Yongchao Liu

Quantitative Stability Analysis of Stochastic Generalized Equations

We consider the solution of a system of stochastic generalized equations (SGE) where the underlying functions are mathematical expectation of random set-valued mappings. SGE has many applications such as characterizing optimality conditions of a nonsmooth stochastic optimization problem or equilibrium conditions of a stochastic equilibrium problem. We derive quantitative continuity of expected value of the set-valued mapping with respect to the variation of the underlying probability measure in a metric space. This leads to the subsequent qualitative and quantitative stability analysis of solution set mappings of the SGE. Under some metric regularity conditions, we derive Aubins property of the solution set mapping with respect to the change of probability measure. The established results are applied to stability analysis of stochastic variational inequality, stationary points of classical one-stage and two-stage stochastic minimization problems, two-stage stochastic mathematical programs with equilibriu...

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 strong duality (zero dual gap) when the ambiguity set is defined through moments. In this paper, we investigate effective ways for verifying the Slater type conditions and introduce other conditions which are based on lower semicontinuity of the optimal value function of the inner maximization problem. Moreover, we propose two discretization schemes for solving the DRO with one for the dualized DRO and the other directly through the ambiguity set of the DRO. In the absence of strong duality, the discretization scheme via Lagrange duality may provide an upper bound for the optimal value of the DRO whereas the direct discretization approach provides a lower bound. Two cutting plane schemes are consequently proposed: one for the discretized dualized DRO and the other for the minimax DRO with discretized ambiguity set. Convergence analysis is presented for the approximation schemes in terms of the optimal value, optimal solutions and stationary points. Comparative numerical results are reported for the resulting algorithms.

Penalized Sample Average Approximation Methods for Stochastic Mathematical Programs with Complementarity Constraints

This paper considers a one-stage stochastic mathematical program with a complementarity constraint (SMPCC), where uncertainties appear in both the objective function and the complementarity constraint, and an optimal decision on both upper-and lower-level decision variables must be made before the realization of the uncertainties. A partially exactly penalized sample average approximation (SAA) scheme is proposed to solve the problem. Asymptotic convergence of optimal solutions and stationary points of the penalized SAA problem is carried out. It is shown under some moderate conditions that the statistical estimators obtained from solving the penalized SAA problems converge almost surely to its true counterpart as the sample size increases. Exponential rate of convergence of estimators is also established under some additional conditions.

Entropic Approximation for Mathematical Programs with Robust Equilibrium Constraints

In this paper, we consider a class of mathematical programs with robust equilibrium constraints represented by a system of semi-infinite complementarity constraints (SICC). We propose a numerical scheme for tackling SICC. Specifically, by relaxing the complementarity constraints and then randomizing the index set of SICC, we employ the well-known entropic risk measure to approximate the semi-infinite constraints with a finite number of stochastic inequality constraints. Under some moderate conditions, we quantify the approximation in terms of the feasible set and the optimal value. The approximation scheme is then applied to a class of two stage stochastic mathematical programs with complementarity constraints in combination with the polynomial decision rules. Finally, we extend the discussion to a mathematical program with distributionally robust equilibrium constraints, which is essentially a one stage stochastic program with semi-infinite stochastic constraints indexed by some probability measures from...

Convergence Analysis Of A Regularized Sample Average Approximation Method For Stochastic Mathematical Programs With Complementarity Constraints

Regularization method proposed by Scholtes (2011) has been a recognized approach for deterministic mathematical programs with complementarity constraints (MPCC). Meng and Xu (2006) applied the approach coupled with Monte Carlo techniques to solve a class of one stage stochastic MPCC and presented some promising numerical results. However, Meng and Xu have not presented any convergence analysis of the regularized sample approximation method. In this paper, we fill out this gap. Specifically, we consider a general class of one stage stochastic mathematical programs with complementarity constraint where the objective and constraint functions are expected values of random functions. We carry out extensive convergence analysis of the regularized sample average approximation problems including the convergence of statistical estimators of optimal solutions, C-stationary points, M-stationary points and B-stationary points as sample size increases and the regularization parameter tends to zero.

Distributionally robust equilibrium for continuous games: Nash and Stackelberg models

We develop several distributionally robust equilibrium models, following the recent research surge of robust game theory, in which some or all of the players in the games lack of complete information on the true probability distribution of underlying uncertainty but they need to make a decision prior to the realization of such uncertainty. We start with a distributionally robust Nash equilibrium model where each player uses partial information to construct a set of distributions and chooses an optimal decision on the basis of the worst distribution rather than the worst scenario to hedge the risk arising from ambiguity of the true probability distribution. We investigate the existence of equilibrium, develop a numerical scheme for its computation, and consider special cases where the distributionally robust Nash equilibrium model can be reformulated as an ordinary deterministic Nash equilibrium. We then extend our modeling scheme to two possible frameworks of distributionally robust Stackelberg setting: a distributionally robust follower model and a distributionally robust leader model. These two frameworks are employed to study an innovative problem of hierarchical competition in a supply chain where a buyer not only invests in its own capacity to supply an end-product market under demand uncertainty but also outsources a certain amount of market supplies to multiply competing suppliers who invest in capacity for obtaining the buyer’s orders. In this application, we show that the buyer has more incentives to invest in capacity whereas the suppliers have less to do so when those suppliers are confronted with more demand uncertainty in the end-product market over the buyer.

Stability Analysis of One Stage Stochastic Mathematical Programs with Complementarity Constraints

We study the quantitative stability of the solution sets, optimal value and M-stationary points of one stage stochastic mathematical programs with complementarity constraints when the underlying probability measure varies in some metric probability space. We show under moderate conditions that the optimal solution set mapping is upper semi-continuous and the optimal value function is Lipschitz continuous with respect to probability measure. We also show that the set of M-stationary points as a mapping is upper semi-continuous with respect to the variation of the probability measure. A particular focus is given to empirical probability measure approximation which is also known as sample average approximation (SAA). It is shown that optimal value and M-stationary points of SAA programs converge to their true counterparts with probability one (w.p.1.) at exponential rate as the sample size increases.