TY - JOUR

T1 - A smooth inexact penalty reformulation of convex problems with linear constraints

AU - TATARENKO, TATIANA

AU - NEDICH, ANGELIA

N1 - Funding Information:
\ast Received by the editors August 22, 2018; accepted for publication (in revised form) May 21, 2021; published electronically August 24, 2021. A preliminary and brief version of section 5 appeared in Proceedings of the 59th IEEE Conference on Decision and Control, 2020. https://doi.org/10.1137/18M1209180 \bfF \bfu \bfn \bfd \bfi \bfn \bfg : The second author received support from the Office of Naval Research, grant N000141612245. \dagger Department of Control Theory and Robotics, Technical University Darmstadt, Darmstadt, 64283, Germany (tatiana.tatarenko@rmr.tu-darmstadt.de). \ddagger School of Electrical, Computer and Energy Engineering (ECEE), Arizona State University, Tempe, AZ 85287-5706 USA (Angelia.Nedich@asu.edu).
Publisher Copyright:
© 2021 Society for Industrial and Applied Mathematics Publications. All rights reserved.

PY - 2021

Y1 - 2021

N2 - In this work, we consider a constrained convex problem with linear inequalities and provide an inexact penalty reformulation of the problem. The novelty is in the choice of the penalty functions, which are smooth and can induce a nonzero penalty over some points in a feasible region of the original constrained problem. The resulting unconstrained penalized problem is parametrized by two penalty parameters which control the slope and the curvature of the penalty function. With a suitable selection of these penalty parameters, we show that the solutions of the resulting penalized unconstrained problem are feasible for the original constrained problem, under some assumptions. Also, we establish that, with suitable choices of penalty parameters, the solutions of the penalized unconstrained problem can achieve a suboptimal value which is arbitrarily close to the optimal value of the original constrained problem. For the problems with a large number of linear inequality constraints, a particular advantage of such a smooth penalty-based reformulation is that it renders a penalized problem suitable for the implementation of fast incremental gradient methods, which require only one sample from the inequality constraints at each iteration. We consider applying SAGA proposed in [A. Defazio, F. Bach, and S. Lacoste-Julien, SAGA: A fast incremental gradient method with support for non-strongly convex composite objectives, in Proceedings of NIPS, 2014, pp. 1646-1654] to solve the resulting penalized unconstrained problem. Moreover, we propose an alternative approach using a sequence of penalized problem. The approach is based on time-varying penalty parameters and, thus, does not require knowledge of some problem-specific constants that may be difficult to estimate. For a strongly convex objective function, under some conditions on the penalty parameters, we prove that a single-loop full gradient-based algorithm applied to the corresponding time-varying penalized problems converges to the solution of the original constrained problem.

AB - In this work, we consider a constrained convex problem with linear inequalities and provide an inexact penalty reformulation of the problem. The novelty is in the choice of the penalty functions, which are smooth and can induce a nonzero penalty over some points in a feasible region of the original constrained problem. The resulting unconstrained penalized problem is parametrized by two penalty parameters which control the slope and the curvature of the penalty function. With a suitable selection of these penalty parameters, we show that the solutions of the resulting penalized unconstrained problem are feasible for the original constrained problem, under some assumptions. Also, we establish that, with suitable choices of penalty parameters, the solutions of the penalized unconstrained problem can achieve a suboptimal value which is arbitrarily close to the optimal value of the original constrained problem. For the problems with a large number of linear inequality constraints, a particular advantage of such a smooth penalty-based reformulation is that it renders a penalized problem suitable for the implementation of fast incremental gradient methods, which require only one sample from the inequality constraints at each iteration. We consider applying SAGA proposed in [A. Defazio, F. Bach, and S. Lacoste-Julien, SAGA: A fast incremental gradient method with support for non-strongly convex composite objectives, in Proceedings of NIPS, 2014, pp. 1646-1654] to solve the resulting penalized unconstrained problem. Moreover, we propose an alternative approach using a sequence of penalized problem. The approach is based on time-varying penalty parameters and, thus, does not require knowledge of some problem-specific constants that may be difficult to estimate. For a strongly convex objective function, under some conditions on the penalty parameters, we prove that a single-loop full gradient-based algorithm applied to the corresponding time-varying penalized problems converges to the solution of the original constrained problem.

KW - Convex minimization

KW - Incremental methods

KW - Inexact penalty

KW - Linear constraints

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U2 - 10.1137/18M1209180

DO - 10.1137/18M1209180

M3 - Article

AN - SCOPUS:85114386517

VL - 31

SP - 2141

EP - 2170

JO - SIAM Journal on Optimization

JF - SIAM Journal on Optimization

SN - 1052-6234

IS - 3

ER -