Convergence Results of the Restricted ERM Model for Nonsmooth Stochastic Generalized Nash Equilibrium Problems

Luo, Meiju; Guo, Fang; Chen, Yi-Zeng

doi:10.1080/01966324.2015.1050564

Cited by 1 publication

(3 citation statements)

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“…see [32, equation (3.8)]. In [36], Luo and Lin minimize the expected residual while convergence analysis of the expected residual minimization (ERM) technique has been carried out in the context of stochastic Nash games [37] and stochastic variational inequality problems [35]. In more recent work, Chen, Wets, and Zhang [7] revisit this problem and present an alternate ERM formulation, with the intent of developing a smoothed sample average approximation scheme.…”

Section: The Expected-residual Minimization (Erm) Formulationmentioning

confidence: 99%

“…Remark: Finally, we emphasize three points regarding the relevance and utility of the provided statements: (i) First, such techniques are of relevance when integration cannot be carried out easily and have less utility when sample spaces are finite; (ii) There are settings where alternate models for incorporating uncertainty have been developed [7,35,37]. Such models assume relevance when the interest lies in robust solutions.…”

Section: Introductionmentioning

confidence: 99%

“…Increasingly, the deterministic model proves inadequate when contending with models complicated by risk and uncertainty. Uncertainty in variational inequality problems has been considered in a breadth of application regimes, ranging from traffic equilibrium problems [7,15], cognitive radio networks [30,49], Nash games [43,37], amongst others. Compared to the field of optimization, where stochastic programming [4,52] and robust optimization [3] have provided but two avenues for accommodating uncertainty in static optimization problems, far less is currently available either from a theoretical or an algorithmic standpoint in the context of stochastic variational inequality problems.…”

Section: Introductionmentioning

confidence: 99%

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On the existence of solutions to stochastic quasi-variational inequality and complementarity problems

Ravat

Shanbhag

2017

Math. Program.

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Variational inequality problems allow for capturing an expansive class of problems, including convex optimization problems, convex Nash games and economic equilibrium problems, amongst others. Yet in most practical settings, such problems are complicated by uncertainty, motivating the examination of a stochastic generalization of the variational inequality problem and its extensions in which the components of the mapping contain expectations. When the associated sets are unbounded, ascertaining existence requires having access to analytical forms of the expectations. Naturally, in practical settings, such expressions are often difficult to derive, severely limiting the applicability of such an approach. Consequently, our goal lies in developing techniques that obviate the need for integration and our emphasis lies in developing tractable and verifiable sufficiency conditions for claiming existence. We begin by recapping almost-sure sufficiency conditions for stochastic variational inequality problems with single-valued maps provided in our prior work [42,43] and provide extensions to multi-valued mappings. Next, we extend these statements to quasi-variational regimes where maps can be either single or set-valued. Finally, we refine the obtained results to accommodate stochastic complementarity problems where the maps are either general or co-coercive. The applicability of our results is demonstrated on practically occuring instances of stochastic quasi-variational inequality problems and stochastic complementarity problems, arising as nonsmooth generalized Nash-Cournot games and power markets, respectively.

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