The Distributionally Robust Optimization Reformulation for Stochastic Complementarity Problems

Abstract

We investigate the stochastic linear complementarity problem affinely affected by the uncertain parameters. Assuming that we have only limited information about the uncertain parameters, such as the first two moments or the first two moments as well as the support of the distribution, we formulate the stochastic linear complementarity problem as a distributionally robust optimization reformation which minimizes the worst case of an expected complementarity measure with nonnegativity constraints and a distributionally robust joint chance constraint representing that the probability of the linear mapping being nonnegative is not less than a given probability level. Applying the cone dual theory and S-procedure, we show that the distributionally robust counterpart of the uncertain complementarity problem can be conservatively approximated by the optimization with bilinear matrix inequalities. Preliminary numerical results show that a solution of our method is desirable.

Robust Optimization Model with Shared Uncertain Parameters in Multi-Stage Logistics Production and Inventory Process

“…The popular way to handle this problem, like in [14,16,18,20,21], is to consider the worst case of objective and each constraint, respectively. That is, min…”

“…That is, the inner maximization problem in Equation ( 3) can be equivalent to its dual form which is a minimization problem under some assumptions and then the minmax model in Equation ( 3) can be converted to a minimization problem. Like the specific problems considered in [14][15][16][17][18][19][20][21][22][23], we focus on affinely adjustable robust optimization with application to a multi-stage logistics production and inventory process problem.…”

Toward Machine Wald

“…Although, as noted in [34], probabilistic methods do not find appreciation among theoreticians and practitioners alike because "probabilistic reliability studies involve assumptions on the probability densities, whose knowledge regarding relevant input quantities is central", the deterministic worst case approach (limited to optimization problems over f ) is sometimes "too pessimistic to be practical" [29,34] because "it does not take into account the improbability that (possibly independent or weakly correlated) random variables conspire to produce a failure event" [113] (which constitutes one motivation for considering ambiguity sets involving both measures and functions). Therefore OUQ and Distributionally Robust Optimization (DRO) [6,48,13,174,177,166,52] could be seen as middle-ground between the deterministic worst case approach of Robust Optimization [6,13] and approaches of Stochastic Programming and Chanced Constrained Optimization [18,23] by defining optimization objectives and constraints in terms of expected values and probabilities with respect to imperfectly known distributions.…”

New Robust Reward‐Risk Ratio Models with CVaR and Standard Deviation

“…Guray Kara et al [16] used robust conditional value-at-risk (RCVaR) under parallelepiped uncertainty to find more robust portfolio allocation and reduce the risk. In addition, more and more theories and methods with incomplete messages have been studied recently in various applications (see [17][18][19][20][21][22][23][24][25][26][27][28][29] for example). In this paper, we also focus on a kind of robust models under some uncertainty set, which refer to robust ratio models with CVaR and standard deviation.…”