Symmetric confidence regions and confidence intervals for normal map formulations of stochastic variational inequalities

Stochastic variational inequalities (SVI) model a large class of equilibrium problems subject to data uncertainty, and are closely related to stochastic optimization problems. The SVI solution is usually estimated by a solution to a sample average approximation (SAA) problem. This paper considers the normal map formulation of an SVI, and proposes a method to build asymptotically exact confidence regions and confidence intervals for the solution of the normal map formulation, based on the asymptotic distribution of SAA solutions. The confidence regions are single ellipsoids with high probability. We also discuss the computation of simultaneous and individual confidence intervals

Symmetric Confidence Regions and Confidence Intervals for Normal Map Formulations of Stochastic Variational Inequalities

Stochastic variational inequalities (SVI) model a large class of equilibrium problems subject to data uncertainty, and are closely related to stochastic optimization problems. The SVI solution is usually estimated by a solution to a sample average approximation (SAA) problem. This paper considers the normal map formulation of an SVI, and proposes a method to build asymptotically exact confidence regions and confidence intervals for the solution of the normal map formulation, based on the asymptotic distribution of SAA solutions. The confidence regions are single ellipsoids with high probability. We also discuss the computation of simultaneous and individual confidence intervals

Analyzing sampling in stochastic optimization: Importance sampling and statistical inference

The objective function of a stochastic optimization problem usually involves an expectation of random variables which cannot be calculated directly. When this is the case, a common approach is to replace the expectation with a sample average approximation. However, sometimes there are difficulties in using such a sample average approximation to achieve certain goals. This dissertation studies two specific problems. In the first problem, we aim to solve a minimization problem whose objective function is the probability of an undesired rare event. To accurately estimate this rare event probability by Monte Carlo simulation, an extremely large sample is required, which is expensive to implement. An importance sampling scheme based on the theory of large deviations is developed to efficiently reduce the sample size and thus reduce the computational cost. The convergence of a sequence of approximation problems is also studied, through which a good initial point to the minimization problem can be found. We also study the buffered probability of exceedance as an alternative risk measure instead of the ordinary probability. Under conditions, the analogous minimization problem can be formulated into a convex problem. In the second problem, we focus on a two-stage stochastic linear programming problem, where the objective function has to be approximated by a sample average function with a random sample of the corresponding random variables. However, such a sample average function is not smooth enough to estimate the Hessian of the objective function which is needed to calculate the confidence intervals for the true solution. To overcome this difficulty, the sample average function is smoothed by its convolution with a kernel function. Methods to compute confidence intervals for the true solution are then developed based on inference methods for stochastic variational inequalities.Doctor of Philosoph

Differential stability of two-stage stochastic programs

Two-stage stochastic programs with random right-hand side are considered. Optimal values and solution sets are regarded as mappings of the expected recourse functions and their perturbations, respectively. Conditions are identified implying that these mappings are directionally differentiable and semidifferentiable on appropriate functional spaces. Explicit formulas for the derivatives are derived. Special attention is paid to the role of a Lipschitz condition for solution sets as well as of a quadratic growth condition of the objective function. (orig.)Available from TIB Hannover: RR 6329(96-36) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDeutsche Forschungsgemeinschaft (DFG), Bonn (Germany)DEGerman

Differential Stability of Two-Stage Stochastic Programs

Two-stage stochastic programs with random right-hand side are considered. Optimal values and solution sets are regarded as mappings of the expected recourse functions and their perturbations, respectively. Conditions are identified implying that these mappings are directionally differentiable and semidifferentiable on appropriate functional spaces. Explicit formulas for the derivatives are derived. Special attention is paid to the role of a Lipschitz condition for solution sets as well as of a quadratic growth condition of the objective function

Differential stability of two-stage stochastic programs

Two-stage stochastic programs with random right-hand side are considered. Optimal values and solution sets are regarded as mappings of the expected recourse functions and their perturbations, respectively. Conditions are identified implying that these mappings are directionally differentiable and semidifferentiable on appropriate functional spaces. Explicit formulas for the derivatives are derived. Special attention is paid to the role of a Lipschitz condition for solution sets as well as of a quadratic growth condition of the objective function. (orig.)Available from TIB Hannover: RR 6329(96-36) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDeutsche Forschungsgemeinschaft (DFG), Bonn (Germany)DEGerman

Differential Stability of Two-Stage Stochastic Programs

Two-stage stochastic programs with random right-hand side are considered. Optimal values and solution sets are regarded as mappings of the expected recourse functions and their perturbations, respectively. Conditions are identified implying that these mappings are directionally differentiable and semidifferentiable on appropriate functional spaces. Explicit formulas for the derivatives are derived. Special attention is paid to the role of a Lipschitz condition for solution sets as well as of a quadratic growth condition of the objective function. Keywords: Two-stage stochastic programs, sensitivity analysis, directional derivatives, semiderivatives, solution sets. 1991 MSC: 90 C 15, 90 C 31 1 Introduction Two-stage stochastic programming is concerned with problems that require a hereand -now decision on the basis of given probabilistic information on the random data without making further observations. The costs to be minimized consist of the direct costs of the here-and-now (or first..