2013/01/08 by Farzad Yousefian, Angelia Nedić, Yousefian, Farzad +3
Computer Science · Decision Sciences · Economics, Econometrics and Finance · Mathematics · #Economic theories and models #FOS: Mathematics #Optimization and Control (math.OC) #Optimization and Variational Analysis #Point processes and geometric inequalities #Risk and Portfolio Optimization #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1301.1711
openalex publication_date 2013/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by problems arising in decentralized control problems and\nnon-cooperative Nash games, we consider a class of strongly monotone Cartesian\nvariational inequality (VI) problems, where the mappings either contain\nexpectations or their evaluations are corrupted by error. Such complications\nare captured under the umbrella of Cartesian stochastic variational inequality\nproblems and we consider solving such problems via stochastic approximation\n(SA) schemes. Specifically, we propose a scheme wherein the steplength sequence\nis derived by a rule that depends on problem parameters such as monotonicity\nand Lipschitz constants. The proposed scheme is seen to produce sequences that\nare guaranteed to converge almost surely to the unique solution of the problem.\nTo cope with networked multi-agent generalizations, we provide requirements\nunder which independently chosen steplength rules still possess desirable\nalmost-sure convergence properties. In the second part of this paper, we\nconsider a regime where Lipschitz constants on the map are either unavailable\nor difficult to derive. Here, we present a local randomization technique that\nallows for deriving an approximation of the original mapping, which is then\nshown to be Lipschitz continuous with a prescribed constant. Using this\ntechnique, we introduce a locally randomized SA algorithm and provide\nalmost-sure convergence theory for the resulting sequence of iterates to an\napproximate solution of the original variational inequality problem. Finally,\nthe paper concludes with some preliminary numerical results on a stochastic\nrate allocation problem and a stochastic Nash-Cournot game.\n