An Incremental Gradient Method for Optimization Problems with Variational Inequality Constraints

We consider minimizing a sum of agent-specific nondifferentiable merely convex functions over the solution set of a variational inequality (VI) problem in that each agent is associated with a local monotone mapping. This problem finds an application in computation of the best equilibrium in nonlinea...

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Bibliographic Details
Published in:IEEE transactions on automatic control Vol. 68; no. 12; pp. 1 - 8
Main Authors: Kaushik, Harshal D., Samadi, Sepideh, Yousefian, Farzad
Format: Journal Article
Language:English
Published: New York IEEE 01.12.2023
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Complexity theory
Computation
Computational complexity
Computational modeling
Convergence
Distributed algorithms
Gradient methods
Iterative methods
Linear programming
Mapping
Mathematical programming
Optimization
Optimization methods
Stochastic processes
Transportation networks
ISSN:0018-9286, 1558-2523
Online Access:Get full text
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Summary:We consider minimizing a sum of agent-specific nondifferentiable merely convex functions over the solution set of a variational inequality (VI) problem in that each agent is associated with a local monotone mapping. This problem finds an application in computation of the best equilibrium in nonlinear complementarity problems arising in transportation networks. We develop an iteratively regularized incremental gradient method where at each iteration, agents communicate over a directed cycle graph to update their solution iterates using their local information about the objective and the mapping. The proposed method is single-timescale in the sense that it does not involve any excessive hard-to-project computation per iteration. We derive non-asymptotic agent-wise convergence rates for the suboptimality of the global objective function and infeasibility of the VI constraints measured by a suitably defined dual gap function. The proposed method appears to be the first fully iterative scheme equipped with iteration complexity that can address distributed optimization problems with VI constraints over cycle graphs.
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ISSN:0018-9286
1558-2523
DOI:10.1109/TAC.2023.3251851