Homotopy Learning of Parametric Solutions to Constrained Optimization Problems
Authors: Shimiao Li
Jan Drgona
Aaron R Tuor
Larry Pileggi
Draguna L Vrabie
Venue: Submitted to ICLR 2023 (2023)
Keywords: homotopy
deep learning
constrained optimization
nonlinear programming
constrained deep learning
differentiable parametric programming

Abstract: Building deep learning (DL) alternatives to constrained optimization problems has been proposed as a cheaper solution approach than classical constrained optimization solvers. However, these approximate learning-based solutions still suffer from constraint violations. From this perspective, reaching a reliable convergence remains an open challenge to DL models even with state-of-the-art methods to impose constraints, especially when facing a large set of nonlinear constraints forming a non-convex feasible set. In this paper, we propose the use of homotopy meta-optimization heuristics which creates a continuous transformation of the objective and constraints during training, to promote a more reliable convergence where the solution feasibility can be further improved. The method developed in this work includes 1) general-purpose homotopy heuristics based on the relaxation of objectives and constraint bounds to enlarge the basin of attraction and 2) physics-informed transformation of domain problem leading to trivial starting points lying within the basin of attraction.  Experimentally, we demonstrate the efficacy of the proposed method on a set of abstract constrained optimization problems and real-world power grid optimal power flow problems with increasing complexity. Results show that constrained deep learning models with homotopy heuristics can improve the feasibility of the resulting solutions while achieving near-optimal objective values when compared with non-homotopy counterparts.

Under review as a conference paper at ICLR 2021
HOMOTOPY LEARNING OF PARAMETRIC SOLUTIONS TO CONSTRAINED OPTIMIZATION PROBLEMS
An Anonymous authors
Paper under double-blind review
ABSTRACT
Building deep learning (DL) alternatives to constrained optimization problems has been proposed as a cheaper solution approach than classical constrained opti- mization solvers. However, these approximate learning-based solutions still suffer from constraint violations. From this perspective, reaching a reliable convergence remains an open challenge to DL models even with state-of-the methods to impose constraints, especially when facing a large set of nonlinear constraints forming a non-convex feasible set. In this paper, we propose the use of homotopy meta-optimization heuristics which creates continuous transformation of the ob- jective and constraints during training, to promote more reliable convergence where the solution feasibility can be further improved. The method developed in this work includes 1) general-purpose homotopy heuristics based on the relax- ation of objectives and constraint bounds to enlarge the basin of attraction and 2) physics-informed transformation of domain problem leading to trivial start- ing points lying within the basin of attraction. Experimentally, we demonstrate the efficacy of the proposed method on set of abstract constrained optimization problems and real-world power grid optimal power flow problems with increasing complexity. Results show that constrained deep learning models with homotopy heuristics can improve the feasibility of the resulting solutions while achieving near-optimal objective values when compared with non-homotopy counterparts.
INTRODUCTION
Recent years have seen a rich literature of deep learning (DL) models for solving constrained opti- mization problems on real-world tasks such as power grid, traffic, or wireless system optimization. These applications can largely benefit from data-driven alternatives enabling fast real-time infer- ence. The problems remain that these problems commonly include a large set of nonlinear system constraints that lead to non-convex parametric nonlinear programming (pNLP) problems which are NP-hard. Earlier attempts simply adopt imitation learning (i.c., supervised learning) to train func- tion approximators via minimization of the prediction error using labeled data of pre-computed solutions. Unfortunately, these models can hardly perform well on unseen data as the outputs are trained satisfy physical constraints, leading ifeasible solutions.
Imitation Learning
Differentiable Parametric Programming or, Constrained Deep Learning, End-to-end NN
Neural Network
|| 112
Neural min (x.5) h(x,)) 0, g(x,5) s
ground truth
Figure 1:
Imitation learning VS end-to-end learning using Differentiable Parametric Programming
To address the feasibility issues, existing methods have explored the imposing of constraints on the output space of deep learning models. Section 2 provides an overview of the existing techniques. The imposing of constraints has inspired the use of end-to-end learning approaches that directly consider the original objectives and constraints in the NN training process without the need expert labeled data.

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However, even the state-of-the-art methods to impose constraints can hardly guarantee a reliable convergence with perfect feasibility on unseen data for large problems. Penalty method which treats the constraints as a form of regularization requires careful selection of penalty weights and such soft-constraint treatment cannot guarantee satisfying constraints to machine precision. Primal-dual Lagrangian-based formulation theoretically provides hard constraint methodology, whereas em- pirical evidence indicate it can perform worse than penalty method (the reason remains unclear, see MMáquue--Nilla et al., 2017)) Another strategy (Donti et al. 202I) adds completion layer after the NN model to reconstruct the complete solution from an incomplete one given by the NN, using the equality constraints. This enables a hard constraint method for equality constraints, whereas when facing nonlinear constraints, the completion layer, as an iterative solver, adds to the com- putation complexity, and can potentially diverge when a bad incomplete output from NN causes a non-existence of feasible solution to be reconstructed.
Due to the lack of consensus in the community, these new approaches are often called by different names such as constrained deep learning, end-to-end neural networks, differentiable optimization layers, or deep declarative networks. In this paper we contribute to this diversity by referring to the proposed method differentiable parametric programming (DPP) to emphasize the connection with sensitivity analysis developed in the context of operations research (Gal & Nedoma 1972] Gal & Greenberg) and later adopted in control theory applications (Bemporad al. 2000 Herceg al., 2013).
As a main contribution, we present novel method that combines homotopy, deep learning and parametric programming formulations into one coherent algorithmic framework. The aim is to obtain a more reliable convergence of constrained deep learning models whose solution feasibility can be further improved. Homotopy based meta-optimization heuristics are developed to create a continuous transformation of objective and constraint sets, making homotopy path that drives the training of NN to gradually learn from easy problems to harder problems. Our contribution includes 2 types of homotopy heuristics which are different ways of utlizing the basin of attraction: 1) homotopy heuristics based on relaxation of objective and constraints to manipulate the basin of attraction, 2) domain-awar homotopy heuristics based on physics-informer transformation of the problem to make it available trivial starting points within the basin of attraction
2 RELATED WORK
2.1 CONSTRAINED NEURAL NETWORKS
Imposing constraints onto the output space of NNs can be done via supervised learning (where labels are used to write the constraints) or unsupervised learning; using either soft constraint (which usually treats the constraints as a regularization) or hard constraint method (which usually means enforcing satisfaction of constraints to machine precision, i.e., perfect satisfaction). We briefly describe the different categories of existing methods, according to the type of constraints to be imposed:
General equality and inequality constraints can be imposed by augmented objective functions, reprojection (as hard constraints), completion layer (as hard constraints), etc. Among augmented objective function methods, penalty method (Yang et al. 2019 Hu et al., 2020 Donti et al., 2021 Pan al. 2019) augments the objective function by additional terms that penalize the violation of constraints, treating the constraints as a regularization with pre-defined weights to control the reg- ularization strength, whereas the primal-dual based formulation (or lagrangian formulation) (Nand- wani et al. 2019 Fioretto et al., 2020; Mārquez-Neila et al. 2017) exploits Lagrangian duality and iteratively updates both primal and dual variables to minimize a Lagrangian loss function. Penalty method, as a soft constraint method, has some theoretical deficits of requiring extra weight tuning for the multi-objective loss function, and no guarantee of satisfying constraints. However, evidence Mārquez-Neila et al.) (017)) has shown that Lagriangian formulation, as a hard constraint method, is empirically worse. Reprojection method makes corrections on out-of-constraint-se outputs by projecting them onto the feasible region, either during the training cycle using different variants of projected gradient descent methods Donti et al. 2021 Märquez-Neit Neila et al. (017), or during the test as a post-processing step (Pan et al. 2019) passed outputs to physical equation solver). A completion layer method (c.g., DC3(Donti et al. 2021), ACnet Beucler et al. (02T)) developed NN to only produce subset of the target output variables, and then extra constraint layer attached
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after NN computes the remaining outputs according to constraints. These methods have pros and cons, as discussed in Section
Domain-specific constraints. Some real-world applications work with graphical structures, neces- sitating the encoding of network topology constraints i) in model architecture as a hard constraint (e.g., Graph Neural Network (GNN) (Kundacina et al. 2022 Donon et al. 2019: Owerko et al., 2020 Diehl 2019) and other graphical models(Li et al. 2022 ). ii) in prior as soft constraint (e.g,
iii) in input features (topology related NN inputs). More attempts to impose dynamic and recursive constraints include i) unrolled neural networks (soft constraint) where recurrent neural network (RNN) and its variants unroll differen- tiable dynamic models (Tuor et al., 2 et al. [02 (Orgona et al. (202I) and iterative physical solvers (Zhang et al. 2019)( Yang et 20.] et al. 2018). or ii) encoding temporal and spatial constraints in latent representations Yuan et al. 2021).
adjacent matrix as prior (Yang et al. 2019 Hu et al. 2020).
2.2 CONSTRAINED OPTIMIZATION
Consider general constrained optimization problem, with & denoting the known parameters repre- senting imput data, and denoting the solution of the corresponding optimization problem. Given paramcter instance ç. the aim is to obtain optimal r by solving:
min (at s.t. g(x, S O, (xx,,))= = a
When there are non-linear objectives or constraints, ( defines a family of parametric non-linear programming (pNLP) problems. pNLP problems are NP-hard and handled poorly with state-of- the-art optimization solvers due to the non-convexities in the solution space. Existing works have explored large number of techniques (Wächter & Biegler 2005 2006 Byrd et al. 2000 Liao 2004), including filter method, line-search, corrections, trust region method, or homotopy methods, to improve on (local) convergence, and also developed many heuristics to allow faster performance of online optimization solvers. The local (and global) convergence properties of optimization meth- ods have also been extensively studied in order to develop tools to improve on convergence guaran- tee for nonlinear programming. However, scalable solution approaches for generic pNLP problems remain a challenge. Reasons include the numerical difficulties (caused by uncertain behaviors of initialization procedures, stopping criteria, etc), ill-conditioning, and the assumptions required to ensure convergence becoming easily violated in practice.
2.3 POWER GRID OPTIMIZATION PROBLEM
One domain-specific problem of interest this paper is the AC optimal power flow (ACOPF) prob- lem (Capitanescu) (Paneey et al. 2020) which hard to solve due to non-convexities. ACOPF is the fundamental optimization problem to determine the optimal control of generator output that can meet the demand with maximal cost-efficiency while safely operating the system within its tech- nical limits. A simple definition of ACOPF is given below, where minimize the generation cost subject to power balance equations and variable bounds:
ng
min Vimag
aq 22 gi
+ B.P +
(2a)
Qs.
s.t.
(2b)
Power balance:
(P' Pa) + j(0, Qa) V @ (Yous V)*
(2c)
reference bus angle: Vimag
(2d)
voltage magnitude bounds: |V[, I |V/ S [V i nb
(2c)
Pgen bounds: PTj < Pgi p+ iE ng
(2f)
Qgen bounds: Q, < Qui s o+ i ng
(2g)
where Pa. Qu are unknown parameters of load demand (given as input information), a, 3,, 7 are generator cost coefficients of the i-th generator, Pgi, Qgi denote real and reactive power output of the i-th generator, yreaa, pimag denote vector of real and imaginary voltage all buses, V is vector of complex bus voltages V = Vreal + +jV jVimeg, |V| denotes the magnitude of voltage, and xt, x denote the upper and lower variable bound.
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2.4 HOMOTOPY METHOD FOR CONSTRAINED OPTIMIZATION
Homotopy method is type of meta-heuristies handle hard problems which can otherwise easily divergence or converge to a bad point. It decomposes the original (nonlinear) problem F(x) to a series of sub-problems H Ar), creating path of optimizers driven by the change of homotopy parameter λς- As A shifts from 0 to 1, the sub-problem H(r,An continuously transforms from simple problem the original one. Commonly, homotopy-based heuristics have been used for local optimization of nonlinear problems. The most popular form of designing H (æ, Aн) is via linear combination of trivial problem (z) and the original one such that (x: AH) (1
λω) (z - Ay Fz) Existing works have developed perturbation techniques for general nonlinear
problems He 1999
Liao 2004). multi-objecttvve problems Hillermeier al. (001). etc.
Domain-specific homotopy heuristics have also been developed to design H (z, Az) whose solu- tions are trivial when X = O. For example, in the domain of circuit simulation (Najm) 2010). Gmin-stepping initially shorts all nodes to ground and gradually remove the short-circuit effect; Tx- stepping initially shorts all transmission lines, and then gradually returns to the original branches. Works in (Pandey et al.] 2018 (Pandey et al.) (eremmnor et al. 2019) further applied the homotopy ideas to power grid simulation and optimization tasks. On the other hand, some works (Dunlavy & O'Leary 2005) also extended homotopy methods to global opti- mization tasks through an ensemble of solution points at each homotopy step, and investigated the probability bound on the convergence global minimizer. Whereas, many homotopy methods empirically slower than most other convergence heuristics (e.g., line search, trust region method, etc). Therefore, the use of homotopy on classical optimization solvers can still suffer from limited time efficiency.
3 HOMOTOPY LEARNING FOR DIFFERENTIABLE PARAMETRIC PROGRAMMING
Here we present novel method that combines homotopy, deep learning and parametric program- ming one coherent algorithmic framework. Specifically, using neural network as an approx- imation of constrained optimization solvers allows fast real-time inference for any new input, and meanwhile, integrating homotopy into the training process facilitates the NN to reach a more reliable convergence with improved feasibility.
3.1 DIFFERENTIABLE PARAMETRIC PROGRAMMING
To deal with the challenge regarding scalability in generic pNLP problems, well as the drawbacks imitation learning, we can build a data-driven alternative to the traditional optimization solver, with the target objectives and constraints directly integrated within the training cycle. This can be achieved by differentiable parametric programming, which adopts an unsupervised learning of NN model, mathematically defined as:
min fonz (x. {) s.t. g(x,() < O, Mxx,E) = 0. == (E),), AA E
(3)
with Te denoting NN model mapping from input & to the output solution and O being the NN weights. Figure T illustrates the difference from imitation learning.
One methodology of interest to this paper is the penalty method, which imposes the constraints by reformulating 3 into an unconstrained form, leading to a NN loss function as below:
min (ot + Em; w. (((h((Te(()EE) + Σ w, (((g((T(())E))
(4)
with P.O) penalizing the violation of constraints, and hyperparameters denoting the pre-defined penalty weights. Popular selections P,() include residual norm penalty, typically L2-norm, for equality constraints and ReLU operator for inequality constraints. Beyond these popular forms, work in (Zhu et al.) 2019) further explored using variational functionals of partial differential equa- tions (PDE) as penalty terms impose PDE constraints.
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3.2 METHOD OVERVIEW AND THEORETICAL FOUNDATIONS
As discussed earlier, when facing non-convex objective function and constraints in (3). even the state-of-the-art methods can have difficulty reaching a good converge with small violation of con- straints. In this paper we develop novel optimization heuristic for these type of problems based on the idea of homotopy (He 1999
Liao
2004).
Specifically, we apply homotopy to the problem by creating continuous transformation of the ob- jecitve and constraints in equation 3 such that a subproblem in the homotopy path can be expressed as:
min fxx (I, E) s.t. 9AN (x, S O, hxn (z. E) = 00
=e((), Ag E
(5)
As the homotopy parameter Aн incrementally changes from 0 to 1, the objective and constraints I.n gradually return back to fabj 9, h, shifting the task from an easy-to-solve problem to the original one. Driven by this transformation, the neural network model me gradually learns to approximate the solution of harder and harder problems.
Next we describe the conceptual idea of our homotopy heuristics along with some theoretical foun- dations. Let (ar, Aω) denote a sub-problem in the homotopy path that transforms the original optimization problem F(x) in (3). We can use a certain local minimization method as the solver of (I, ) to get zg local minimizer of H (a, Aς). For that local minimization method, we can define the basin of attraction B(Ax) of local minimizer of H (x, Aι) as the set of points such that the local minimization method started at x E B(AØ) will converge to that minimizer xxx Dunlavy & O' O'Leary 2005).
Assuming that the original problem F(x) has a global minimizer x that is unique and isolated. According to studies in (Dunlavy & O'Leary 2005), , the Implicit Function Theorem guarantees that there exists of unique curve of isolated minimizers that passes through (x", 1). meaning the desirable optimum is theoretically accessible through a curve of homotopy problem minimizers (x), Ar).
Importantly. when taking small enough homotopy step, i.e., { A (k-1) + Ad with small enough Adн, there is a high likelihood that the minimizer of a subproblem Hxx, A 1(111) is in the
basin of attraction of the next subproblem (xx, x ), i..., (*--1) ))) This motivates us to carefully design (xx λн) and select Aλи to generate a easy homotopy path where, starting with a proper (x, 0). each following sub-problem Hx, A (8)) is solved without difficulty due good starting point given from (ar, X (k-1))) Figure illustrates such homotopy path.
Basin of attraction
B (( (0)
0)
B(2 (1)
B(2 1)
A local minimizer of the original problem
x
(
x (t-1) E B(2
Return to original problem F F(x)
H(x,0)
H(x, 1)
Figure 2: A desirable theoretical homotopy path that gradually leads to a minimizer of the original problem: each subproblem easily solvable by starting from point in the basin of attraction ((())
Based on these findings, the main idea of our homotopy heuristics include:
1. creating proper transformation of the objective and constraints via a linear combiniation (xx, Aa) = (1 - A Hz) + D F(z), with the initial problem (xr, Au = 0) Ho(r) being easily solvable. The illustration in Figure reveals 2 different ways to design Ho(r):
TYPE I: expanding the basin of attraction ((xzz) ( to a larger volume in the initial homotopy step Ho(z). In this case, the initial problem is easy solve in way that there exist more
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choices of starting point that can enable solution trajectories to (local) minimizer. The basin of attraction gradually shrinks it as λy increases.
TYPE II, designing (a) such that trivial starting point zu B(A 0) is available (before any training starts). In this case, Ho(z) is easy in way that we have a starting point within its basin of attraction, which guarantees a solution trajectory to a minimizer.
2. selection of proper homotopy step AAи that trades off between the likelihood of (k-1) (x ()) and an acceptable time complexity.
Eventually, the overall training process can be described as Algorithm
Algorithm Homotopy Learning of Neural Network
1: Initialize: homotpy parameter λи f O, NN model ITe
2: while Aн $ do
3: 1. Update objective and constraints fxu= ha. (.) = 0, (1) 00
4: 2. Train NN using penalty method
5: while not converged do for all E:
6:
Forward pass: I <- (E)
7: Get loss: LAd SAx (x) E, Wey 11(ha (x))]1 +, Winey relulgx (x))
8: Backward pass: mee + arg ming LAw
9:
end while
10: 3. Update homotpy parameter An < An + AdH
11:
end while
3.3 TYPE I: RELAXATION BASED HEURISTICS FOR OBJECTIVE AND CONSTRAINTS
Let's first focus TYPE homotopy methods which enlarge the basin of attraction create easily solvable problems at the initial homotopy steps. We propose the use of relaxation based heuristics, which work by convexifying the objective and expanding the feasible constraint set at the beginning of the homotopy process.
We briefly illustrate the idea behind using relaxation. Mathematically, consider minimizing a non- convex smooth objective function faly subject to a constraint set S, using penalty method (or barrier method, etc) to reformulate the problem. Let I. denote the (smooth) augmented loss function, denote one of the (local) minimizers x² E S, and let M(x") denote a local neighborhood of x where a generic (Lipschitz smooth) convex function. Now consider using gradient descent as the local optimization method with its learning rate lower than twice the smallest optimal learning rate for any component (1) < min "hoppt studies have shown that otherwise learning will diverge), then starting from any _((z)), ).
guarantee reach x² the final solution. Therefore, N(**) can be considered a lower bound of the basin attraction for x², i..., B(x²) 2 N(x*))
As relax (convexify) the objective function, the resulting augmented loss function L has higher convexity, making the optimization landscape more convex. Then for a certain local minimizer x", there's 's likelihood that x= has larger neighborhood. A (x*) where L is generic convex, in which case the lower bound of B(x*) increases:
(+(x*)
Further when relaxing the constraint set from S to St, with S S+ and let x" (S) denote the set of all local minimizers in the constraint set S, then the expanded constrained set likely to contain more local minimizers, i.c.,
(8) (S)
Therefore the total basin of attraction (the set of starting points that will lead to any local minimizer), as the union of B(x*) for each individual minimizer x², will expand after relaxation:
Bit = (s+ 2 Uxrex" (s+ B(x*) (sx)(x")) = Biotal
(8)
Below we introduce the relaxation heuristics. To handle non-convex objective functions during the homotopy process, apply the heuristics of Convexify Objective (CObj):

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Convexify Objective (CObj): Given non-convex objective function fobj, the homotopy path starts from minimizing a convex objective function feuz (which close to the original foty). and gradually insert non-convexity via a linear combination with the original objective:
fAa = λи * fobs + Aeu
For inequality constraints g(-) < 0 during the homotopy process, we propose 2 heuristics:
1) Shrink Bounds (SBnds): the homotopy process, the constraints transforms by g S (1 Au brt As Aи increases from 0 to 1, the inequality constraints gradually transforms from S e+ to the original constraints g S O. This is intuitive for variable bounds x" < S x which can be rewritten in the homotopy process as:
A (1 - AH)E S S AH + (1 (1-x
(10)
where e+, are some pre-defined relaxation of upper and lower bounds making it easier to satisfy. During the homotopy process, the bounds are gradually tightened as 2 increases.
2) Grow Penalty (GPen): Another homotopy heuristic exploits the penalty strength of the violation of constraints. The transformation of constraints can be expressed as:
Agg S
(11)
such that the increasing Ay leads a growing penalty of the constraint violations when the con- straints are included in the augmented loss function.
Finally, we propose the "Split and Shrink" (SaS) heuristics for equality constraints h(-) - 0:
Split and Shrink (SaS); Any equality constraint h(-) O can be equivalently split into two in- equality constraints
S These constraints can be relaxed via perturbation of the bounds: S h < with > 0. A larger perturbation makes the constraints easier to satisfy. This moti- vates us to design homotopy path where the perturbation gradually decreases. more detail, with predefined large perturbation E and tiny perturbation e,, the split constraints are perturbed by:
E - (1 - A S S c + (1 Aн)
(12)
As the homotopy parameter λи shifts from 0 to 1, the decreasing perturbation leads to a tighter bounds that enforces h(-) = 0 more closely.
3.4 TYPE II DOMAIN-AWARE TRANSFORMATION WITH TRIVIAL SOLUTIONS AVAILABLE
This section further explores TYPE II homotopy heuristics which makes it available a trivial starting point within the basin of attraction B(AH 0). Specifically, consider minimizing an objective function fobj subject to a constraint set S, we aim transform the problem via carefully designed manipulation of the objective (if non-convex) and the constraint functions, such that the purturbed problem (xx, Ah 0) min pertturbed, Spurttrrdd easy-to-solve in a way that, before any training starts, trivial starting point E B(AH = 0) is available to guarantee solution trajectory towards a minimizer.
Instead of simple relaxation of the constraint bounds, transforming an entire problem into one which has trivial solutions (or trivial starting points) often requires some domain knowledge. this paper, we work in the context of power grid, and show the design of 2 domain-specine homotopy heuristics: load-stepping and Tx-stepping to impose highly non-linear equality constraints. We consider the power grid optimization control problem defined in ( and the power system related symbols used here are also based on the definitions in Section
3.4.1
The homotopy method of load stepping creates path of power flow balance constraints induced by a gradual increase of load demand:
hag (P, Pa) + + Au Qa) v O (Ybusv)"
(13)
From a domain perspective, when ' = 0, all load demands are zeroed (A Pa = An = 0)
and all variable bounds can be removed/relaxed using the heuristies desgined in Section 3.3 for
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inequalities. A trivial solution to this problem exists: zo [V,,, V, Pg. Qgl 0, representing that the power grid is closed off everywhere with no supply and demand. We make use of this trivial solution via warm- hom-ooy loss: wwarm Wwarm (x - mo) mo)
(14)
to guide the update of the deep learning model towards a quick convergence to feasible point for (only) the first homotopy step. A physics-informed homotopy optimization using penalty method can therefore be written as (x, λн) min LA + fo (AHH) * waarm with (o())) being an indicator function, and LAx is augmented loss function defined in Alogrithm
3.4.2 TX-STEPPING
Unlike load stepping which manipulates load demand, Tx-stepping manipulates the branches in- stead, through a continuous transformation of the bus admittance matrix Y:
has = (Pg - Pa) J(2, Qa)
(Yaι v)"
(15)
where YAN - Yous +1 Yo, From domain perspective, the first homotopy step A 0. we can replace all branches with zero-resistance low-impedance lines (e.g. impedance = 0+ 10 41), giving a bus admittance matrix Yo. This creates all nearly shorted branches with nice properties that 1) lossless lines lead to no real power loss during transmission, i.e., Pg Σ Pa. and 2) 02 oref for any bus specifically, all bus voltage magnitudes are close to the reference bus values due the low voltage drops, and all bus voltage angles will lic within a e-small radius around the refernce bus angle. These nice properties make the problem easily solvable and trivial solution 2o available.
NUMERICAL RESULTS
We evaluate the efficacy of homotopy heuristics on both general non-convex constrained optimiza- tion problems and the real-world problem of power grid optimal power flow. The use of homotopy expected to enable a more reliable convergence of the neural network models to outperform non- homotopy results on the following criteria:
Optimality: the objective function sabj achieved by the solution. For ACOPF problem represents the per-hour cost (S/h) of the generation dispatch.
Feasibility: how much the solution & violates the equality and inequality con- straints. Feasibility is quantified by the mean and maximum violation of con- straints: mean(h(ā)), max(h($)). mean(relu(g($))
max(relu(g(i))). In real-world
tasks, smaller violation of constraints means the neural network outputs more practical solution for real-world optimization and control.
We compare the different versions of homotopy optimization with non-homotopy settings. The non- homotopy baseline is vanilla penalty method as formulated in (4) whereas the homotopy settings combinations of different heuristics added to the vanilla method. Appendix A describes the details on experiment settings and hyper-paramete tuning, in order for a fair comparison.
4.1 NON-CONVEX OPTIMIZATION WITH RANDOM LINEAR CONSTRAINTS
First consider problem with non-convex objective and random linear inequality constraints:
(1 1--xj x,)² + 2( - x2)² s.t. Ax b
(16)
n is the problem size (complexity), r is the solution vector representing n variables to solve, & is an round(0.4 (0.4 x vector representing the (known) input parameter that varies across instances, A pax1 diim(E) are randomly generated matrices representing random linear constraints.
Table shows results on test data for problems with varying complexity. Results demonstrate that adding homotopy heuristics onto the penalty method enables neural network outputs to have a smaller violation constraints, which means better fcasibility. See Appendix A for details on experiment settings.
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Method
Complexity 7
100
25 (Δλ# = 0.05)
50
(AAr = 0.005)
(AA# 0.05)
(AAu 0.05)
Winng = 500
obj
0,48
31.71
62.03
276.16
CObj, SaS, SBnds
mean viol
0.0000X0.0000)
0.0001(0.0002)
0.0003(0.001)
0,0002(0.00M) (009))
viol
0.00000000000
0.0016(0.0048)
0.0129(0.0526)
0.0246(0.0947)
obj
0,48
31.69
62.59
253.42
CObj, SaS, GPen
viol
0.00000000000)
0.0001(0.0005)
0.00022 0005))
0.0004(0.0018)
max viol
0.00000000000
0.0029(0.0121)
0.0061 (0.0141)
0.0297(0.1032)
obj
0.48
33.35
62.33
272.36
SaS, SBnds
mean viol
0.0000(0.0000)
(0.003))
0.0002 0009)
0.0002(0.0000
max viol
0.000X00.0000)
0.0023(0.0066)
0.0082(0.0222)
0,0180(0.071))
obj mean viol
0.47
32.15
62.20
447.32
SaS, GPen
0.0000X0.0000)
(.000((0.0004)
0.0006
0.003 (0.0067)
max viol
0.0000X0.0000)
0.0024(0.0080)
0.0095(0.0347)
0. 0.99((0..588))
obj viol
0,46
31.84
62.44
208.71
Vanilla
0.0000X0.0000)
0.0005(0.0010)
0.0005(0.0011)
0.0015(0.0023)
(penalty method)
max viol
0.0000(0.0000)
0.0095(0.0160)
0.0193(0.0470)
0.1249(0.18) 18777)
Table I: Non-convex problem with n variables and n random linear constraints as varies as 5, 25, 50, 100. Results over 100 test instances are listed, using metries of objective, mean and max inequality constraint violations. The violations are formatted as average value (std) in this Table. Results show that the homotopy heuristics enable smaller violations than vanilla penalty method.
4.2 POWER GRID AC OPTIMAL POWER FLOW PROBLEM
These subsection evaluates on the real-world task ACOPF problem (defined in Section 2.3 Table 2 shows ACOPF results on 30-bus system. Homotopy methods improve the feasibility of NN outputs with smaller violations of equality and inequality constraints. See Appendix for our experiment settings and hyper-parametes tuning.
Method
Obj
Mean eq.
Max eq.
Mean ineq.
Max ineq.
SaS, SBnds
666
0.0027 (0.0015)
0.010 (0.005)
0.0000 (0.0000)
0.000 (0.001)
SaS, GPen
666
0,0023 (0.0014)
0.008 (0.005)
0,0000 (0.0000)
0.000 (0.000)
Tx-stepping, SBnds
665
0.0024 (0.0013)
0.008 (0.005)
0.0000 (0.0000)
0.000 (0.000)
Tx-stepping, GPen
665
0.0017 (0.0009)
0.006 (0.003)
0.0000 (0.0000)
0.000 (0.000)
Load-stepping, SBnds
666
0.0020 (0.0012)
0.007 (0.004)
0.0000 (0.0000)
0.000 (0.000)
Load-stepping, GPen
667
0.0024 (0.0016)
0.009 (0.005)
0.0000 (0.0000)
0.000 (0.000)
vanilla penalty
673
0.0036 (0.0027)
0.013 (0.010)
0.0000 (0.0000) 0.001 (0.003)
Table 2: Results of ACOPF problem on case30, over 100 test instances. Mean and max violations are analyzed across the test instances and we list the average value (std). Vanilla penalty method has larger violations of constraints, whereas homotopy methods have smaller violations.
5 CONCLUSION
This work proposed the use homotopy optimization for the unsupervised learning of deep learning models constrained by a large set of (nonlinear) equality and inequality constraints. The homotopy heuristics developed in this paper include general-purpose homotopy heuristics based on relaxation of constraint bounds enlarge the basin of attraction, as well as physics-informed transformation of domain problem leading to trivial starting points lying within the basin of attraction. Our numerical studies including family of abstract and real-world problems indicate that the developed ho- motopy heuristics achieve a more reliable convergence, giving predictions with improved feasibility on unseen data.

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Under review as a conference paper at ICLR 2021
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A APPENDIX
A.1 EXPERIMENT SETTINGS AND HYPER PARER--RRMEEER TUNING
To create a fair comparison in each optimization problem, experiments with and without homotopy heuristics will train with the same NN architecture, Adam optimizer, and learning rate scheduler (StepLR with step=100, gamma=0.1, and a minimal learning rate 10-5; in homotopy methods, scheduler is only applied to the last homotopy step). Early stopping is also applied in each periment to avoid overfitting: each vanilla method trains for 1,000 epochs with warmup=50, pa- lience=200, and each homotopy method trains 100 epochs in each homotopy step with warmup=50, patience=50. All NNs are trained on PyTorch.
Experiment settings for the non-convex problem with random linear constraints, see problem defi- nition (16) in Section 4.1 are listed below:
Dataset: 50,000 instances (with train/val hhii//vaiiaaion/testt ratio 8:1:1)
NN architecture: cylinder NN with 4 layers, hidden layer size increases with problem size n by hiddenlayersize = 30/n//
Penalty weights: Wey 50, Wineg = 50
Homotopy settings: CObj has feez = Σm xz)² 2(12 + +x)), SaS has 0.01, EL = 0, SBnds has e = 1, = 0.
These hhperyyppraaaee apper-parameters are tuned and kept fixed across all experiments of this non-convex problem.
For the ACOPF problem, we propose an additional trick of P, pull-up and used it on all experiments.
(Pg pull-up) Based on power system domain knowledge, any decision with supply lower than de- mand is always technically infeasible. To avoid bad predictions of this type. we apply the heuristics of Pg pull-up where an additional Nomain-speccif constraint Σ Pg the generation up and thus promote convergence This constraint is not subject to homotopy heuristics and remains the same in the homotopy path. experiment settings are as follows. Similarly as for other constraints, we have a weight Wyallup to impose the additional constraint in penalty method.
Pa 2 is added to pull point with total supply higher than demand.
Dataset: 50,000 instances (with train/validation/tsst ratio 8:1:1), data are generated by ran- domly sampling load profiles in the range of 75% 150% of the base load profile (base load is the load in case data).
Batchsize: 1024
NN architecture: cylinder NN with layers and hidden layer size = 200,
Penalty weights: Weg 105 Wineg
10
General homotopy settings: Aλи 0.05, SaS has eн 0.01, ez SBnds has e
1 = 0.
Domain specific homotopy settings: warm homotopy loss has Wwarm = 106
others: Pg pull up has upultup = 106 < 0.01
The hyper-parameters are kept fixed across all experiments.
12


Reviews:
[{"review_id": "Vo-hEdKlt42", "reviewer": "Reviewer_H3TT", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "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": "The paper considers approximating solutions of constrained optimization problems using neural networks.  A neural network is trained to approximate a parametric family of such problems, mapping the parameters to the optimal solution (i.e. arg-max).  The objective for training the neural net is a combination of the objective of optimization plus constraint violations scaled by some weight (penalty method).  The innovation in the paper is to consider a homotopy of optimization problems -- i.e. connecting a continuum of optimization problems interpolating between the hard problem one ultimately wants to solve and an easy problem, and attempting to follow a solution path.  The setting is a fully general non-convex optimization with non-convex equality and inequality constraints (which is essentially hopeless in its full generality). An application to power-grid is considered. ", "summary_of_the_review": "The idea of using homotopy continuation to train neural nets to approximate solutions of constraint optimization problems is sensible.  Experiments are weak, and while the do show evidence that homotopy may be marginally helpful -- they are not very convincing.  Very informal language, including problem definition, and important missing detail.  If you do attempt formal arguments supporting homotopy methods -- you may need to limit the scope, impose clear assumptions, and cite specific results in existing literature in convex optimization / homotopy methods. Othwerwise it appears to be loosely-motivated intuitive mathematical musings, rather than formal argument. ", "strength_and_weaknesses": "Strength:  the homotopy approach that creates a path of problems connecting hard problems to easy problems is a very established general class of methods, which works well in certain domains.  Approximating optimization problems via neural networks is an interesting and active line of work.  So the idea to use homotopy methods to improve training such neural nets seems sensible.  Also application to power-grid is interesting, and the particular physically-motivated relaxations to make the problem easier are maybe a useful highlight of the paper.   \n\nWeaknesses: 1. by considering the fully general problem (non-convex objectives with non-convex equality and inequality constraints) basically you ensure that nothing useful can be said,  as the problem is not solvable.  There are some non-rigorous mathematical musings that if the problem has a single stable isolated global optima, and is \"locally convex or Lipschitz\" (which is easy to understand for unconstrained optimization, but requires considerably more care for constrained optimization) and for a slow-enough step size starting in the vicinity of the optimum that maybe it should-work.  Rigor is at pictorial level -- this picture of basin-attraction seems to make sense.  In general I'm not sure if there are any results that if the origin problem is convex then following the homotopy will reach some arbitrary local optima of the non-convex one -- if there are such results the authors should cite them explicitly.  To reach global or even good solutions I believe is hopeless without very strong assumptions as homotopy will have various issues -- for example path bifurcations, and path jumps/discontinuities: the mapping from the homotopy parameter to local optima could be highly discontinuous.  Consider a star-shaped constraint set and a linear objective function.  So essentially the method is a heuristic method -- which could in practice be useful for some problems -- if the results were very thorough and convincing. \n\n2.  The scope is unclear, and connection to parametric programming is poorly described.  The authors come up with yet another name for the problem 'differentiable parametric programming' -- but do not really explain the link to parametric programming.  E.g. in parametric LP there are usually 1 (or a small number) of parameters that can vary, and the others are fixed. The goal is to characterize how the optimal solution varies w.r.t. to these small set of parameters.   In your description it seems it could range from 1 parameter to all parameters?  Do you think your method is able to solve say arbitrary QPs via neural nets,  or is it limited to small-dimensional parameters?  For 1D parametric programming it may be cheaper to have a grid of the parameter, and solve the optimization problem, and interpolate, avoiding expensive training of DNNs.  For convex parametric programming, the solution path can be characterized (e.g. parametric LPs and some QPs have piecewise linear solution paths).   It's unclear that the power-grid problem  has any structure that makes the problem easier to solve than generic non-convex constrained problems. \n\n3. Use of very informal language and unsupported claims, while attempting to appear mathematically rigorous. a) What does \"reaching reliable convergence\" mean?  Do you mean converging to a local optimum without constraint violations? Such terms used throughout the paper have to be clearly defined. (b) \"There's a high-likelihood that the minimizer will be in basic of attraction\".  Is this fully informal, or is there some probabilistic model maybe?  (c)   pNLP methods are NP-hard.  non-convex problems are in general not efficiently solvable.  Being parametric doesn't not itself create the problem.  E.g. 1-dimentional parametric LPs are efficiently solvable. \n(d) \"Penalty methods have no guarantees of satisfying constraints\" -- there's existing literature on \"exact penalty methods\" started by Prof. Bertsekas,  that do have rigorous guarantees in the convex setting. \n\"Necessary and sufficient conditions for a penalty method to be exact\", Dimitri P. Bertsekas. Mathematical Programming volume 9, pages 87–99 (1975) . For non-convex setting no guarantees are possible essentially for any methods.  \n(e) Smaller violation of constraints in real world means more practical solution. \"Better feasibility\"...  in some settings it's binary: either feasible or infeasible.  (f) \"Studies have shown that\" -- is used a few times.  It's unclear if this is experimental studies or formal mathematical argument -- please say, and provide references. \n\n4. No discussion of how to select your penalty weights w.  It seems pretty important.   No discussion of feasibility.  How do you know if some parameter setting is even feasible? \n\n5. Weak experiments.   The results are not well explained -- e.g. in Table 1 and Table 2 it's not clear what the different methods are, which are homotopy methods and which aren't.   Details of the experiments are missing (some are in the appendix), but others are not discussed -- e.g. how to set w-weights (penalty weights).   Results are mostly incremental increases (I'm assuming the bottom case 'vanilla' is non-homotopy). The others are incrementally better -- especially for the power-grid problem.  Making constraint violations an order of magnitude smaller may be qualitatively different, but significance of small improvements   (0.13 to 0.10) -- I'm not clear about. \n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "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": "7IDD2WuQc", "reviewer": "Reviewer_JJRB", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "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": "The paper proposes a new deep learning approach for solving constrained optimization with the help of  homotopy heuristics. The key idea is to create a continuous transformation of objective and constraint with gradually increasing complexity. While there is no theoretical analysis provide, the experimental results show that the heuristics can greatly improve the performance of penalty methods  on both  synthetic problem and the real-world AC optimal power flow problem. ", "summary_of_the_review": "Overall, this paper provides an interesting idea to make learning based method more effective in constrained optimization. However, as far as I know, penalty method is as practical as Lagrangian-based methods, as the Lagrangian method  can automatically adjust the penalty (Lagrangian multipliers). For a fair comparison, I suggest the author compares with the state-of-the-art algorithm in these areas. I also have some concerns about the soundness of the algorithm as stated above. ", "strength_and_weaknesses": "The paper is well-written and easy to follow. It clearly illustrated the intuition of the homotopy method and the empirical performance. \n\nWeakness\nThe soundness of the proposed homotopy method is not fully convincing. While I can see the intuition from the paper, it is hard to convince me that one can easily find the right hyper-parameters to efficiently use homotopy method. \n\nMoreover, I think the paper lacks comparison with many other important algorithms for constrained optimization, such as augmented Lagrangian and primal-dual method, which all have theoretically justifiable ways to tune the penalty weight. \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": "-LRh8Hrz24", "reviewer": "Reviewer_Rnhn", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 3, "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 homotopy learning method for solving constrained optimization problems. The idea of homotopy is pretty intuitive: we have a parameter lambda_h \\in [0,1], indicating the difficulty of the current problem. lambda_h=0 means a trivial initial problem H0 and lambda_h=1 means the target difficult problem F. The intermediate problem is defined as H= (1-\\lambda_H) H0 + lambda_H F. \n\nTherefore, the main challenge here is to design a good H0, so that the whole optimization process will easy proceed by gradually increasing lambda_h from 0 to 1, and optimize the function H along the way. \n\nStarting from the penalty method (which is classical), the authors introduce two types of methods for designing H0. Type I is general techniques, including CObj, SBnds, GPen, and Sas. Type II is domain-aware techniques, including Load-stepping and Tx-stepping, specifically designed for power system. \n\nThe authors conduct two experiments, showing that the proposed techniques can outperform the vanilla penalty method. ", "summary_of_the_review": "I think this paper does not provide convincing evidence supporting the significance of the newly proposed method, both in terms of novelty and experimental results. ", "strength_and_weaknesses": "I am not familiar with constraint optimization problems, so my review might be biased.\n\nStrength: \nIf I understand correctly, the main contribution of this paper is combining homotopy (lambda_H), deep learning (using neural network to do optimization) and diffentiable parametric programing (penalty method) together. And the novelty here is proposing the two types of methods. \n\nWeakness: \nI think the homotopy method is intuitive and interesting, but at the current moment, I am not sure how significant the method is. I have the following claims about this paper, but I will be happy to change them after the rebuttal:\n1. It seems that combining homotopy, deep learning and DPP is not a very novel idea. According to the related work section, all of these methods were used previously, for the constraint optimization problems. \n2. It seems that the proposed methods in Type I are pretty straightforward. Are there anything special about SBnds, GPen, etc? For example, people previously never used such tricks before (which is hard for me to believe), or there are some theoretical supporting evidences behind these methods. Moreover, why do we need epsilon_L and epsilon_H in (12)? The authors may need to provide further clarifications. \n3. It seems that there are no general principles for Type II methods, because the authors only provide the methods for power system. If I am solving for another problem, how shall we design new Type II methods? \n\n4. The experimental results do not look convincing. \n\n----4a. I do not really undrstand the numbers here. It seems that all methods achieve pretty good performance, with very small differences. \n\n----4b. The authors are just comparing their method with the penalty method, not other methods. It is hard for me to judge whether their methods are better than other state-of-the-art methods. \n\n----4c. In the related work, the authors mentioned that the existing use of homotopy on classical optimization solvers can be slow. I think it is worthwhile to compare the methods in the paper, with the previous homotopy method, in terms of running time. \n\n\n\n\n\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": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject