A class of smoothing SAA methods for a stochastic mathematical program with complementarity constraints

A class of smoothing sample average approximation (SAA) methods is proposed for solving the stochastic mathematical program with complementarity constraints (SMPCC) considered by Birbil et al. [S.I. Birbil, G. Gürkan, O. Listes, Solving stochastic mathematical programs with complementarity constraints using simulation, Math. Oper. Res. 31 (2006) 739–760]. The almost sure convergence of optimal solutions of the smoothed SAA problem to that of the true problem is established by the notion of epi-convergence in variational analysis. It is demonstrated that, under suitable conditions, any accumulation point of Karash–Kuhn–Tucker points of the smoothed SAA problem is almost surely a kind of stationary point of SMPCC as the sample size tends to infinity. Moreover, under a strong second-order sufficient condition for SMPCC, the exponential convergence rate of the sequence of Karash–Kuhn–Tucker points of the smoothed SAA problem is investigated through an application of Robinson?s stability theory. Some preliminary numerical results are reported to show the efficiency of proposed method.     

A note on the sample average approximation method for stochastic mathematical programs with complementarity constraints

Meng and Xu (2006) [3] proposed a sample average approximation (SAA) method for solving a class of stochastic mathematical programs with complementarity constraints (SMPCCs). After showing that under some moderate conditions, a sequence of weak stationary points of SAA problems converge to a weak stationary point of the original SMPCC with probability approaching one at exponential rate as the sample size tends to infinity, the authors proposed an open question, that is, whether similar results can be obtained under some relatively weaker conditions. In this paper, we try to answer the open question. Based on the reformulation of stationary condition of MPCCs and new stability results on generalized equations, we present a similar convergence theory without any information of second order derivative and strict complementarity conditions. Moreover, we carry out convergence analysis of the regularized SAA method proposed by Meng and Xu (2006) [3] where the convergence results have not been considered.     

On the Convergence of Coderivative of SAA Solution Mapping for a Parametric Stochastic Generalized Equation

The aim of this paper is to investigate the convergence properties for Mordukhovich’s coderivative of the solution map of the sample average approximation (SAA) problem for a parametric stochastic generalized equation. It is demonstrated that, under suitable conditions, both the cosmic deviation and the ρ-deviation between the coderivative of the solution mapping to SAA problem and that of the solution mapping to the parametric stochastic generalized equation converge almost surely to zero as the sample size tends to infinity. Moreover, the exponential convergence rate of coderivatives of the solution maps to the SAA parametric generalized equations is established. The results are used to develop sufficient conditions for the consistency of the Lipschitz-like property of the solution map of SAA problem and the consistency of stationary points of the SAA estimator for a stochastic mathematical program with complementarity constraints.     

Stochastic Multiobjective Optimization: Sample Average Approximation and Applications

We investigate one stage stochastic multiobjective optimization problems where the objectives are the expected values of random functions. Assuming that the closed form of the expected values is difficult to obtain, we apply the well known Sample Average Approximation (SAA) method to solve it. We propose a smoothing infinity norm scalarization approach to solve the SAA problem and analyse the convergence of efficient solution of the SAA problem to the original problem as sample sizes increase. Under some moderate conditions, we show that, with probability approaching one exponentially fast with the increase of sample size, an ϵ-optimal solution to the SAA problem becomes an ϵ-optimal solution to its true counterpart. Moreover, under second order growth conditions, we show that an efficient point of the smoothed problem approximates an efficient solution of the true problem at a linear rate. Finally, we describe some numerical experiments on some stochastic multiobjective optimization problems and report preliminary results.     

On the Convergence of Coderivative of SAA Solution Mapping for a Parametric Stochastic Variational Inequality

The aim of this paper is to investigate the convergence properties for Mordukhovich’s coderivative of the solution map of the sample average approximation (SAA) problem for a parametric stochastic variational inequality with equality and inequality constraints. The notion of integrated deviation is introduced to characterize the outer limit of a sequence of sets. It is demonstrated that, under suitable conditions, both the cosmic deviation and the integrated deviation between the coderivative of the solution mapping to SAA problem and that of the solution mapping to the parametric stochastic variational inequality converge almost surely to zero as the sample size tends to infinity. Moreover, the exponential convergence rate of coderivatives of the solution maps to the SAA parametric stochastic variational inequality is established. The results are used to develop sufficient conditions for the consistency of the Lipschitz-like property of the solution map of SAA problem and the consistency of stationary points of the SAA estimator for a stochastic bilevel program.     

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.     

A Smoothing SAA Method for a Stochastic Linear Complementarity Problem

Utilizing the well-known aggregation technique, we propose a smoothing sample average approximation (SAA) method for a stochastic linear complementarity problem, where the underlying functions are represented by expectations of stochastic functions. The method is proved to be convergent and the preliminary numerical results are reported.     

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.     

CVaR-constrained stochastic programming reformulation for stochastic nonlinear complementarity problems

We reformulate a stochastic nonlinear complementarity problem as a stochastic programming problem which minimizes an expected residual defined by a restricted NCP function with nonnegative constraints and CVaR constraints which guarantee the stochastic nonlinear function being nonnegative with a high probability. By applying smoothing technique and penalty method, we propose a penalized smoothing sample average approximation algorithm to solve the CVaR-constrained stochastic programming. We show that the optimal solution of the penalized smoothing sample average approximation problem converges to the solution of the corresponding nonsmooth CVaR-constrained stochastic programming problem almost surely. Finally, we report some preliminary numerical test results.     

Convergence analysis of sample average approximation for a class of stochastic nonlinear complementarity problems: from two-stage to multistage

In this paper, we consider the sample average approximation (SAA) approach for a class of stochastic nonlinear complementarity problems (SNCPs) and study the corresponding convergence properties. We first investigate the convergence of the SAA counterparts of two-stage SNCPs when the first-stage problem is continuously differentiable and the second-stage problem is locally Lipschitz continuous. After that, we extend the convergence results to a class of multistage SNCPs whose decision variable of each stage is influenced only by the decision variables of adjacent stages. Finally, some preliminary numerical tests are presented to illustrate the convergence results.

     

Convergence analysis of stationary points in sample average approximation of stochastic programs with second order stochastic dominance constraints

Sample average approximation (SAA) method has recently been applied to solve stochastic programs with second order stochastic dominance (SSD) constraints. In particular, Hu et al. (Math Program 133:171–201, 2012) presented a detailed convergence analysis of $\epsilon $ -optimal values and $\epsilon $ -optimal solutions of sample average approximated stochastic programs with polyhedral SSD constraints. In this paper, we complement the existing research by presenting convergence analysis of stationary points when SAA is applied to a class of stochastic minimization problems with SSD constraints. Specifically, under some moderate conditions we prove that optimal solutions and stationary points obtained from solving sample average approximated problems converge with probability one to their true counterparts. Moreover, by exploiting some recent results on large deviation of random functions and sensitivity analysis of generalized equations, we derive exponential rate of convergence of stationary points.     

A log-exponential smoothing method for mathematical programs with complementarity constraints

In this paper a log-exponential smoothing method for mathematical programs with complementarity constraints (MPCC) is analyzed, with some new interesting properties and convergence results provided. It is shown that the stationary points of the resulting smoothed problem converge to the strongly stationary point of MPCC, under the linear independence constraint qualification (LICQ), the weak second-order necessary condition (WSONC), and some reasonable assumption. Moreover, the limit point satisfies the weak second-order necessary condition for MPCC. A notable fact is that the proposed convergence results do not restrict the complementarity constraint functions approach to zero at the same order of magnitude.     

Sample average approximation of stochastic dominance constrained programs

In this paper we study optimization problems with second-order stochastic dominance constraints. This class of problems allows for the modeling of optimization problems where a risk-averse decision maker wants to ensure that the solution produced by the model dominates certain benchmarks. Here we deal with the case of multi-variate stochastic dominance under general distributions and nonlinear functions. We introduce the concept of ${\mathcal{C}}$ -dominance, which generalizes some notions of multi-variate dominance found in the literature. We apply the Sample Average Approximation (SAA) method to this problem, which results in a semi-infinite program, and study asymptotic convergence of optimal values and optimal solutions, as well as the rate of convergence of the feasibility set of the resulting semi-infinite program as the sample size goes to infinity. We develop a finitely convergent method to find an ${\epsilon}$ -optimal solution of the SAA problem. An important aspect of our contribution is the construction of practical statistical lower and upper bounds for the true optimal objective value. We also show that the bounds are asymptotically tight as the sample size goes to infinity.     

The Sample Average Approximation Method Applied to Stochastic Routing Problems: A Computational Study

The sample average approximation (SAA) method is an approach for solving stochastic optimization problems by using Monte Carlo simulation. In this technique the expected objective function of the stochastic problem is approximated by a sample average estimate derived from a random sample. The resulting sample average approximating problem is then solved by deterministic optimization techniques. The process is repeated with different samples to obtain candidate solutions along with statistical estimates of their optimality gaps.We present a detailed computational study of the application of the SAA method to solve three classes of stochastic routing problems. These stochastic problems involve an extremely large number of scenarios and first-stage integer variables. For each of the three problem classes, we use decomposition and branch-and-cut to solve the approximating problem within the SAA scheme. Our computational results indicate that the proposed method is successful in solving problems with up to 2¹⁶⁹⁴ scenarios to within an estimated 1.0% of optimality. Furthermore, a surprising observation is that the number of optimality cuts required to solve the approximating problem to optimality does not significantly increase with the size of the sample. Therefore, the observed computation times needed to find optimal solutions to the approximating problems grow only linearly with the sample size. As a result, we are able to find provably near-optimal solutions to these difficult stochastic programs using only a moderate amount of computation time.     

Smoothing approach to Nash equilibrium formulations for a class of equilibrium problems with shared complementarity constraints

The equilibrium problem with equilibrium constraints (EPEC) can be looked on as a generalization of Nash equilibrium problem (NEP) and the mathematical program with equilibrium constraints (MPEC) whose constraints contain a parametric variational inequality or complementarity system. In this paper, we particularly consider a special class of EPECs where a common parametric P-matrix linear complementarity system is contained in all players?? strategy sets. After reformulating the EPEC as an equivalent nonsmooth NEP, we use a smoothing method to construct a sequence of smoothed NEPs that approximate the original problem. We consider two solution concepts, global Nash equilibrium and stationary Nash equilibrium, and establish some results about the convergence of approximate Nash equilibria. Moreover we show some illustrative numerical examples.     

Global Convergence of a Smooth Approximation Method for Mathematical Programs with Complementarity Constraints

A new smoothing approach was given for solving the mathematical programs with complementarity constraints (MPCC) by using the aggregation technique. As the smoothing parameter tends to zero, if the KKT point sequence generated from the smoothed problems satisfies the second-order necessary condition, then any accumulation point of the sequence is a B-stationary point of MPCC if the linear independence constraint qualification (LICQ) and the upper level strict complementarity (ULSC) condition hold at the limit point. The ULSC condition is weaker than the lower level strict complementarity (LLSC) condition generally used in the literatures. Moreover, the method can be easily extended to the mathematical programs with general vertical complementarity constraints.     

Solving stochastic mathematical programs with equilibrium constraints via approximation and smoothing implicit programming with penalization

In this paper, we consider the stochastic mathematical programs with linear complementarity constraints, which include two kinds of models called here-and-now and lower-level wait-and-see problems. We present a combined smoothing implicit programming and penalty method for the problems with a finite sample space. Then, we suggest a quasi-Monte Carlo approximation method for solving a problem with continuous random variables. A comprehensive convergence theory is included as well. We further report numerical results with the so-called picnic vender decision problem.     

Barzilai–Borwein method with variable sample size for stochastic linear complementarity problems

A smoothing method for solving stochastic linear complementarity problems is proposed. The expected residual minimization reformulation of the problem is considered, and it is approximated by the sample average approximation (SAA). The proposed method is based on sequential solving of a sequence of smoothing problems where each of the smoothing problems is defined with its own sample average approximation. A nonmonotone line search with a variant of the Barzilai–Borwein (BB) gradient direction is used for solving each of the smoothing problems. The BB search direction is efficient and low cost, particularly suitable for nonmonotone line search procedure. The variable sample size scheme allows the sample size to vary across the iterations and the method tends to use smaller sample size far away from the solution. The key point of this strategy is a good balance between the variable sample size strategy, the smoothing sequence and nonmonotonicity. Eventually, the maximal sample size is used and the SAA problem is solved. Presented numerical results indicate that the proposed strategy reduces the overall computational cost.     

Analysis of stochastic dual dynamic programming method

In this paper we discuss statistical properties and convergence of the Stochastic Dual Dynamic Programming (SDDP) method applied to multistage linear stochastic programming problems. We assume that the underline data process is stagewise independent and consider the framework where at first a random sample from the original (true) distribution is generated and consequently the SDDP algorithm is applied to the constructed Sample Average Approximation (SAA) problem. Then we proceed to analysis of the SDDP solutions of the SAA problem and their relations to solutions of the “true” problem. Finally we discuss an extension of the SDDP method to a risk averse formulation of multistage stochastic programs. We argue that the computational complexity of the corresponding SDDP algorithm is almost the same as in the risk neutral case.     

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.