Quantitative stability of mixed-integer two-stage quadratic stochastic programs

For our introduced mixed-integer quadratic stochastic program with fixed recourse matrices, random recourse costs, technology matrix and right-hand sides, we study quantitative stability properties of its optimal value function and optimal solution set when the underlying probability distribution is perturbed with respect to an appropriate probability metric. To this end, we first establish various Lipschitz continuity results about the 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 linear constraints. The obtained results extend earlier results about quantitative stability properties of stochastic integer programming and stability results for mixed-integer parametric quadratic programs.     

Relations between stochastic and parametric programming for decision problems with a random objective function

In this paper we study the relation between the general concept for an optimal solution for stochastic programming problems with a random objective function-the concept of an £-efficient solution-and the associated parametric problem, We show that it is possible under certain assumptions to obtain some or even all £-efficient solutions of the stochastic problem by solving the parametric problem with respect to a certain parameter set.     

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.     

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.     

Concavity and efficient points of discrete distributions in probabilistic programming

We consider stochastic programming problems with probabilistic constraints involving integer-valued random variables. The concept of a p-efficient point of a probability distribution is used to derive various equivalent problem formulations. Next we introduce the concept of r-concave discrete probability distributions and analyse its relevance for problems under consideration. These notions are used to derive lower and upper bounds for the optimal value of probabilistically constrained stochastic programming problems with discrete random variables. The results are illustrated with numerical examples. Received: October 1998 / Accepted: June 2000?Published online October 18, 2000     

Stochastic optimal control and algorithm of the trajectory of horizontal wells

This paper presents a nonlinear, multi-phase and stochastic dynamical system according to engineering background. We show that the stochastic dynamical system exists a unique solution for every initial state. A stochastic optimal control model is constructed and the sufficient and necessary conditions for optimality are proved via dynamic programming principle. This model can be converted into a parametric nonlinear stochastic programming by integrating the state equation. It is discussed here that the local optimal solution depends in a continuous way on the parameters. A revised Hooke–Jeeves algorithm based on this property has been developed. Computer simulation is used for this paper, and the numerical results illustrate the validity and efficiency of the algorithm.     

Scenario reduction in stochastic programming with respect to discrepancy distances

Discrete approximations to chance constrained and mixed-integer two-stage stochastic programs require moderately sized scenario sets. The relevant distances of (multivariate) probability distributions for deriving quantitative stability results for such stochastic programs are ℬ-discrepancies, where the class ℬ of Borel sets depends on their structural properties. Hence, the optimal scenario reduction problem for such models is stated with respect to ℬ-discrepancies. In this paper, upper and lower bounds, and some explicit solutions for optimal scenario reduction problems are derived. In addition, we develop heuristic algorithms for determining nearly optimally reduced probability measures, discuss the case of the cell discrepancy (or Kolmogorov metric) in some detail and provide some numerical experience.     

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 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.     

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 and sensitivity-analysis for stochastic programming

Stability and sensitivity studies for stochastic programs have been motivated by the problem of incomplete information about the true probability measure through which the stochastic program is formulated and in connection with the development and evaluation of algorithms. The first part of this survey paper briefly introduces and compares different approaches and points out the contemporary efforts to remove and weaken assumptions that are not realistic (e.g., strict complementarity conditions). The second part surveys recent results on qualitative and quantitative stability with respect to the underlying probability measure and describes the ways and means of statistical sensitivity analysis based on Gâteaux derivatives. The last section comments on parallel statistical sensitivity results obtained in the parametric case, i.e., for probability measures belonging to a parametric family indexed by a finite dimensional vector parameter.     

A stochastic programming model to find optimal sample sizes to estimate unknown parameters in an LP

An LP is considered where the technology coefficients are unknown and random samples are taken to estimate them. A stochastic programming problem is formulated to find the optimal sample sizes where it is required that a confidence interval should cover the unknown deterministic optimum value by a given probability and the cost of sampling be minimum.     

Assessing solution quality in stochastic programs

Determining whether a solution is of high quality (optimal or near optimal) is fundamental in optimization theory and algorithms. In this paper, we develop Monte Carlo sampling-based procedures for assessing solution quality in stochastic programs. Quality is defined via the optimality gap and our procedures' output is a confidence interval on this gap. We review a multiple-replications procedure that requires solution of, say, 30 optimization problems and then, we present a result that justifies a computationally simplified single-replication procedure that only requires solving one optimization problem. Even though the single replication procedure is computationally significantly less demanding, the resulting confidence interval might have low coverage probability for small sample sizes for some problems. We provide variants of this procedure that require two replications instead of one and that perform better empirically. We present computational results for a newsvendor problem and for two-stage stochastic linear programs from the literature. We also discuss when the procedures perform well and when they fail, and we propose using ɛ-optimal solutions to strengthen the performance of our procedures.     

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.     

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.     

Distribution sensitivity for certain classes of chance-constrained models with application to power dispatch

Using results from parametric optimization, we derive for chance-constrained stochastic programs quantitative stability properties for locally optimal values and sets of local minimizers when the underlying probability distribution is subjected to perturbations in a metric space of probability measures. Emphasis is placed on verifiable sufficient conditions for the constraint-set mapping to fulfill a Lipschitz property which is essential for the stability results. Both convex and nonconvex problems are investigated. For a chance-constrained model of power dispatch, where the power demand enters as a random vector with incompletely known probability distribution, we discuss consequences of our general results for the stability of optimal generation costs and optimal generation policies.The authors thank P. Kleinmann (formerly with the Humboldt-Universität, Berlin, Germany) for his active cooperation in designing the power dispatch model and J. Mayer (MTA SZTAKI, Budapest, Hungary) for his insight into energy optimization. Further thanks are due to the referees for their constructive criticism.This research was developed in the course of a contract study between the International Institute for Applied Systems Analysis, Laxenburg, Austria and the Humboldt-Universität, Berlin, Germany.     

A note on second-order stochastic dominance constraints induced by mixed-integer linear recourse

We introduce stochastic integer programs with second-order dominance constraints induced by mixed-integer linear recourse. Closedness of the constraint set mapping with respect to perturbations of the underlying probability measure is derived. For discrete probability measures, large-scale, block-structured, mixed- integer linear programming equivalents to the dominance constrained stochastic programs are identified. For these models, a decomposition algorithm is proposed and tested with instances from power optimization.     

Robustness in stochastic programs with risk constraints

This paper is a contribution to the robustness analysis for stochastic programs whose set of feasible solutions depends on the probability distribution?P. For various reasons, probability distribution P may not be precisely specified and we study robustness of results with respect to perturbations of?P. The main tool is the contamination technique. For the optimal value, local contamination bounds are derived and applied to robustness analysis of the optimal value of a portfolio performance under risk-shaping CVaR constraints. A?new robust portfolio efficiency test with respect to the second order stochastic dominance criterion is suggested and the contamination methodology is exploited to analyze its resistance with respect to additional scenarios.     

Solving generalized semi-infinite programs by reduction to simpler problems

The article intends to give a unifying treatment of different approaches to solve generalized semi-infinite programs by transformation to simpler problems. In particular dual-, penalty-, discretization-, reduction-, and Karush-Kuhn-Tucker (KKT)-methods are applied to obtain equivalent problems or relaxations of a simpler structure. The relaxations are viewed as a perturbation P _τ of the original problem P, depending on a perturbation parameter τ > 0, and are analyzed by using parametric programming techniques. We give convergence results and results on the rate of convergence for the minimal values and the optimal solutions of P _τ when τ tends toward 0. We review earlier studies and present new ones.     

Stochastic Material and Topology Design

In order to find a robust optimal topology or material design with respect to stochastic variations of the model parameters of a mechanical structure, the basic optimization problem under stochastic uncertainty must be replaced by an appropriate deterministic substitute problem. Starting from the equilibrium equation and the yield/strength conditions, the problem can be formulated as a stochastic (linear) program “with recourse”. Hence, by discretization the design space by finite elements, linearizing the yield conditions, in case of discrete probability distributions the resulting deterministic substitute problems are linear programs with a dual decomposition data structure.