Continuity of parametric mixed-integer quadratic programs and its application to stability analysis of two-stage quadratic stochastic programs with mixed-integer recourse

For mixed-integer quadratic program where all coefficients in the objective function and the right-hand sides of constraints vary simultaneously, we show locally Lipschitz continuity of its optimal value function, and derive the corresponding global estimation; furthermore, we also obtain quantitative estimation about the change of its optimal solutions. Applying these results to two-stage quadratic stochastic program with mixed-integer recourse, we establish quantitative stability of the optimal value function and the optimal solution set with respect to the Fortet-Mourier probability metric, when the underlying probability distribution is perturbed. The obtained results generalize available results on continuity properties of mixed-integer quadratic programs and extend current results on quantitative stability of two-stage quadratic stochastic programs with mixed-integer recourse.     

Quantitative stability of full random two-stage stochastic programs with recourse

In this paper, we consider quantitative stability analysis for two-stage stochastic linear programs when recourse costs, the technology matrix, the recourse matrix and the right-hand side vector are all random. For this purpose, we first investigate continuity properties of parametric linear programs. After deriving an explicit expression for the upper bound of its feasible solutions, we establish locally Lipschitz continuity of the feasible solution sets of parametric linear programs. These results are then applied to prove continuity of the generalized objective function derived from the full random second-stage recourse problem, from which we derive new forms of quantitative stability results of the optimal value function and the optimal solution set with respect to the Fortet–Mourier probability metric. The obtained results are finally applied to establish asymptotic behavior of an empirical approximation algorithm for full random two-stage stochastic programs.     

Continuity of the optimal value function and optimal solutions of parametric mixed-integer quadratic programs

To properly describe and solve complex decision problems, research on theoretical properties and solution of mixed-integer quadratic programs is becoming very important. We establish in this paper different Lipschitz-type continuity results about the optimal value function and optimal solutions of mixed-integer parametric quadratic programs with parameters in the linear part of the objective function and in the right-hand sides of the linear constraints. The obtained results extend some existing results for continuous quadratic programs, and, more importantly, lay the foundation for further theoretical study and corresponding algorithm analysis on mixed-integer quadratic programs.     

Quantitative Stability Analysis of Two-Stage Stochastic Linear Programs with Full Random Recourse

Abstract

In this paper, we apply the parametric linear programing technique and pseudo metrics to study the quantitative stability of the two-stage stochastic linear programing problem with full random recourse. Under the simultaneous perturbation of the cost vector, coefficient matrix, and right-hand side vector, we first establish the locally Lipschitz continuity of the optimal value function and the boundedness of optimal solutions of parametric linear programs. On the basis of these results, we deduce the locally Lipschitz continuity and the upper bound estimation of the objective function of the two-stage stochastic linear programing problem with full random recourse. Then by adopting different pseudo metrics, we obtain the quantitative stability results of two-stage stochastic linear programs with full random recourse which improve the current results under the partial randomness in the second stage problem. Finally, we apply these stability results to the empirical approximation of the two-stage stochastic programing model, and the rate of convergence is presented.     

Continuity and Stability of a Quadratic Mixed-Integer Stochastic Program

For the two-stage quadratic stochastic program where the second-stage problem is a general mixed-integer quadratic program with a random linear term in the objective function and random right-hand sides in constraints, we study continuity properties of the second-stage optimal value as a function of both the first-stage policy and the random parameter vector. We also present sufficient conditions for lower or upper semicontinuity, continuity, and Lipschitz continuity of the second-stage problem's optimal value function and the upper semicontinuity of the optimal solution set mapping with respect to the first-stage variables and/or the random parameter vector. These results then enable us to establish conclusions on the stability of optimal value and optimal solutions when the underlying probability distribution is perturbed with respect to the weak convergence of probability measures.     

Quantitative stability of fully random mixed-integer two-stage stochastic programs

Mixed-integer two-stage stochastic programs with fixed recourse matrix, random recourse costs, technology matrix, and right-hand sides are considered. Quantitative continuity properties of its optimal value and solution set are derived when the underlying probability distribution is perturbed with respect to an appropriate probability metric.     

Continuity and stability of fully random two-stage stochastic programs with mixed-integer recourse

In order to derive continuity and stability of two-stage stochastic programs with mixed-integer recourse when all coefficients in the second-stage problem are random, we first investigate the quantitative continuity of the objective function of the corresponding continuous recourse problem with random recourse matrices. Then by extending derived results to the mixed-integer recourse case, the perturbation estimate and the piece-wise lower semi-continuity of the objective function are proved. Under the framework of weak convergence for probability measure, the epi-continuity and joint continuity of the objective function are established. All these results help us to prove a qualitative stability result. The obtained results extend current results to the mixed-integer recourse with random recourse matrices which have finitely many atoms.     

Problem-based optimal scenario generation and reduction in stochastic programming

Scenarios are indispensable ingredients for the numerical solution of stochastic programs. Earlier approaches to optimal scenario generation and reduction are based on stability arguments involving distances of probability measures. In this paper we review those ideas and suggest to make use of stability estimates based only on problem specific data. For linear two-stage stochastic programs we show that the problem-based approach to optimal scenario generation can be reformulated as best approximation problem for the expected recourse function which in turn can be rewritten as a generalized semi-infinite program. We show that the latter is convex if either right-hand sides or costs are random and can be transformed into a semi-infinite program in a number of cases. We also consider problem-based optimal scenario reduction for two-stage models and optimal scenario generation for chance constrained programs. Finally, we discuss problem-based scenario generation for the classical newsvendor problem.

     

Postoptimality for mean-risk stochastic mixed-integer programs and its application

The mean-risk stochastic mixed-integer programs can better model complex decision problems under uncertainty than usual stochastic (integer) programming models. In order to derive theoretical results in a numerically tractable way, the contamination technique is adopted in this paper for the postoptimality analysis of the mean-risk models with respect to changes in the scenario set, here the risk is measured by the lower partial moment. We first study the continuity of the objective function and the differentiability, with respect to the parameter contained in the contaminated distribution, of the optimal value function of the mean-risk model when the recourse cost vector, the technology matrix and the right-hand side vector in the second stage problem are all random. The postoptimality conclusions of the model are then established. The obtained results are applied to two-stage stochastic mixed-integer programs with risk objectives where the objective function is nonlinear with respect to the probability distribution. The current postoptimality results for stochastic programs are improved.     

Stability analysis for stochastic programs

For stochastic programs with recourse and with (several joint) probabilistic constraints, respectively, we derive quantitative continuity properties of the relevant expectation functionals and constraint set mappings. This leads to qualitative and quantitative stability results for optimal values and optimal solutions with respect to perturbations of the underlying probability distributions. Earlier stability results for stochastic programs with recourse and for those with probabilistic constraints are refined and extended, respectively. Emphasis is placed on equipping sets of probability measures with metrics that one can handle in specific situations. To illustrate the general stability results we present possible consequences when estimating the original probability measure via empirical ones.     

Distribution sensitivity in stochastic programming

In this paper, stochastic programming problems are viewed as parametric programs with respect to the probability distributions of the random coefficients. General results on quantitative stability in parametric optimization are used to study distribution sensitivity of stochastic programs. For recourse and chance constrained models quantitative continuity results for optimal values and optimal solution sets are proved (with respect to suitable metrics on the space of probability distributions). The results are useful to study the effect of approximations and of incomplete information in stochastic programming.This research was presented in parts at the 4th International Conference on Stochastic Programming held in Prague in September 1986.     

Quantitative stability of full random two-stage problems with quadratic recourse

ABSTRACT

In this paper, we discuss quantitative stability of two-stage stochastic programs with quadratic recourse where all parameters in the second-stage problem are random. By establishing the Lipschitz continuity of the feasible set mapping of the restricted Wolfe dual of the second-stage quadratic programming in terms of the Hausdorff distance, we prove the local Lipschitz continuity of the integrand of the objective function of the two-stage stochastic programming problem and then establish quantitative stability results of the optimal values and the optimal solution sets when the underlying probability distribution varies under the Fortet–Mourier metric. Finally, the obtained results are applied to study the asymptotic behaviour of the empirical approximation of the model.     

On distributionally robust optimization problems with k-th order stochastic dominance constraints induced by full random quadratic recourse

In this paper, we consider the optimization problems with k-th order stochastic dominance constraint on the objective function of the two-stage stochastic programs with full random quadratic recourse. By establishing the Lipschitz continuity of the feasible set mapping under some pseudo-metric, we show the Lipschitz continuity of the optimal value function and the upper semicontinuity of the optimal solution mapping of the problem. Furthermore, by the Hölder continuity of parameterized ambiguity set under the pseudo-metric, we demonstrate the quantitative stability results of the feasible set mapping, the optimal value function and the optimal solution mapping of the corresponding distributionally robust problem.     

Improving constants of strong convexity in linear stochastic programming

We derive formulas for constants of strong convexity (CSCs) of expectation functions encountered in two-stage stochastic programs with linear recourse. One of them yields a CSC as the optimal value of a certain quadratically constrained quadratic program, another one in terms of the thickness of the feasibility polytope of the dual problem associated to the recourse problem. CSCs appear in Hoelder-type estimates relating the distance of optimal solution sets of stochastic programs to a suitable distance of underlying probability distributions.     

On the Glivenko-Cantelli problem in stochastic programming: Mixed-integer linear recourse

Expected recourse functions in linear two-stage stochastic programs with mixed-integer second stage are approximated by estimating the underlying probability distribution via empirical measures. Under mild conditions, almost sure uniform convergence of the empirical means to the original expected recourse function is established.     

Stability in stochastic programming with recourse-estimated parameters

In this paper, stability of the optimal solution of stochastic programs with recourse with respect to parameters of the given distribution of random coefficients is studied. Provided that the set of admissible solutions is defined by equality constraints only, asymptotical normality of the optimal solution follows by standard methods. If nonnegativity constraints are taken into account the problem is solved under assumption of strict complementarity known from the theory of nonlinear programming (Theorem 1). The general results are applied to the simple recourse problem with random right-hand sides under various assumptions on the underlying distribution (Theorems 2–4).     

Solving stochastic programs with integer recourse by enumeration: A framework using Gröbner basis

In this paper we present a framework for solving stochastic programs with complete integer recourse and discretely distributed right-hand side vector, using Gröbner basis methods from computational algebra to solve the numerous second-stage integer programs. Using structural properties of the expected integer recourse function, we prove that under mild conditions an optimal solution is contained in a finite set. Furthermore, we present a basic scheme to enumerate this set and suggest improvements to reduce the number of function evaluations needed.     

Distribution functions in stochastic programs with recourse: A parametric analysis

This paper studies the behavior of the optimum value of a two-stage stochastic program with recourse (random right-hand sides) as the mean and covariance matrices defining the random variables in the program are perturbed. Several results for convex programs are developed and are used to study the effect such perturbations have on the regularity properties of the stochastic programs. Cost associated with incorrectly specifying the mean and covariance matrices are discussed and estimated. A stochastic programming model in which the random variable is dependent on the first-stage decision is presented.     

Constructive characterizations of the value-function of a mixed-integer program I

Recent work has provided explicit formulas for the value function of a pure integer program, and for indicator functions for the set of feasible right-hand sides. These formulas are based on linear functions, next-higher integer operations, sums, and maxima. In the present paper, we investigate possible extensions to mixed-integer programs.