Stability of solutions to chance constrained stochastic programs

Perturbations of convex chance constrained stochastic programs are considered the underlying probability distributions of which are r-concave. Verifiable sufficient conditions are established guaranteeing Hölder continuity properties of solution sets with respect to variations of the original distribution. Examples illustrate the potential, sharpness and limitations of the results

A mixed-integer stochastic nonlinear optimization problem with joint probabilistic constraints

We illustrate the solution of a mixed-integer stochastic nonlinear optimization problem in an application of power management. In this application, a coupled system consisting of a hydro power station and a wind farm is considered. The objective is to satisfy the local energy demand and sell any surplus energy on a spot market for a short time horizon. Generation of wind energy is assumed to be random, so that demand satisfaction is modeled by a joint probabilistic constraint taking into accountthe multivariate distribution. The turbine is forced to either operate between given positive limits or to be shut down. This introduces additional binary decisions. The numerical solution procedure is presented and results are illustrated

On stability in multiobjective programming - a stochastic approach

We assume that a deterministic multiobjective programming problem is approximated by surrogate problems based on estimations for the objective functions and the constraints. Making use of a large deviations approach, we investigate the behavior of the constraint sets, the sets of efficient points and the solution sets if the underlying sample tends to infinity. The results are illustrated by applying them to stochastic programming with chance constraints where (i) the distribution function of the random variable is estimated by the empirical distribution function and (ii) certain parameters are estimated

Stabilita a aproximace pro úlohy stochastického programování

Matematicko-fyzikální fakultaFaculty of Mathematics and Physic

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

Deviation measures in stochastic programming with mixed-integer recourse

Stochastic programming offers a way to treat uncertainty in decision problems. In particular, it allows the minimization of risk. We consider mean-risk models involving deviation measures, as for instance the standard deviation and the semideviation, and discuss these risk measures in the framework of stochastic dominance as well as in the framework of coherent risk measures. We derive statements concerning the structure and the stability of the resulting optimization problems whereby we emphasize on models including integrality requirements on some decision variables. Then we propose decomposition algorithms for the mean-risk models under consideration and present numerical results for two stochastic programming applications