Monotonicity and Complexity of Multistage Stochastic Variational Inequalities

In this paper, we consider multistage stochastic variational inequalities (MSVIs). First, we give multistage stochastic programs and multistage multi-player noncooperative game problems as source problems. After that, we derive the monotonicity properties of MSVIs under less restrictive conditions. Finally, the polynomial rate of convergence with respect to sample sizes between the original problem and its … Read more

Convergence Analysis and a DC Approximation Method for Data-driven Mathematical Programs with Distributionally Robust Chance Constraints

In this paper, we consider the convergence analysis of data-driven mathematical programs with distributionally robust chance constraints (MPDRCC) under weaker conditions without continuity assumption of distributionally robust probability functions. Moreover, combining with the data-driven approximation, we propose a DC approximation method to MPDRCC without some special tractable structures. We also give the convergence analysis of … Read more

Decomposition Methods for Solving Two-Stage Distributionally Robust Optimization Problems

Decomposition methods have been well studied for solving two-stage and multi-stage stochastic programming problems, see [29, 32, 33]. In this paper, we propose an algorithmic framework based on the fundamental ideas of the methods for solving two-stage minimax distributionally robust optimization (DRO) problems where the underlying random variables take a finite number of distinct values. … Read more

Convergence Analysis of Sample Average Approximation of Two-stage Stochastic Generalized Equations

A solution of two-stage stochastic generalized equations is a pair: a first stage solution which is independent of realization of the random data and a second stage solution which is a function of random variables. This paper studies convergence of the sample average approximation of two-stage stochastic nonlinear generalized equations. In particular an exponential rate … Read more

Discrete Approximation of Two-Stage Stochastic and Distributionally Robust Linear Complementarity Problems

In this paper, we propose a discretization scheme for the two-stage stochastic linear complementarity problem (LCP) where the underlying random data are continuously distributed. Under some moderate conditions, we derive qualitative and quantitative convergence for the solutions obtained from solving the discretized two-stage stochastic LCP (SLCP). We ex- plain how the discretized two-stage SLCP may … Read more

Subdifferentiation and Smoothing of Nonsmooth Integral Functionals

The subdifferential calculus for the expectation of nonsmooth random integrands involves many fundamental and challenging problems in stochastic optimization. It is known that for Clarke regular integrands, the Clarke subdifferential equals the expectation of their Clarke subdifferential. In particular, this holds for convex integrands. However, little is known about calculation of Clarke subgradients for the … Read more

Distributionally Robust Optimization with Matrix Moment Constraints: Lagrange Duality and Cutting Plane Methods

A key step in solving minimax distributionally robust optimization (DRO) problems is to reformulate the inner maximization w.r.t. probability measure as a semiinfinite programming problem through Lagrange dual. Slater type conditions have been widely used for zero dual gap when the ambiguity set is defined through moments. In this paper, we investigate effective ways for … Read more

Asymptotic Convergence Analysis for Distributional Robust Optimization and Equilibrium Problems

In this paper, we study distributional robust optimization approaches for a one stage stochastic minimization problem, where the true distribution of the underlying random variables is unknown but it is possible to construct a set of probability distributions which contains the true distribution and optimal decision is taken on the basis of worst possible distribution … Read more

Asymptotic Analysis of Sample Average Approximation for Stochastic Optimization Problems with Joint Chance Constraints via CVaR/DC Approximations

Conditional Value at Risk (CVaR) has been recently used to approximate a chance constraint. In this paper, we study the convergence of stationary points when sample average approximation (SAA) method is applied to a CVaR approximated joint chance constrained stochastic minimization problem. Specifically, we prove, under some moderate conditions, that optimal solutions and stationary points … Read more

Exact Penalization, Level Function Method and Modified Cutting-Plane Method for Stochastic Programs with Second Order Stochastic Dominance Constraints

Level function methods and cutting plane methods have been recently proposed to solve stochastic programs with stochastic second order dominance (SSD) constraints. A level function method requires an exact penalization setup because it can only be applied to the objective function, not the constraints. Slater constraint qualification (SCQ) is often needed for deriving exact penalization. … Read more