Huifu Xu

Stochastic Approximation Approaches to the Stochastic Variational Inequality Problem

Stochastic approximation methods have been extensively studied in the literature for solving systems of stochastic equations and stochastic optimization problems where function values and first order derivatives are not observable but can be approximated through simulation. In this paper, we investigate stochastic approximation methods for solving stochastic variational inequality problems (SVIP) where the underlying functions are the expected value of stochastic functions. Two types of methods are proposed: stochastic approximation methods based on projections and stochastic approximation methods based on reformulations of SVIP. Global convergence results of the proposed methods are obtained under appropriate conditions.

A Stochastic Multiple-Leader Stackelberg Model: Analysis, Computation, and Application

We study an oligopoly consisting of M leaders and N followers that supply a homogeneous product (or service) noncooperatively. Leaders choose their supply levels first, knowing the demand function only in distribution. Followers make their decisions after observing the leader supply levels and the realized demand function. We term the resulting equilibrium a stochastic multiple-leader Stackelberg-Nash-Cournot (SMS) equilibrium. We show the existence and uniqueness of SMS equilibrium under mild assumptions. We also propose a computational approach to find the equilibrium based on the sample average approximation method and analyze its rate of convergence. Finally, we apply this framework to model competition in the telecommunication industry.

Stochastic mathematical programs with equilibrium constraints, modelling and sample average approximation

In this article, we discuss the sample average approximation (SAA) method applied to a class of stochastic mathematical programs with variational (equilibrium) constraints. To this end, we briefly investigate the structure of both–the lower level equilibrium solution and objective integrand. We show almost sure convergence of optimal values, optimal solutions (both local and global) and generalized Karush–Kuhn–Tucker points of the SAA program to their true counterparts. We also study uniform exponential convergence of the sample average approximations, and as a consequence derive estimates of the sample size required to solve the true problem with a given accuracy. Finally, we present some preliminary numerical test results.

Sample Average Approximation Methods For A Class Of Stochastic Variational Inequality Problems

In this paper we apply the well known sample average approximation (SAA) method to solve a class of stochastic variational inequality problems (SVIPs). We investigate the existence and convergence of a solution to the sample average approximated SVIP. Under some moderate conditions, we show that the sample average approximated SVIP has a solution with probability one and with probability approaching one exponentially fast with the increase of sample size, the solution converges to its true counterpart. Finally, we apply the existence and convergence results to SAA method for solving a class of stochastic nonlinear complementarity problems and stochastic programs with stochastic constraints.

Smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications

Inspired by a recent work by Alexander et al. (J Bank Finance 30:583–605, 2006) which proposes a smoothing method to deal with nonsmoothness in a conditional value-at-risk problem, we consider a smoothing scheme for a general class of nonsmooth stochastic problems. Assuming that a smoothed problem is solved by a sample average approximation method, we investigate the convergence of stationary points of the smoothed sample average approximation problem as sample size increases and show that w.p.1 accumulation points of the stationary points of the approximation problem are weak stationary points of their counterparts of the true problem. Moreover, under some metric regularity conditions, we obtain an error bound on approximate stationary points. The convergence result is applied to a conditional value-at-risk problem and an inventory control problem.

Stochastic Nash equilibrium problems: sample average approximation and applications

This paper presents a Nash equilibrium model where the underlying objective functions involve uncertainty and nonsmoothness. The well-known sample average approximation method is applied to solve the problem and the first order equilibrium conditions are characterized in terms of Clarke generalized gradients. Under some moderate conditions, it is shown that with probability one, a statistical estimator (a Nash equilibrium or a Nash-C-stationary point) obtained from sample average approximate equilibrium problem converges to its true counterpart. Moreover, under some calmness conditions of the Clarke generalized derivatives, it is shown that with probability approaching one exponentially fast by increasing sample size, the Nash-C-stationary point converges to a weak Nash-C-stationary point of the true problem. Finally, the model is applied to stochastic Nash equilibrium problem in the wholesale electricity market.

Single and multi-period optimal inventory control models with risk-averse constraints

This paper presents some convex stochastic programming models for single and multi-period inventory control problems where the market demand is random and order quantities need to be decided before demand is realized. Both models minimize the expected losses subject to risk aversion constraints expressed through Value at Risk (VaR) and Conditional Value at Risk (CVaR) as risk measures. A sample average approximation method is proposed for solving the models and convergence analysis of optimal solutions of the sample average approximation problem is presented. Finally, some numerical examples are given to illustrate the convergence of the algorithm.

Convergence Analysis of Sample Average Approximation Methods for a Class of Stochastic Mathematical Programs with Equality Constraints

In this paper we discuss the sample average approximation (SAA) method for a class of stochastic programs with nonsmooth equality constraints. We derive a uniform Strong Law of Large Numbers for random compact set-valued mappings and use it to investigate the convergence of Karush-Kuhn-Tucker points of SAA programs as the sample size increases. We also study the exponential convergence of global minimizers of the SAA problems to their counterparts of the true problem. The convergence analysis is extended to a smoothed SAA program. Finally, we apply the established results to a class of stochastic mathematical programs with complementarity constraints and report some preliminary numerical test results.

An Implicit Programming Approach for a Class of Stochastic Mathematical Programs with Complementarity Constraints

In this paper, we consider a class of stochastic mathematical programs in which the complementarity constraints are subject to random factors and the objective function is the mathematical expectation of a smooth function which depends on both upper and lower level variables and random factors. We investigate the existence, uniqueness, and differentiability of the lower level equilibrium defined by the complementarity constraints %and its dependence using a nonsmooth version of implicit function theorem. We also study the differentiability and convexity of the objective function which implicitly depends upon the lower level equilibrium. We propose numerical methods to deal with difficulties due to the continuous distribution of the random variables and intrinsic nonsmoothness of lower level equilibrium solutions due to the complementarity constraints in order that the treated programs can be readily solved by available numerical methods for deterministic mathematical programs with complementarity constraints.

Necessary and Sufficient Conditions for Optimal Offers in Electricity Markets

In this paper, we consider the optimal policy for a generator offering power into a wholesale electricity market operating under a pool arrangement. Anderson and Philpott [Math. Oper. Res., 27 (2002), pp. 82--100] recently discussed necessary conditions for an optimal offer curve when there is uncertainty in the demand and in the behavior of other participants in the market. They show that the objective function in these circumstances can be expressed as a line integral along the offer curve of a profit function integrated with respect to a market distribution function. In this paper, we prove the existence of an optimal offer stack, and we extend the analysis of [Math. Oper. Res., 27 (2002), pp. 82--100] to include necessary conditions of a higher order in the presence of horizontal and/or vertical sections in an offer curve. Finally, we establish sufficient conditions for an offer curve to be locally optimal.