Stochastic Programming with Equilibrium Constraints

Abstract

In this paper we discuss here-and-now type stochastic programs with equilibrium constraints. We give a general formulation of such problems and study their basic properties such as measurability and continuity of the corresponding integrand functions. We also discuss consistency and rates of convergence of sample average approximations of such stochastic problems.

Robust bi-level optimization models in transportation science

“…Fletcher et al 2006;Ralph 2008; and references therein). Shapiro (2006) establishes that convergence in some cases is exponential, which implies that relatively few samples may be needed to reach at least a nearoptimal (or, near-stationary) solution; numerical experiments in Shapiro & Xu (2005) and Birbil et al (2006) confirm this belief at least in small-scale examples. Therefore, there is a hope that SMPEC traffic models can be approximately solved with a computational effort that is not several orders of magnitude larger than, for one instance, that of a discretized problem.…”

“…Since stochastic programming is an instance of (SMPEC U ) (e.g. Patriksson & Wynter 1999;Shapiro 2006), the latter clearly contains robust models. As remarked in Evgrafov & Patriksson (2004), however, no generic conditions on Z and S under which one can expect a continuous behaviour of the optimal solutions to non-convex models within the framework of (SMPEC U ) (or at least the continuous behaviour of the optimal values of the upper-level variables x) yet exist.…”

On M-stationary points for a stochastic equilibrium problem under equilibrium constraints in electricity spot market modeling

“…Hence, the pair (q, y) of generation and flow has to be considered as a Ê N +m -valued random vector on (Ω, F , È) and the ISO has to minimize the expected overall costs, i.e., (26) min…”

“…We refer to the monographs [14], [19], [5], the survey [12] and to [25], [6]. Extensions to stochastic MPECs (SMPECs) can be found in [26], [27] and applications to electricity markets are discussed, e.g., in [9], [21].…”

Combined Monte Carlo sampling and penalty method for Stochastic nonlinear complementarity problems

“…Note that problem (1.2) is actually a stochastic mathematical program with equilibrium constraints (SMPEC), which plays an important role in many fields such as engineering design, economic equilibrium, and multilevel games. See [20,1,15,16,18,19,21,22,26,27,28,29] for details. Problem (1.2) has the following main difficulties:…”