research-article

A Method with Convergence Rates for Optimization Problems with Variational Inequality Constraints

Authors:

Harshal D. Kaushik,

Farzad YousefianAuthors Info & Claims

SIAM Journal on Optimization, Volume 31, Issue 3

Pages 2171 - 2198

https://doi.org/10.1137/20M1357378

Published: 01 January 2021 Publication History

Abstract

We consider a class of optimization problems with Cartesian variational inequality (CVI) constraints, where the objective function is convex and the CVI is associated with a monotone mapping and a convex Cartesian product set. This mathematical formulation captures a wide range of optimization problems including those complicated by the presence of equilibrium constraints, complementarity constraints, or an inner-level large scale optimization problem. In particular, an important motivating application arises from the notion of efficiency estimation of equilibria in multi-agent networks, e.g., communication networks and power systems. In the literature, the iteration complexity of the existing solution methods for optimization problems with CVI constraints appears to be unknown. To address this shortcoming, we develop a first-order method called averaging randomized block iteratively regularized gradient (aRB-IRG). The main contributions include the following: (i) In the case where the associated set of the CVI is bounded and the objective function is nondifferentiable and convex, we derive new nonasymptotic suboptimality and infeasibility convergence rate statements in an ergodic sense. We also obtain deterministic variants of the convergence rates when we suppress the randomized block-coordinate scheme. Importantly, this paper appears to be the first work to provide these rate guarantees for this class of problems. (ii) In the case where the CVI set is unbounded and the objective function is smooth and strongly convex, utilizing the properties of the Tikhonov trajectory, we establish the global convergence of aRB-IRG in an almost sure and a mean sense. We provide the numerical experiments for computing the best Nash equilibrium in a networked Cournot competition model.

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Latafat PThemelis AVilla SPatrinos P(2025)On the Convergence of Proximal Gradient Methods for Convex Simple Bilevel OptimizationJournal of Optimization Theory and Applications10.1007/s10957-024-02564-6204:3Online publication date: 1-Mar-2025
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Cao JJiang RAbolfazli NHamedani EMokhtari AOh ANaumann TGloberson ASaenko KHardt MLevine S(2023)Projection-free methods for stochastic simple bilevel optimization with convex lower-level problemProceedings of the 37th International Conference on Neural Information Processing Systems10.5555/3666122.3666390(6105-6131)Online publication date: 10-Dec-2023
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Jalilzadeh AYousefian FEbrahimi M(2023)Stochastic Approximation for Estimating the Price of Stability in Stochastic Nash GamesACM Transactions on Modeling and Computer Simulation10.1145/363252534:2(1-24)Online publication date: 11-Nov-2023
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cover image SIAM Journal on Optimization

SIAM Journal on Optimization Volume 31, Issue 3

DOI:10.1137/sjope8.31.3

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© 2021, Society for Industrial and Applied Mathematics.

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Society for Industrial and Applied Mathematics

United States

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Published: 01 January 2021

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Cited By

Latafat PThemelis AVilla SPatrinos P(2025)On the Convergence of Proximal Gradient Methods for Convex Simple Bilevel OptimizationJournal of Optimization Theory and Applications10.1007/s10957-024-02564-6204:3Online publication date: 1-Mar-2025
https://dl.acm.org/doi/10.1007/s10957-024-02564-6
Cao JJiang RAbolfazli NHamedani EMokhtari AOh ANaumann TGloberson ASaenko KHardt MLevine S(2023)Projection-free methods for stochastic simple bilevel optimization with convex lower-level problemProceedings of the 37th International Conference on Neural Information Processing Systems10.5555/3666122.3666390(6105-6131)Online publication date: 10-Dec-2023
https://dl.acm.org/doi/10.5555/3666122.3666390
Jalilzadeh AYousefian FEbrahimi M(2023)Stochastic Approximation for Estimating the Price of Stability in Stochastic Nash GamesACM Transactions on Modeling and Computer Simulation10.1145/363252534:2(1-24)Online publication date: 11-Nov-2023
https://dl.acm.org/doi/10.1145/3632525
Xie ZCai GDong Q(2023)Strong convergence of Bregman projection method for solving variational inequality problems in reflexive Banach spacesNumerical Algorithms10.1007/s11075-022-01414-893:1(269-294)Online publication date: 1-May-2023
https://dl.acm.org/doi/10.1007/s11075-022-01414-8
Shen LHo-Nguyen NKılınç-Karzan F(2023)An online convex optimization-based framework for convex bilevel optimizationMathematical Programming: Series A and B10.1007/s10107-022-01894-5198:2(1519-1582)Online publication date: 1-Apr-2023
https://dl.acm.org/doi/10.1007/s10107-022-01894-5

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