Vlasta Kaňková

A Note on Quantitative Stability and Empirical Estimates in Stochastic Programming

The paper deals with a stability of stochastic programming problems considered with respect to a probability measure space. In particular, the paper deals with the stability of the problems in which the operator of mathematical expectation appears in the objective function, constraints set is “deterministic” and the probability measure space is equipped with the Kolmogorov or the Wasserstein metric. The stability results are furthermore employed to statistical estimates in the stochastic programming problems. Some results on a consistence and a rate of convergence are presented.

A Note on Multifunctions in Stochastic Programming

Two-stage stochastic programming problems and chance constrained stochastic programming problems belong to deterministic optimization problems depending on a probability measure. Surely, the probability measure can be considered as a parameter of such problems and, moreover, it is reasonable to investigate stability with respect to it. However, to investigate the stability of the above mentioned problems it means, mostly, to investigate simultaneously behaviour of the objective functions and multifunctions corresponding to the constraints sets. The aim of the paper is to investigate stability of the multifunctions and summarize by it (from the constraints point of view) problems with the stable behaviour.

A Remark on Multiobjective Stochastic Optimization Problems: Stability and Empirical Estimates

We consider a multiobjective optimization problem in which objective functions are in the form of mathematical expectation of functions depending on a random element and a constraints set can depend on the probability measure. Evidently then probability measure can be considered as a parameter of the problem. The aim of this note is to present a survey of some assertions on a stability and statistical estimates of the set of (properly) efficient points. To this end, already known results from one-objective stochastic programming theory are employed.

Multistage Stochastic Programs via Stochastic Parametric Optimization

Multistage stochastic programming problems can be defined as a finite system of (mostly parametric) one-stage stochastic programming problems with an inner type of dependence (for details see e.g. [1], [2], [6]). Employing this approach we can introduce the multistage (M+1-stage, M ≥ 1) stochastic programming problem as the problem.

A Note on the Relationship between Strongly Convex Functions and Multiobjective Stochastic Programming Problems

We consider multiobjective optimization problems in which objective functions are in the form of mathematical expectation of functions depending on a random element and a constraints set can depend on a probability measure. An efficient points set characterizes the multiobjective problems very often instead of the solution set in one objective case. A stability of the efficient points set (w.r.t. a probability measures space) and empirical estimates have been already investigated in the case when all objective functions were assumed to be strongly convex. The aim of the contribution is to present a modified assertions under rather weaker assumptions.

A note on objective functions in multistage stochastic nonlinear programming problems

Multistage stochastic programming problems well correspond to many practical situations in which a random element exists and moreover it is reasonable to treat them with respect to some discrete time interval. In particular, this type of problems correspond to practical situations that can be considered with respect to some time interval and simultanously decomposed with respect to the individual time points. The aim of this paper is to investigate the objective functions corresponding to the individual problems belonging to the one multistage stochastic programming problem. A special attention is paid to the Lipschitz property.

On Stability in Two-Stage Stochastic Nonlinear Programming

Two-stage stochastic programming problems are very often assigned to practical optimization problems with random elements. Especially, these models are employed if the basic solution should be determined without knowing the random parameter realization and if the obtained effect can be corrected by a new optimization problem (called the recourse problem) depending on the random elements realization. It is well-known that then the total problem depends on the random elements only through the corresponding probability measure. Consequently, the probability measure can be treated as a parameter in such problems and it is surely reasonable to study the stability with respect to it. The aim of this paper is to study the stability of two-stage nonlinear programming problem with respect to the distribution function. Of course, the linear case is also included in our consideration.