due to the absence of a general rule for setting $\rho$, it is often tuned manually or heuristically. In this paper, we propose CA-ADMM (Context-aware Adaptive ADMM)) which learns to adaptively adjust $\rho$ to accelerate ADMM. CA-ADMM extracts the spatio-temporal context, which captures the dependency of the primal and dual variables of QP and their temporal evolution during the ADMM iterations. CA-ADMM chooses $\rho$ based on the extracted context. Through extensive numerical experiments, we validated that CA-ADMM effectively generalizes to unseen QP problems with different sizes and classes (i.e., having different QP parameter structures). Furthermore, we verified that CA-ADMM could dynamically adjust $\rho$ considering the stage of the optimization process to accelerate the convergence speed further.

Under review as a conference paper at ICLR 2023
LEARNING CONTEXT-AWARY ADAPTIVE SOLVERS TO ACCELERATE CONVEX QUADRATIC PROGRAMMING
Anonymous authors Paper under double-blind review
ABSTRACT
Convex quadratic programming (QP) is an important sub-field of mathematical optimization. The alternating direction method of multipliers (ADMM) is a suc- cessful method to solve QP. Even though ADMM shows promising results in solv- ing various types of QP, its convergence speed is known to be highly dependent on the step-size parameter p. Due to the absence of a general rule for setting p. it is often tuned manually heuristically. In this paper, we propose CA-ADMM (Context-aware Adaptive ADMM)) which learns to adaptively adjust p to acceler- context, which captures the
ate ADMM. CA-ADMM extracts the apatio-tempor dependency of the primal and dual variables of QP and their temporal evolution during the ADMM iterations. CA-ADMM chooses p based the extracted con- text. Through extensive numerical experiments, we validated that CA-ADMM effectively generalizes to unseen QP problems with different sizes and classes (i.e.. having different QP parameter structures). Furthermore, we verified that CA-ADMM could dynamically adjust p considering the stage of the optimization process to accelerate the convergence speed further.
INTRODUCTION
Among the optimization classes, quadratic program (QP) is widely used due its mathematical tractability, e.g. convexity, in various fields such as portfolio optimization (Boyd et al., 2017; Comuéjols et al., 2018; Boyd al., 2014; Markowitz, 1952), machine learning (Keeman et al., 2001; Sha et al., 2002), control, (Buijs et al., 2002; Krupa et al., 2022; Bartlett et al., 2002), and communication applications (Luo & Yu, 2006; Hons, 2001). As the necessity to solve large opti- mization problems increases, it is becoming increasingly important to ensure the scalability of QP for achieving the solution the large-sized problem accurately and quickly.
Among solution methods to QP, first-order methods (Frank & Wolfe, 1956) owe their popularity due to their superiority in efficiency over other solution methods, for example active set (Wolfe, 1959) and interior points methods (Nesterov & Nemirovskii, 1994). The alternating direction method of multipliers (ADMM) (Gabay & Mercier, 1976; Mathematique et al., 2004) is commonly used for returning high solution quality with relatively small computational expense (Stellato et al., 2020b). Even though ADMM shows satisfactory results in various applications, its convergence speed is highly dependent on both parameters of QP and user-given step-size p. In an attempt to resolve these issues, numerous studies have proposed heuristic (Boyd et al., 2011; ;; He et al., 2000; Stellato al., 2020a) or theory-driven (Ghadimi al., 2014)) methods for deciding p. But strategy for selecting the best performing P still needs to be found (Stellato et al., 2020b). Usually is tuned in a case-dependent manner (Boyd et al., 2011: Stellato al., 2020a; Ghadimi et al., 2014).
Instead of relying on hand-tuning learning (RL) learn a policy that adaptively adjusts p to accelerate the convergence of ADMM. They model the iterative procedure of ADMM as the Markov decision process (MDP) and apply the generic RL method to train the policy that maps the current ADMM solution states to a scalar value of This approach shows relative effectiveness over the heuristic method (e.g.. OSQP (Stellato et al., 2020a)), but it has several limitations. It uses a sealar value of p that cannot adjust differently depending individual constraints. And, only considers the current state without its history to determine p. and therefor cannot capture the non-stationary aspects of ADMM iterations. This method inspired us to model the iteration of ADMM as non-stationary networked system. We
recent study (Ichnowski et al., 2021) utilizes reinforcement

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developed more flexible and effective policy for adjusting p for all constraints simultaneously and considering the evolution of ADMM states.
In this study, we propose Context-aware Adaptive ADMM (CA-ADMM), an RL-based adaptive ADMM algorithm, to increase the convergence speed of ADMM. To overcome the mentioned limi- tations of other approaches, we model the iterative cllution-finding process of ADMM the MDP whose context determined by QP structure (or parameters). We then utilize a graph recurrent neural network (GRNN) extract (I) the relationship among the primal and dual variables of the QP problem, i.e., its spatial context and (2) the temporal evolutions of the primal and dual variables, i.e. its temporal context. The policy network then utilizes the extracted spatio-temporal context to adjust p.
From the extensive numerical experiments, we verified that CA-ADMM adaptively adjusts o in consideration of QP structures and the iterative stage of the ADMM to accelerate the convergence speed further. We evaluated CA-ADMM in various QP benchmark datasets and found to be significantly more efficient than the heuristic and learning-based baselines in number of iterations until convergence. CA-ADMM shows remarkable generalization to the change of problem sizes and, more importantly, benchmark datasets. Through the ablation studies, we also confirmed that both spatial and temporal context extraction schemes are crucial to learning a generalizable p policy. The contributions of the proposed method are summarized below:
Spatial relationships: We propose heterogeneous graph representation of QP and ADMM state that captures spatial context and verifies its effect on the generalization of the learned policy.
Temporal relationships: We propose to use a temporal context extraction scheme that captures the relationship of ADMM states over the iteration and verifies its effect on the generalization of the learned policy.
CA-ADMM outperforms state-of-the-art heuristics (i.e., OSQP) and learning-bast based basclines on the training QP problems and, more importantly, out-of-training QP problems, which include large size problems from a variety of domains.
RELATED WORKS
Methods for selecting of ADMM. In the ADMM algorithm, step-size p plays a vital role in determining the convergence speed and accuracy. For some special cases of QP, there a method to compute optimal (Ghadimi et al., 2014). However, this method requires linear independence of the constraints, e.g., nullity of A is nonzero, which is difficult to apply in general QP problems. Thus, various heuristics have been proposed to choose p (Boyd et al.. 20 He et al., 2000; Stellato et al., 2020a). Typically, the adaptive methods that utilize state-dependent step-size pe show a relatively faster convergence speed than non-adaptive methods. He et al. (2000); Boyd et al. (2011) 1) suggest a rule for adjusting P: depending on the ratio of residuals. OSQP (Stellato et al., 2020a) extends the heuristic rule by adjusting P with the values of the primal and dual optimization variables. Even though OSQP shows improved performance, designing such adaptive rules requires tremendous effort. Furthermore, the designed rule for specific QP problem class is hard to generalize to different QP classes having different sizes, objectives and constraints. Recently, Ichnowski et al. (2021) employed RI learn policy for adaptively adjusting p depending iterations. This method outperforms other baselines, showing the potential that an effective rule for adjusting o can be learned without problem-specific knowledge using data. However, this method still does not sufficiently reflect the structural characteristics of the QP and the temporal evolution of ADMM iterations. Both limitations make capturing the proper problem context challenging, limiting its generalization capability to unseen problems of different sizes and with alternate objectives and constraints.
the states of ADMM
Graph neural network for optimization problems. An optimization problem comprises objec- tive function, decision variables, and constraints. When the optimization variable is a vector, there is typically interaction among components in the decision vector with respect to an objective or constraints. Thus, to capture such interactions, many studies have proposed to use graph represen- tation to model such interactions in optimization problems. Gasse et al. (2019) expresses mixed integer programming (MIP) using a bipartite graph consisting of two node types, decision variable
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nodes, and constraint nodes. They express the relationships among variables and the relationships between decision variables and constraints using different kinds of edges whose edge values are associated with the optimization coefficients. Ding et al. (2019) extends the bipartite graph with new node type, "objective'. The objective node represents the linear objective term and is connected to the variable node, with the edge features and coefficient value of the variable component in the objective term. They used the tripartite graph as an input of GCN to predict the solution of MIP.
ML accelerated scientific computing. ML models are employed to predict the results of complex computations, e.g., the solution of ODE and PDE, and the solution of optimization problems, and trained by supervised learning. As these methods tend to be sample inefficient, other approaches learn operators that expedite computation speeds for ODE (Poli et al., 2020; Berto et al., 2021). fixed point iterations (Bai et al., 2021; Park et al., 2021b), and matrix decomposition (Donon et al., 2020). Poli et al. (2020) uses ML to predict the residuals between the accurate and inaccurate ODE solvers and use the predicted residual to expedite ODE solving. Bai et al. (2021); Park et al. (2021b) trains network to learn alternate fixed point iterators, trained to give the same solution while requiring less iterations to converge. However, the training objective all the listed methods is minimizing the network prediction with the ground truth solution, meaning that those methods are risk find invalid solutions. CA-ADMM learns to provide the proper that minimize ADMM iterations, rather than directly predicting the solution of QP. This approach, i.c., choose p for ADMM, still guarantees that the found solution remains equal to the original onc.
3 BACKGROUND
Quadratic programming (QP) A quadratic program (QP) associated with N variables and M constraints is an optimization problem having the form of the following:
1 Px q
min IF
2
(1)
subject to Ar
where x E RR-V the decision variable, P E S is the positive semi-definite cost matrix, q E R is the linear cost term, A E IR xxN is a MM x N matrix that describes the constraints, and and u are their lower and upper bounds. QP generally has no closed-form solution except some special cases, for example A = 0. Thus, QP is generally solved via some iterative methods.
First-order QP solver Out of the iterative methods to solve QP, the alternating direction method of multipliers (ADMM) has attracted considerable interest due to its simplicity and suitability for various large-scale problems including statistics, machine learning, and control applications. (Boyd et al., 201 1). ADMM solves a given QP(P; q, 4, A, u) through the following iterative scheme. At each step f, ADMM solves the following system of equations:
P ol A
AT
x++1
0fe -9
diag(p)
Pet1
21 diag(p)
where a > is the regularization parameter that assures the unique solution of the linear system, P R. is the step-size parameter, and diag(p) E MM is diagonal matrix with elements p. By solving Eq. (2), ADMM finds Z1+1 and 1441. Then, updates y/s and 2, with the following
equations:
ž++ + 2t + diag(p) (1141 ( - 3c)
(3)
2++1 + IT + diag(p)
(4)
V++1 2, + (uag(p)(22)
1).
(5)
where 11 the projection onto the hyper box [L. u.. ADMM proceeds the iteration until the primal primal T = Azz 24 M and dual residuals rduual = Px, + q A A
yt E RV are sufficiently small.
For instance, the termination criteria are as follows:
primmll |/ S primal || 1100 chull
where primal and dual are sufficiently small positive numbers.
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Enxiruement (09gP sclver)
Capture Temporal Contest
x'
QF Pamnacoo
P.g. A.
K GAU (.-a)
s.t. I S Ag
ADM tasstion tsop functoo)
Actor
In S. Arpt Mex
26-2
Gr
Pt
be A Mes
Spataa Contant
Updattd primal var, Apet Updated dual var, vor-
IGAA
HGA
Histary
Current
mag q'x
max p'x
x Px p'x
Ar
primal Re-2 Past dual var Yi-s
Past primal var, Ke-1 Past dual var Ye-1
Curnent primal var Current durl var.y:
Figure 1: An overview of CA-ADMM. CA-ADMM uses the spatio-temporal context, which is extracted by GRNN from the ADMM state observations, to compute for ADMM algorithm.
METHOD
In this section, we present Context-aware Adaptive ADMM (CA-ADMM) that automatically ad- justs P parameters considering the spatial and temporal contexts of the ADMM to accelerate the convergence of ADMM. We first introduce the contextual MDP formulation of ADMM iterations for solving QP problems and discuss two mechanisms devised to extract the spatial and temporal context from the ADMM iterations.
4.1 CONTEXTUAL MDP FORMULATION
As shown in Eqs. (2) to (5), ADMM solves QP by iteratively updating the intermediate variables Tr.ltt. and ze. Also, the updating rule is parameterized with QP variables P.q. 1, A,
We aim to learn a QP solver that can solve general QP instances whose structures are different from QPs used for training the solver. As shown in Eqs. (2) to (5), ADMM solves QP by iteratively up- dating the intermediate variables Zz. Yr, and žt. Also, the updating rule is parameterized with QP variables P. 1. A, In this perspective, a different MDP whose dynamics can be characterized by the structure of QP and the ADMM's up- dating rule. Based this perspective, instances, is equivalent to learning an adaptive policy (ADMM operator) that can solve a contextual MDP (QP instance with different parameters).
consider the iteration of ADMM for each QP as
objective, learning QP solver that solve general QP
Assuming we have QP(P, q, A, u), where P. L, u are the parameters of the QP problem, and (DM where & is the parameters of the ADMM algorithm. In our study, is p, but it can be any configurable parameter that affects the solution updating procedure. We then define the contextual MDP MM- (x, U, T% R, 7), where X (Ix,, St, 1=1,,,, is the set of states, x' Ir, x, Pt p, is the transition function whose behaviour is contextualized with the context c (P. q, A, u. Φ). R is the reward function that returns O when He is the terminal state (i.c.. the QP problem is solved) and -1 otherwise, and 7 [0, 1) is discount factor. We set MDP transition made every 10 ADMM steps to balance computational cost and acceleration performance. That is, we observe MDP states at every 10 ADMM steps and change Pt; thus, during the ten steps, the same actions Pt are used for ADMM. As shown in the definition of TV. the context cltters the solution updating procedure of ADMM (i.e.,
U U= (p:)) (- 1,.... the set of actions, T5 x') =P(
Te A Te' to the given state x). Therefore, it is crucial to effectively process the context information e deriving the policy no (1r.) to accelerate the ADMM iteration.
One possible approach to accommodate e is manually designing feature that serves as a (minimal) sufficient statistic that separates T from the other Tç'. However, designing such feature can involve enormous efforts; thus, it derogates the applicability of learned solvers to the new solver classes. Alternatively, we propose learn context encoder that extracts the more useful context information from not only the structure of a target QP but also the solution updating history of the ADMM so that can context- adaptive policy for adjusting p. To successfully
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n +
Raaty duyky is
by Rects the Sugel Si8t
io
Figure 2: Heterograph representation of QP.
extract the context information from the states, we consider the relationships among the primal and dual variables (i.e., the spatial relationships) and the evolution of primal/dual variables with ADMM iteration (i.e., temporal relationship). To consider such spatio-temporal correlations, we propose an encoder network that is parameterized via a GNN followed by RNN. Fig. visualizes the overall network architecture.
4.2 EXTRACTING SPATIAL CONTEXT INFORMATION VIA HETEROGENEOUS GRAPH NEURAL NETWORK
Heterogeneous graph representation of QP. The primal and dual y/: variables of QP are lated via the QP parameters (P, q, A, I, u). As shown Fig. 2, their relationship can be modeled as heterogeneous graph where the nodes represent the primal and dual variables, and the edges represent the relationship among these variables. As the roles of primal and dual variables in solv- ing QP are distinctive, it necessitates the graph representation that reflects the different variable type information. Regarding that, we represent the relationships of the variables at t-th MDP step with directed heterogeneous graph G, = (V.. E.) As the graph construction is only associated with the current I, we omit the step-index for notational brevity. The node set consist of the primal primal W primal and dual node sets Vana The 7ith primal and m-th dual nodes have node feature s and chual defined as:
m
primal n
Iog 12
dual log ||
loo, Il you < ccoul
(7)
Encoding ADMM state
dual m
1oggo
primall m
logiu primal
piina Ceptuns Vmn Pezes min (Zm (ms Um
km
Encuding ADMM
(8)
1 equality innequalit)
Encoding QP problem
primall and dual denotes the m-th and n-th element of primal and dual residuals, respectively. where r'm
||I Ioo denotes the infinity norm, 1 1_min and 1. ped the indicators whether the m-th primal and r--hh dual residual is smaller than geimal and duaa respectively. eequlityy and innquulttyy are the indicators whether the m-th constraint equality and inequality, respectively.
The primal and dual node features are design to capture ADMM state information and the QP struc- ture. However, those node features do not contain information of g. or A. We encode such QP problem structure in the edges. The edge set IE consist of
The primal-to-prinn
edge set Er defined as {ef (P, + O4(i,)) E [1, x [1, where [1, NJ = (1, ...., .N. i.e., the edge from the primal node to j th primal node exists when p2p
.,
the corresponding P is not zero. The edge Ei
pip has the feature % as [Poj. q..
The primal-to-dual edge set ggpad is defined as {eigh AA + OV(i,) E [1,N] x [I] The .2dd has the feature .2d [A
edge ci
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The dual-to-primal edge set E is defined as feılAi # 0V(1.5) [1, x [1. The edge 'ig
d2p d2p has the feature sij [Ay]]
Graph preprocessing for unique QP representation. QP problems have the same optimal deci- sion variables for scale transformation of the objective function and constraints. However, such transformations alter the QP graph representation. To make graph representation invariant to the scale transform, we preprocess QP problems by dividing the elements of P, q, a, A, with problem-dependent constants. The details the preprocessing steps are given in Appendix A.
Grpah embedding with Hetero GNN. The heterogeneous graph & models the various type of information among the primal and dual variables. To take into account such type information for the node and edge embedding, aware graph attention (Park et al., a0) to embed G. HGA employs separate node and edge en- coders for each node and edge type. For each edge type, computes edge embeddings, applies the attention mechanism to compute the weight factors, and then aggregates the resulting edge embed- dings via weighted summation. Then, HGA sums the different-typed aggregated edge embeddings to form a eypeinndependent edge embedding. Lastly, HGA updates node embeddings by using the type-independent edge embeddings as an input to node encoders. We provide the details of HGA in Appendix B.
propose heterogeneous graph attention (HGA), variant of type-
4.3 EXTRACTING TEMPORAL CONTEXT VIA RNN
The previous section discusses extracting context by considering the spatial correlation among the variables within the MDP step. We then discuss extracting the temporal correlations. As verified from numerous optimization literature, deciding P based on the multi-stage information (e.g., mo- mentum methods) helps increase the stability and solution quality of solvers. Inspired by that, we consider the time evolution of the ADMM states to extract temporal context. To accomplish that we use an RNN to extract the temporal context from the ADMM state observations. At the eh MDP step, we consider / historical state observations to extract the temporal context. Specifically, we first apply HGA I. 9t-1 and then apply GRU (Chung al., 2014) as follows:
G4-148 HGA(G.-1+5)
for ő 0,
1-1
(9)
(10) where P/-145 is updated graph at t + 8, V1-148 is the set of updated nodes of 94 and H4-145 are the hidden embeddings. We define the node-wise context C2 at time as the last hidden embedding H4-1.
H1-145 GRU(V H4-148-1)
for = 1.2. ......,-11 ,11
4.4 ADJUSTING P: USING EXTRACTED CONTEXT
We parameterize the policy Te Ct) via the HGA followed by MLP. Since action Pr is not defined at t, we exclude Pt from the node features G, and instead node-wise concatenate Ca to the node features. For the action-value network Qo Pt, C.), puts. Qo(G,, Pt, C.) has a similar network architecture as To(G,, C.). We train (0 C.) and (G, Pt. C.) on the QP problems of size 10~15 (i.e., N E [10, 15)) with DDPG (Lillicrap al., 2015) algorithm. Appendix C details the used network architecture and training procedure further.
Gt, Pt, and C. in-
Comparison RLQP. Our closest baseline RLQP (Ichnowski al., 2021) also trained using RL. At step f, it utilizes the state z = (f mm) with
dual m
[logio Primaal |ooo log || |occ), Um, Pm:
(11)
min(zm 2m Im Um 2m); 2m - (A.)
then uses an shared MLP The (sst)))) ) to compute Px- We observe that the state representation and policy network are insufficient to capture the contextual information for differentiating MDPs. The state representation does not include information P and and the policy network does not explic- itly consider the relations among the dual variables. However, shown in Eq. (2), ADMM iteration results different (Iz, 14) with changes in P. and A, meaning that T also changes. Therefore, such state representations and network architectures may result in inability capture contextual

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Table 1: In-domain results (in # iterations). Smaller is better (L). Best in bold. We measure the average and standard deviation of ADMM iterations for each QP benchmark, All QP problems are
generated to have S N S 15.
Randon QP
EqOP
Porfolio
SVM
Huber
Control
Laaso
OSQP
81.12 + 21.93
25.52 + 0.73
185, + 28.73
183,79 + 40.88
54.38 + 5.37
30.38 + 24.98
106.17 + 45.70 70
RLOP
69.42 27.35
20.27 + 0.49
28.45 = 5.52
34.14 + 3.84
25.16 = 0.66
26.78 + 15.84
25.61 0.68
CA-ADMM
27.44 + 4.86
19.61 + 0.61
20.12 0.47
25.05 2.30
20.32 = 0.69
17.52 3.88
19.08 1.01
ADMM iters
CA-ADMM
RLQP
OSQP
200
0
10 15
15 100
200
300
- 400
500
600
700
800
900
1000
100
200
300
400
500
600
700
800
900
Random QP size
Figure 3: QP size generalization results. Smaller is better (4). We plot the average and standard
deviation of the iterations as the bold lines and error bars, respectively. The gray area indicates the training QP sizes.
changes among QP problems. Additionally, using single state observation as the input for To may also hinder the possibility of capturing temporal context.
5 EXPERIMENTS
We evaluate CA-ADMM in various canonical QP problems such as Random QP, EqQP, Porfo oorfo SVM, Huber, Control, and Lasso. We use RLQP, another learned ADMM solver (Ichnowski al., 2021), and OSQP, a heuristic- solver (Stellato et al., 2020a), as baselines. Training and testing problems are generated according to Appendix D.
In-domain results. It is often the case that similar optimization is repeatedly solved with only parameter changes. We train the solver using a particular class of QP instances and apply the trained solver solve the same class of QP problems but with different parameters. We use the number of ADMM iterations until it's convergence as a performance metric for quantifying the solver speed. Table 1 compares the averages and standard deviations of the ADMM iterations required by different baseline models in solving the QP benchmark problem sets. As shown in Table 1, CA-ADMM exhibits 2 8x acceleration compared OSQP, the one of the best heuristic method. We also observed that ADMMDMM consistently outperforms RLQP with generally smaller standard deviations.
We evaluate the tree---aafrrabitttyyyyooo and scalability of CA-ADMM by examining whether the models trained using smaller QP problems still perform well on the large QPs. We trained CA-ADMM using small-sized Random QP instances having 10 ~ 15 variables and applied the trained solve to solve large-scaled Random QP instances having up to 1000 variables. Fig. 3 shows how the number of ADMM iterations varies with the size of QP. As shown in the figure, CA-ADMM requires the least iterations for all sizes of test random QP instances despite having been trained on the smallest setting of 10 ~ 15. The gap between our method and RLQP becomes larger as the QP size increases. is interesting to note that OSQP requires relatively few ADMM iterations when the problem size becomes larger. It is possibly because the parameters of DSQP are tuned to solve the medium- sized QPs. We conclude that CA- learn how effectively adjust ADMM parameters such that the method can effectively solve even large-sized QP instances.
Cross-domain results. We hypothesize that our learned QP solvers can transfer to new domains more easily than other baselines. To test this hypothesis, we train CA-ADMM and RLQP on Random QP and EqQP datasets, which do not assume specific QP parameter values or constraints. The trained models evaluated on different QP domains without additional training, i.e., which is equivalent to a zero-shot transfer setting. As shown in Table 2, CA-ADMM consistently outperforms RLQP in absolute ADMM steps and shows less performance degradation when transferring new domains.
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Table 2: Cross-domain results. zero-shot transfer (Random QP EqQP) + X, where e [Random QP, EqQP, Porfolio, SVM, Huber, Control, Lasso}: Smaller is better (L). Best in bold. We measure the average and standard deviation of ADMM iterations for each QP benchmark. The gray colored cell ( denotes the training problems. Green colored values are the case where the transferred model outperforms the in-domain model. All QP problems are generated to have
40 < N + M < 70.
Randon QP
EqQP
Porfolio
SVM
Huber
Control
Laa80
DSqP
160.99 1: 92.21
25.52 1: 0.73
185,18 28.73
183.79 40.88
54.38 5.37
30.38 =: 24.98
106.17 1: 45.70
(transferred)
BLGP
65.28 57.75
28.48 + 0.70
31.80 7.97
37.96 = 10.16
28.70 0.67
74.60 = 49.71
29.18 + 29.18
RLGP (in (in-oomain)
69.42 27.35
20.27 + 0.49
28.45 + 5.52
34.14 3.84
25,16 0.66
26.78 = 15.84
25.61 + 0.68
CA-ADVN
(tranfferee)
28.52 = 7.13
19.38 + 0.64
25,38 7.03
31.26 = 13:44
19.56 + 0.92
27.16 4.97
23.84 = 2.10
CA-ADVN
(in-domain)
27.44 = 4,86
19.61 + 0.61
20.12 + 0.47
25.05 2.30
20.32 = 0.69
17.52 3.88
19.08 = 1.01
We also observed that for EqQP and Huber CA-ADM exhibits better performance than in-domain cases. The experiment results indicate that the context encoder plays a crucial role in transferring to different problem elasses.
Benchmark QP results. From the above results, we confirm that CA-ADMM outperforms the baseline algorithms in various synthetic datasets. We then evaluate CA-ADMM on the 134 public QP benchmark instances (Maros & Mészáros, 1999), which have different distributions of generat- ing QP parameters (i.e., P,q,A, and u). For these tasks, we trained CA-ADMM and RLQP on Random QP, EqQP, Portfolio, SVM, Huber, Control and Lasso. As shown in Table A.8, CA-ADMM exhibits the lowest iterations for 85 instances. These results indicate that CA-ADMM can generalize to the QP problems that follow completely different generating distributions.
Application to linear programming. Linear programming (LP) is an important class of mathe- matical optimization. By setting P as the zero matrix, a QP problem becomes an LP problem. To further understand the generalization capability of CA-ADMM, we apply CA-ADMM trained with 10 < N < 15 Random QP instances solve 100 random LP of size 10 S N 15. As shown in Table A.7, we observed that CA-ADMM shows ~ 1.75 and ~ 30.78 times smaller iterations than RLQP and OSQP while solving all LP instances within 5000 iterations. These results again highlight the generalization capability of CA-ADMM to the different problem classes.
Computational times. We measure the performance of solver with the number of iteration to converge (i.e., solve) it is independent from the implementation and computational resources. However, having lower computation time is also vital. With a desktop computer equips an AMD Threadripper 2990WX CPU and Nvidia Titan X GPU, we measure the compuational times of CA- ADMM, OSQP on GPU and OSQP on CPU. At each MDP step, CA-ADMM and the baseline algo- rithms compute p via the following three steps: (1) Configuring the linear system and solving it linear system solving, (2) constructing the state (e.g.. graph construction) state construction, and (3) computing p from the state p computation. We first measure the average unit computation times (e.g., per MDP step computational time) of the three steps on the various-sized QP problems. As shown in Table A.5, the dominating factor is linear system solving step rather than state con- struction and p computation steps. We then measure the total computation times of each algorithm. As shown Table A.6, when the problem size is small, CA-ADMM takes longer than other baselines, especially OSQP. However, as the problem size increases, CA-ADMM takes less time than other baselines due to the reduced number of iterations.
6 ABLATION STUDY
Effect of spatial context extraction methods. To understand the contribution of spatial and tem- poral context extraction schemes the performance of our networks, conduct ablation study. The variants are HGA that uses spatial context extraction schemes and RLQP that does not use a spatial context extracting scheme. As shown in Table 3, the models using the spatial context (i.e., CA-ADMM
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Table 3: Ablation study results. Best in bold. We measure the average and standard deviation of ADMM iterations.
Congrooents
Probless 400
Gmahi rege.
15-
503
630
KD3P
22.33
09.42
2 20.00
101.4
-00.9
137.60
+12.16
105,40
186.20
+.8.22
2:9.44
214.00
225.20
3.5.19
234.20
3.4.3
243,80
=7.08
252.40
2.155
2: 16615
260.50
CA-AINC
2.4.6
27.44
5.229
31,70
$4.70
3007
48.80
4.60
2:2.44
43.40
$.3.9
46.70
2.366
47.80
1.90
49.10
= 2.00
48.70
1.44
50.10
50,50
1.28
CA-ADMM
HGA
10910Pt
-2
7
1
3
4
6
7
MDP step
MDP step
Figure 4: Example of 1og over MDP steps on a QP of N = 2, 000 and M = 6000. The gray lines visualize the time evolution of logio Pt. The black lines and markers visualize the averages of log10 Pt over the dual variables. ADMM terminates at the 6-th MDP (i.e., solving QP in 60 ADMM steps). HGA takes additional two MDP steps solve the given QP.
and HGA) outperform the model without the spatial context extraction. The results indicate that the spatial context significantly contributes to higher performance.
Effect of temporal context extraction methods. To further investigate the effects of temporal context, we evaluate CA- CA-ADMM HGA that exclude GRU (i.e.. temporal context extraction scheme)
from our model. We also consider RLQP. Table shows that for N S 15 HGA produces better performance than CAMADMM, while for N 2 15 the opposite is true. We conclude that the temporal- extraction mechanism crucial to generalizing to larger QP problems.
Qualitative analysis. Additionally, we visualize the Pt of CA-ADMM and HGA during solving QP problem size of 2,000. As shown in Fig. 4, both of the models gradually decrease Pt over the course of optimization. This behavior is similar the well-known step-size scheduling heuristics implemented in OSQP. We also observed that Pt suggested by CA-ADMM generally decreases over the optimization steps. On the other hand, HGA shows the plateau on scheduling Pt. We observed similar patterns from the different QP instances with different sizes.
Effect of MDP step interval. We update every N = 10 ADMM iteration. In principle, we can adjust p at every iteration while solving QP. In general, smaller N allows the algorithm to change p more often and potentially further decrease the number of iterations. However, it is prone to increase the computation time due to frequent linear system solving (i.e., solving Eq. (1)). Having higher N shows the opposite tendency in the iteration and computational time trade-off. Table Table A.9 summarizes the effect of N on the ADMM iterations and computational times. From the results, we confirm that N 10 attains nice balance of iterations and computational time.
CONCLUSION
To enhance the convergence property of ADMM, we introduced CA-ADMM, a learning- pol- icy that adaptively adjusts the step-size parameter p. We employ spatio-temporal context encoder that extracts the spatial context, i.c., the correlation among the primal and dual variables, and tem- poral context, i.e., the temporal evolution of the variables, from the ADMM states. The extracted context contains information about the given QP and optimization procedures, which are used as input to the policy for adjusting P for the given QP. We empirically demonstrate that CA-ADMM significantly outperforms the heuristics and learning- based baselines and that CA-ADMM general- izes the different QP class benchmarks and large-size QPs more effectively.
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A GRAPH PREPROCESSING
In this section, we provide the details of the graph preprocessing scheme that imposes the scale invariant to the heterograph representation. The preprocessing step is done by scaling the objective function, and the constraints. Algorithm and explain the preprocessing procedures.
Algorithm 1: Objective scaling
Input: Quadratic cost matrix E ST. linear cost terms E RV
Output: Scaled quadratic cost matrix P' E S scale linear cost terms q' E R for n 1,2,... do
p" max ((Pm))) (1)) (Psl))
// Compute scaler
P' P/pp
q' q/p"
end
Algorithm 2: Constraints scaling
Input:
constraint matrix A E RMxN lower bounds E RR,, upper bound u E R.M. Output: Scaled constraint matrix A' E MxN, Scaled lower bounds l' E RMM, Scaled upper bound u' E RMM
for m = 1, do
an max, (Iax,((An (|Amm))
// Compute scaler for each
A m
Am/mm
I' Lm/ am
10
um Um /n
I
end
B DETAILS OF HGA
In this section, we provide the details of the HGA layer.
HGA is composed of edge update, edge aggregation, node update and hetero node aggregation to extract the informative context. This is done by considering each edge type and computing the updated node, edge embedding.
We denote source and destination nodes with the index and j and the edge type with k. h and h denote the source and destination node embedding and ht denotes the embedding of the & type edge between the node and j.
Edge update HGA computes the edge embedding hyy and the corresponding attention logit zy as:
M/ HGA (1h,, h2, n,,))
(A.1)
2;j HGA ([h, h2, hty)),
(A.2)
where HGA and HGA are the hetero edge update function and the hetero attention function for edge type k. respectively. Both functions are parameterized with Multilayer Perceptron (MLP).
Edge aggregation HGA aggregates the typed edges as follows:
exp(2f))
aij = Eie exp(=
(A.3)
where (i) (u.ltype of egg = k, ny E (i)) ViJ is the set of all neighborhood of ey, which type of edge from the element of the set to y is Second, HGA aggregates the messages and produces the type message m for node uy.
m
Σ
anyh'y
(A.4)
iEN
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Table A.1: The MLP sturcture of functions in HGA's first layer
input dimensions
hidden dimensions
oulput dimensions
hidden activations
oulput activation
HGAP P3p
E
HGA p2d
12
HGA E
d2p
12
HGA
HGA A
12
(64, 16]
LeakyReL.U
HGA d2p
A
HGA
11
H
16
H d2p
11
HGAN
(64, 32]
Edge type-aware node update The aggregated message with edge type k. my, is used to compute the updated node embedding with edge type k, by" as:
by HGA (lh,, m)))
(A.5)
where HGA/ is the hetero node update function, which is composed of MLP.
The above three steps (edge update, edge aggregation, edge type-aware node update) are performed separately each edge type preserve the characteristics of the edge type and extract meaningful
embeddings.
Hetero Node update HGA aggregates the updated hetero node feature by produce the updated node embedding h2 as follows.
First, HGA computes the attention logit d corresponding to 6f as:
= #HGAA((b))
(A.6)
where HGAN is the hetero node update function, which is composed of MLP. Second, HGA computes attention 3 using the softmax function with dy:
exp(d
(A.7)
ExERAm (i) exp(d?)
where (a)s(() is the edge types that has destination node at j. Finally, HGA aggregates the per-type the updated messages to compute updated node feature h2 for Uj as:
h2
(A.8)
C DETAILS OF NETWORK ARCHITECTURE AND TRAINING
In this section, we provide the architecture of pocliy network Ty and action-value network Q
Policy netowrk architecture. As explained in Section 4.4, the policy network is consist HGA and MLP. We parameterize my as follows:
P: MLP (HAAACCONCATIGGGG))
(A.9)
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Table A.2: The MLP stureture of functions in HGA's layer(n 2 2)
input dimensions
hidden dimensions
oulput dimensions
hidden activations
oulput activation
HGAP P3p E
HGA p2d
HGAB E
d2p
HGAR
HGA p2d
A
664, 16
LeakyReL.U
HGA d2p
H
H
HGAy d2p
HGAN
[64, 32]
where MLP, is two-layered MLP with the hidden dimension [64, 32]. the LeakyReLU hidden acti- vation, the ExpTanh (Eq. (A.10)) last activation, and HGA, is HGA layer.
ExpTanh(2:) = (Lanh(x:) + 1) x 3 (-3)
(A.10)
Q network 2x architecture. parameterize Q.r as follows:
As mentioned in Section 4.4, Qx has similar architecture to ma. We
P: MLP RREADOUT (HGA) (HCA
(A.11)
where MLP two-layered MI.P with the hidden dimension [64, 32]. the LeakyRel.U hidden ac- tivation, the Identity last activation, HGA, is HGA layer, and READOUT is the weighted sum, min, and amx readout function that summarizes node embeddings into single vector.
Training detail. We train all models with mini-batches of size 128 for 5,000 epochs by using Adam with the fixed learning rate. For TT. we set learning rate as 10-1 and, for Qo, we set learning rate as 10-3 We set history length as
D DETAILS OF PROBLEM GENERATION
In this section, we provide the generation scheme problem classes in OSQP, which is used to train and test our model. Every problem class is based OSQP (Stellato al., 2020a), but modified some settings, such as the percentage of nonzero elements in the matrix or matrix size, to match graph sizes among the problem classes.
D.1 RANDOM QP
1 2 xx
Pr + q Tx
min
(A.12)
subject to Ar
Problem instances We set random positive semidefinite matrix P Ruxn by using matrix mul- tiplication on random matrix M and its transpose, whose element My NN(0 1) has only 15% being nonzero elements. We generate the constraint matrix E m with Ag (0, 1) with only 45% nonzero elements. We chose upper bound u with u, ~ (Az), + (0 1), where æ is randomly chosen vector, and the upper bound I, is same
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D.2 EQUALITY CONSTRAINED QP
min πt Pr + qxx
(A.13)
subject 1 Ar u.
Problem instances We random positive semidefinite matrix P R by using matrix mul- tiplication on random matrix M and its transpose, whose element My ~ (O, 1) has only 30% of being nonzero elements. We generate the constraint matrix A E Rmxx with A, NO, 1) with only 75% nonzero elements. We chose upper bound u with u, ~ (Ax);, where z is randomly chosen vector, and the upper bound 4 E is same u,
D.3 PORTFOLIO OPTIMIZATION
1 DE 2 y
min
2 1
subject to
y F x.
(A.14)
1,
2 0,
Problem instances We set the factor loading matrix F E RR"x" whose element Fg- ~ NNO0 1) as 90% nonzero elements. We generate diagonal matrix D E R whose element D; C(O.1) x ym is the asset-specific risk. We chose a random vector E R", whose element (4; ~ (0, and the expected returns.
D.4 SUPPORT VECTOR MACHINE (SVM)
min T A
subject
t diag b) Ar +
(A.15)
t20 0.
Problem instances We set the matrix A E guxn which has elements A. as follows:
SN 1
m 2)
S otherwise
Ai =
W 4)
We generate the random vector b E Rmm ahead as follows:
+1, i
b =
1, otherwise
choose the scalar À as 1
D.5 HUBER FITTING
min
T u 2M1 T (r+ +
subject to Ar bu r 8,
(A.16)
r 2 0,
Problem instances We set the matrix A Txx with element A,j ~ A/(0, 1) to have 50% nonzero elements. To generate the vector b E Rm, we choose the random vector E R. ahead as follows:
[0 }) with probability p 0.95
U.
[10 10], otherwise
Then, set the vector b = A(u <), where e; NN 2)
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Table A.3: Range of n, m which uses for each problem class generation
Randon QP
Egue
Porfolio
Buber
Control
Lasao
Ea_Randon QP
10 15
30 40
50 ~0 60
r 6
5 6
2 2~ G
5
20 30
10 15
15 20
5 6
50 60
50 60
1
50 ~ 60
45 ~ 70
Relation
3
10
L
10
D.6 OPTIMAL CONTROL
T-1
min 27 QTQTTT +
Σ
Ti Qui + ut Ru
(A.17)
subject 1 Ax, Buc.
E X, us E U, 30 Finit-
Problem instances We set the dynamic A R A I + A, where Di 0.000. We generate the matrix B E R=xm, whose element B.j ~ (O, We choose state cost mQ as diag(g). where q. (0, 10) with 30% zeros elements in eq. We generate the input cost R 0. O. We set time 7 as 5.
D.7 LASSO
1 リ y + 71
min subject to Ar b.
(A.18)
< x < t
Problem instances We set the matrix A R whose element A,, (O 1) with 90% nonzero elements. To generate the vector b E Rm, we set the sparse vector E IR" ahead as follows:
0, [W 1),
with probability p = 0.5
=
otherwise
Then, we chose the vector Ae + where 6; ~ NN(, 1). We set the weighting parameter yy as [ blloc.
For the seven problem types mentioned above, the range of variables n and m can be referred to in the Table A.3. The relationship between and is also described to facilitate understanding.
D.8 ENTIRE RANDOM QP
1
min
2 r Pr. + q x
(A.19)
subject to 1. Ar u.
Br=
Problem instances We set random positive semidefinite matrix P E RAXX by using ma- trix multiplication on random matrix M and its transpose, whose element Mj; ~ ~N(0,1) has only 15% being nonzero elements. We generate the inequality constraint matrix
E mm with A. (0, 1) with only 60% nonzero elements. We chose upper bound u E Im with u4 ~ (Az), + +O 1), where x is randomly chosen vector and L; as -00 We generate the equal- ity constraint matrix B E I mpm2x x with B, _N(0,,) 1) with only 60% nonzero elements, We set constant b R with b, = (Bx).. where x is a randomly chosen vector at the generating process of vector We m. and ma as m and 6m respectively.
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Table A.4: QP size generalization results in table format. Smaller better (4). Best in bold. We measure the average and standard deviation of the iterations for each QP sizes.
10
-00 200
200
300
100
500
500
600
600 - 700
800
900 -M 1000
27.44
31.70
34,70
40.90
43.40)
46.70
47,80
49.10
48.70
50.10
50.50
CA-ADXX
+ 4.30
8.29
=307
4.66
=: 2.24
3.09
+ 2,56
= 1.997
#: 2.00
1.44
1.25
BL2P
69.42
27.35
101.44
20.00
137.60
169.40
186.20
214.60
226.20
5.19
234.20
4.83
243,80
7,08
252.40
260,00
16.15
120.00
142.20
137.60
130.40
DeqP
+.21.93
+ 31.96
13.54
19.78
13.18
15.11
+9733
E DETAIL OF THE EXPERIMENT RESULTS
E.1 DETAILS OF THE IN-DOMAIN EXPERIMENT
In this section, we provide the problem generation, and additional experimental results for the In- domain experiments.
Training data generation For the training set generation, the problems for each class are gen- erated randomly with instance generating rules (Appendix D). Each problem's size of QP is described in Table A.3.
Evaluation data generation Table I's evaluation data generated in the same way as training data generation. To construct empty intersection between train and test set, each set generated from different random seed. To verify the QP size generalization test (Fig. 3, Table A.4), problems are generated randomly for the problem class Random QP with dimension of in [100, 1000].
Additional results We provide the following Table A.4 to show the numerical results difficult to visualize in Fig. 3.
E.2 DETAILS OF THE BROSS-DOMAIN EXPERIMENT
We provide the problem generation for the cross-domain experiments.
Training data generation We create a new problem class named Entire Random QP that con- tains both the randomly generated inequality and equality constraints there is problem class in OSQP. The form of the QP problem class and instance generating rule are described in Ap- pendix D.8. With the new problem class, we generate the training with size in Table A.3.
Evaluation data generation For the training set generation, the problems for each class are gener- ated randomly with instance generating rules in (Appendix D). the size of each problem is described in Table A.3.
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F ADDITIONAL EXPERIMENTS
F.1 RESULT FOR UNIT COMPUTATIONAL TIME OF THE SIZE GENERALIZATION EXPERIMENT
Table A.5: Unit computational time results for each QP sizes in table format. Smaller is better ()). Best in bold. We measure the average of the unit time per a MDP step for each QP sizes. The unit of every element Second.
Unit timse per
nspp
Problezz sine
State Conuraztinn
Toul limz
State Canattation
system volring
Total
wilsing
Constmationn
compataiion
compatiioo
linc
15 100
0,000
0.025
0.000
0.088
0.000
0.003
0.001
0.008
0,005
0.003
0,071
0,085
0.005
0,010
300
0.036
0.50
0.015
0.045
0.130
0.309
100 400
03001
01779
0.001
0.100
01000
0146
0001
01070
0.006
0.00
0.156
0.90
0.000
0.179
0.184
0451
0,324
0.055
0.340
0,800
(00 100
03041
0.999
0.007
0.597
01000
0,377
0101
0.336
0.068
0.507
0,251
0,90
100
0.00
0.266
0.010
0.811
0001
0.10
0.543
1.20
900
1000
0.000
1339
0.066
1.7%
0.000
0.15
0.01
1.45
0.150
1.590
0.358
2363
F.2 RESULT FOR TOTAL COMPUTATIONAL TIME OF THE SIZE GENERALIZATION EXPERIMENT
Table A.6: Total computational time results for each QP sizes in table format. Smaller is better (1). Best in bold. We measure the average of the total time and the MDP steps for each QP sizes.
OSQP
BLOP
CA-AINM
Problem size
Total MDP steps
(Second))
MDP steps
Total time (Secord)
Tocal MDP stops (Second)
MDP stess
Total time (Second)
Total /DP steps (Second)
MDP steps
Total time (Secona))
15 100
0.008
11.2
D.087
0.008
10.4
0,082
D,885
3.6
0.306
100 200
0.014
12.6
0.179
0,015
14.2
0.213
0.127
4.2
0,535
200
0.734
0.636
0.209
1.291
490
0.220
14.4
3.172
22.0
3.158
0.454
5.0
2.269
500 - 600
0.392
13.8
5.416
0.229
23.2
5.312
0.000
5.2
3.588
600 - 700
0.597
14.6
8.715
0.334
24.0
8.012
0.933
5.6
5.226
700 800
0.881
14.4
12,693
0.510
34.8
12,636
1,250
5.2
6,499
800 900
1,204
13.8
16.609
0.679
25.8
17.527
L.661
5.8
9.654
23.472
28.007
2.363
13,233
F.3 RESULT FOR LP EXPERIMENT
Table A.7: LP experiment results.: Smaller is better (L). Best in bold. We measure the average and standard deviation of the iterations and average total computational time for Random LP problems.
CA-ADMM
RLQP
DSQP
Total time (Second)
0.367
0.092
1.228
Solve r
1.00
1.00
0.58
Iterations
79.94
139.28
2432.74
Solve ratio is ratio between solved problems and total generated problems.
19

---

Under review as a conference paper at ICLR 2023
F.4 RESULT FOR MAROS & MESZAROS PROBLEMS
Table A.8: QP-henchamrk readdos QP + EqQP+ Portfoliot SVM+ Huber+ Control+ Lasso) X, where X E (Maros & Meszaros]: Smaller is better (4). Best in bold. We measure the iterations and the Total Computation time for each QP benchmark.
Problem information
Iterations
Computation time (sec.)
Instance
nonzeros
CA ADMM
RLQP
OSQP
CA-ADMM
RLQP
OSQP
AUG2D
20200
30200
80000
41
28
0.617
0.878
0.528
AUG2DC
20200
30200
80400
16
33
27
0.551
0.668
0.46
AUG2DCQP
20200
30200
80400
82
failed
580
2.679
failed
11.133
AUG2DQP
20200
30200
80000
failed
3.518
failed
10.082
AUG3D
3873
4873
13092
19
42
19
0.32
0.179
0.072
AUG3DC
3873
4873
14292
15
0.24
0.145
0.062
AUG3DCQP
3873
4873
14292
28
38
37
0.39
0.152
0.129
AUG3DQP
3873
4873
13092
31
70
40
0.56
0.281
0.167
CONT-050
2597
4998
17199
failed
0.273
failed
0.137
CONT-100
10197
19998
69399
20
19
37
0.7
0.576
0.9
CONT-101
10197
20295
62496
failed
1607
53.217
failed
45.805
CONT-200
40397
79998
278799
21
33
44
4.94
8.974
7.03
CONT-201
40397
80595
249996
3761
failed
3154
817.087
failed
793.884
CVXQP1 M
1000
1,500
9466
41
328
61
0.965
2.445
0.476
CVXQP1_8
100
150
920
28
56
50
0.335
0.042
0.035
CVXQP2.L
10000
12500
87467
11.636
78.098
18.781
CVXQP2_M
1000
1250
8717
25
175
41
0.378
0.53
0.13
CVXQP2_S
100
125
846
126
34
0.308
0.092
0.024
CVXQP3 L
10000
17500
102465
85
failed
95
306.795
failed
346.485
CVXQP3JM M
1000
1750
10215
212
failed
153
5.46
failed
1.53
CVXQP3_S
175
994
0.368
0.047
0,031
DPKL01
133
210
1785
20
36
31
0.176
0.033
0.031
DTOC3
14999
24997
64989
failed
failed
248
failed
failed
2.037
DUAL1
85
86
7201
17
26
29
0.172
0.024
0.023
DUAL2
96
97
9112
26
21
0.204
0.03
0.024
DUAL3
111
112
12327
21
0.181
0.032
0.027
DUAL4
75
76
5673
17
25
42
0.161
0.021
0.032
DUALC1
224
2025
0.308
0.076
0.044
DUALC2
7
236
1659
40
210
59
0.324
0.142
0.038
DUALC5
8
286
2296
61
48
0.309
0.047
0.033
DUALCS
8
511
4096
44
102
64
0.402
0.082
0.052
EXDATA
3000
6001
2260500
85
3004
86
53.737
1544.541
41.597
GENHS28
18
0.151
0.019
0.01
GOULDQP2
699
1048
2791
20
33
20
0.165
0.032
0.018
GOULDQP3
699
1048
3838
0.17
0.026
0.037
HS118
15
32
69
769
479
202
6.739
0.27
0.098
20

---

Under review as a conference paper at ICLR 2023
Problem information
Iterations
Total Computation time (Second)
name
nonzeros
CA-ADMM
RLQP
OSQP
CA-ADMM
RLQP
OSQP
2
22
0.193
0.031
0.016
HS268
5
10
.55
15
161
15
0.142
0.107
0.011
HS35
3
4
13
17
19
29
0.143
0.013
0.019
HS35MOD
3
13
14
13
0.143
0.024
0.01
HS51
5
8
21
18
32
19
0.167
0.025
0.023
HS52
5
8
21
13
29
12
0.164
0.02
0.01
HS53
5
17
13
0.146
0.02
0.009
HS76
4
7
22
15
46
39
0.145
0.028
0.022
HUES-MOD
10000
10002
40000
1830
306
31
524.61
19.032
244.839
HUESTIS
10000
10002
40000
1830
522. 189
18.395
246.329
KSIP
20
1021
19938
75
1993
53
1,385
6.166
0.053
LASER
1002
2002
9462
25
49
44
0.315
0.068
0.055
LISWET1
10002
20002
50004
27
56
0.409
0.169
0.212
LISWET10
10002
20002
50004
23
31
56
0.375
0.164
0.23
LISWET11
10002
20002
50004
26
31
56
0.413
0.17
0.21
LISWET12
10002
20002
50004
26
56
0.37
0.166
0.235
LISWET2
10002
20002
50004
21
31
56
0.399
0.174
0,218
LISWET3
10002
20002
50004
21
31
56
0.431
0.172
0.215
LISWET4
10002
20002
50004
21
56
0.402
0.173
0.214
LISWETS
10002
20002
50004
21
15
50
0.429
0.087
0.184
LISWET6
10002
20002
50004
26
31
56
0.415
0.172
0.229
LISWET7
10002
20002
50004
21
56
0.396
0.17
0.211
LISWETS
10002
20002
50004
21
31
56
0.406
0.175
0,214
LISWET9
10002
20002
50004
21
31
56
0.394
0.171
0.215
LOTSCHD
12
72
21
0.285
0.122
0.032
MOSARQP1
2500
3200
8512
41
56
0.618
0.163
0.099
KOSARQP2
900
1500
4820
24
131
55
0.331
0.204
0.069
POWELL20
10000
20000
40000
failed
1055
failed
8.85
3.873
PRIMAL1
325
410
6464
23
59
31
0.913
0.083
0.047
PRIMAL2
649
745
9339
14
72
30
0.614
0.184
0.062
PRIMAL3
745
856
23036
15
26
0.794
1.662
0.182
PRIMAL4
1489
1564
19008
17
815
24
0.769
9.351
0.312
PRIMALC1
230
239
2529
58
failed
failed
0.501
failed
failed
PRIMALC2
231
238
2078
160
failed
failed
1.335
failed
failed
PRIMALCS
287
295
2869
225
failed
4764
2.153
failed
5.028
PRIMALC8
52()
528
5199
79
failed
failed
0,856
failed
failed
Q25FV47
1571
2391
130523
202
failed
544
13.749
failed
8.678
QADLITTL
97
153
637
1611
failed
138
30.348
failed
0.079
QAPIRO
32
59
124
39
1104
failed
0.415
0.605
failed
QBANDM
472
777
3023
113
failed
failed
3.78
failed
failed
QBEACONF
262
435
3673
failed
failed
56
failed
failed
0.057
QBORE3D
315
548
1872
failed
failed
847
failed
failed
0.667
QBRANDY
249
469
2511
901
failed
failed
31.157
failed
failed
QCAPRI
353
624
3852
421
failed
829
13.648
failed
0.737
QE226
282
505
4721
173
3100
failed
6.079
2.971
failed
21

---

Under review as a conference paper at ICLR 2023
Problem information
Iterations
Total Computation time (Second)
name
nonzeros
CA-ADMM
RLQP
OSQP
CA-ADMM
RLQP
OSQP
QETAMACR
688
1088
11613
failed
3303
1290
failed
6.886
2.701
QFFFFF80
854
1378
10635
131
failed
694
6,949
failed
3.034
QFORPLAN
421
582
6112
failed
1950
740
failed
2.654
1.021
QGFRDXPK
1092
1708
3739
1991
failed
33
40.741
failed
0.037
QGROW15
645
945
7227
failed
failed
failed
failed
failed
failed
QGROW22
946
1386
10837
failed
failed
failed
failed
failed
failed
QGROW7
301
441
3597
failed
failed failed
failed
failed
failed
QISRAEL
142
316
3765
49
2518
83
1.705
2.239
0.07
QPCBLEND
83
157
657
33
19
79
0.621
0.014
0.051
QPCROEI1
384
735
4253
failed
failed
87
failed
failed
0.11
QPCBOEI2
143
309
1482
failed
failed
51
failed
failed
0.038
QPCSTAIR
467
823
4790
failed
failed
739
failed
failed
0.815
QPILOTNO
2172
3147
16105
361
failed
20
23.219
failed
0.152
QPTEST
2
4
10
12
43
33
0.138
0.031
0.024
QRECIPE
180
271
923
271
62
0.504
0.213
0.04
QSC205
203
408
785
19
7
64
0.206
0.007
0.059
QSCAGR25
500
971
2282
failed
failed
259
failed
failed
0.215
QSCAGR7
140
269
602
failed
failed
543
failed
failed
0.331
QSCFXM1
457
787
4456
failed
failed
382
failed
failed
0.387
QSCFXM2
914
1574
8285
failed
failed
1176
failed
failed
1.729
QSCFXM3
1371
2361
11501
1361
failed
729
57.445
failed
1.37
QSCORPIO
358
746
1842
348
failed
1.226
0.314
failed
QSCRS8
1169
1659
4560
83
failed
failed
1.854
failed
failed
QSCSD1
760
837
4584
41
118
215
0.799
0.195
0.331
QSCSD6
1350
1497
8378
49
289
285
0.818
0.478
0.67
QSCSD8
2750
3147
16214
54
2256
248
1.143
6.356
0.814
QSCTAP1
480
780
2442
483
failed
0.91
0.464
failed
QSCTAP2
1880
2970
10007
42
280
failed
0.987
0.757
failed
QSCTAP3
2480
3960
13262
324
failed
1.468
1.063
failed
QSEBA
1028
1543
6576
failed
failed
failed
failed
failed
failed
QSHARE1B
225
342
1436
failed
failed
574
failed
failed
0.38
QSHARE2B
79
175
873
56
failed
2277
0.892
failed
1.276
QSHELL
1775
2311
74506
failed
failed
71
failed
failed
0.55
QSHIPO4L
2118
2520
8548
failed
115
2.916
failed
0.363
QSHIP04S
1458
1860
5908
111
failed
111
2.713
failed
0.23
QSHIPO8L
4283
5061
86075
failed
132
4.887
failed 1.375
QSHIPO8S
2387
3165
32317
278
failed
178
9,257
failed
0.566
QSHIP12L
5427
6578
144030
741
failed
150
33.934
failed
1.872
QSHIP12S
2763
3914
44705
failed
failed
144
failed
failed
0.487
QSIERRA
2036
3263
9582
failed
failed
206
failed
failed
0.671
QSTAIR
467
823
6293
failed
353
6.397
failed
0.436
QSTANDAT
1075
1434
5576
88 failed
failed
2.129
failed
failed
$268
10
55
161
15
0.138
0.01
STADAT1
2001
6000
13998
failed
failed
failed
failed
failed
failed
STADAT2
2001
6000
13998
failed
failed 558
failed
failed
0.979
22

---

Under review as a conference paper at ICLR 2023
Problem information
Iterations
Total Computation time (Second)
name
n
m
nonzeros
CA-ADMM
RLQP
OSQP
CA-ADMM
RLQP
OSQP
STADAT3
4001
12000
27998
failed
failed
702
failed
failed
2.145
STCQP1
4097
6149
66544
22
110
39
6.154
19.142
6.738
STOQP2
4097
6149
66544
21
1.529
3.519
1.191
TAME
2
3
8
19
26
21
0.138
0.019
0.016
UBH1
18009
30009
72012
93
4757
92
3.348
39.363
0.787
VALUES
202
203
7846
34
0.682
0.019
0.031
YAO
2002
4002
10004
764
528
797
8.107
0.627
0.853
ZECEVIC2
2
4
7
22
232
36
0.204
0.124
0.021
F.5
MDP STEP INTERVAL ABLATION
Step interval ablation results.: Smaller is better (4). Best in bold. We measure the average and standard deviation of the iterations for each step-interven intervlls.....
Table A.9:
Computational time
Step-interval
Iterations
(sec.)
27.02
0.194
10
30.42
0.111
50
53.39
0.065
100
65.72
0.050
23


Reviews:
[{"review_id": "fbu1YOMQfz", "reviewer": "Reviewer_L3nw", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper proposes a context-aware adaptive mechanism to adjust the step-size parameter rho in ADMM for solving convex quadratic programming problems, denote as CA-ADMM. It extracts the spatio-temporal context during the ADMM iterations. Numerical experiments on various type of QP sets are reported.", "summary_of_the_review": "The paper proposes a new method for using ML techniques to accelerate ADMM for convex quadratic programming. The merit of this paper is more from the practical point of view. However, the reported experimental result has two main issues. First, the experiment is only conducted on synthetic/randomly generated data set. Second, only number of iterations is reported for comparing the efficiency between algorithms. ", "strength_and_weaknesses": "Strength:\ni) The proposed method seems to have a substantial improvement over RLQP and OSQP in terms of number of iterations\n\nWeakness:\ni) The reported experimental results are on number of iterations. It would be much more meaningful to report the CPU time as well. In particular, the time for finding a good rho in CA-ADMM is supposed to take more time than that in OSQP. It is important to note if the decrease in number of iterations can compensate the overhead in finding a good rho.\nii) The paper is more on the empirical side. Therefore, it is very important to see the real performance of the proposed method. However, all the experiments are done on synthetic data sets (randomly generated). It is necessary to compare the proposed method with OSQP on public available real QP data set, for example, the Meszaros' QP set (http://old.sztaki.hu/~meszaros/public_ftp/qpdata/). \niii) If the paper only considers convex quadratic programming, please emphasize this in the title and abstract. There is no experiments reported on non-convex quadratic programming problems.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "axwflWigcC", "reviewer": "Reviewer_nZJ9", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper introduces a machine learning based heuristic method to adaptively choose an important parameter of ADMM when applying to solve quadratic programming problems. The proposed method added a temporal component via a gated recurrent unit, which allows the model to incorporate consecutive information from previous iterations. In addition, the authors proposed heterogeneous graph attention for the embedding. The experiments show some improvements.", "summary_of_the_review": "Overall, this paper has some non-trivial contributions but not significant ones. Thus it needs to be significantly improved to achieve the standard of ICLR.\n\n=======After rebuttal=====\n\nThanks for answering my questions.\n\nAfter reading your rebuttal, I still think using spatial and temporal with RL is not innovative. In addition,  only given the definition of contextual MDP is not enough to support the intuition of using the graph attention layer. Therefore, I still believe the contribution of this work is limited.\n\nFor the new experiments, it is only compared with QP ADMM methods by setting matrices to be zeros on some random data. I think it will be more convincing to compare with ADMM algorithms designed for LP on some real datasets.\n\nThus, I would like to maintain my original score.", "strength_and_weaknesses": "Strengths:\n1. This paper tests out a clear hypothesis that attention and temporal information can improve NN based parameter selection heuristics, and provide an empirical evaluation of their new approach.\n2. The experiments are thorough and help ascertain the importance of each new contribution independently. The ablation study also does a reasonably good job of including necessary experiments for ascertaining the importance of different parameters and other aspects needed for reproducibility.\n\nWeakness:\n1. The technical contribution seems relatively incremental since both GRU and attention are commonly used. In addition, the intuition why GRU and attention can improve performance is not clear.\n2. The authors only showed improvement in the iteration. How about the total time? In my opinion, using a graph neural network to get the parameter $\\rho$ may increase the time complexity and resource consumption.\n3. Why doesn't the author also apply similar algorithms for using ADMM to solve linear programming problems? If the idea works well for QP, it should work on LP as well. If not, providing reasons why it doesn't work on LP may help the community to understand the limitations and strengths of the proposed method.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "-aEqzTcdnZL", "reviewer": "Reviewer_1ZMW", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper considers the problem of QP in which the authors propose an adaptive scheme for ADMM step size selection (CA-ADMM). CA-ADMM obtains the step sizes using an MDP. The context for the MDP is arrived at using spatial and temporal means, wherein they use a graph neural network defined over the ADMM variables to generate the former and use the previous history to generate the latter. They do comprehensive experiments on various datasets illustrating that the proposed method leads to lesser number of admm iterations than the compared baselines. ", "summary_of_the_review": "The paper proposes a novel scheme for ADMM weights using an MDP based updates. The novelty of the scheme and the experimental results are convincing. ", "strength_and_weaknesses": "Strengths\n- Novelty of the proposed architecture for ADMM updates\n- The experimentatal results are strong. \n-- The proposed method performs convincingly better than the baselines.\n-- The dependency on the problem size is also studied in the experiments. \n-- The ability to transfer trained model from smaller sizes to larger settings.\n\n\nQuestions\n- (Minor) Table 2. What do the rows with labels \"transferred\" and \"in-domain\" refer? Some clarification may help understanding this better. \n- Was there any experimentation done comparing how frequently \\rho should be updated ? Or is there any objective justification for the update once in 10 iterations. \n- How might the performances when we compare well-conditioned vs ill-conditioned matrix P?\n\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject