## Stochastic optimization

### Field characteristic

#### Stochastic programming and Decision in Economy

Main attention was focused firstly on a special type of multistage stochastic programming problems in which an underlying random element follows (generally) nonlinear autoregressive sequence and constraints sets are given by systems of individual probability constraints. There were introduced assumptions under which the above mentioned multistage problems are equivalent to the "classical" multistage problems with deterministic constraints. Moreover, the stability results has been introduced.

Secondly, problems in the stochastic programming with recourse were analyzed. The stochastic programming problems with recourse are a composition of an inner and an outer problems. A solution of the inner problem can depend on the random element realization and on a solution of the outer problem while the solution of the outer problem can depend on the random element only through the corresponding probability measure. According to this fact the solution of the inner problem also depends actually on the underlying probability measure. The aim is to investigate the stability of the inner problem w.r.t. the probability measure. This results can find an application (for example) in a production process in which managers corresponding to the outer and the inner problems are different.

Thirdly, a relationship between integrated empirical process and empirical estimates in the stochastic programming was investigated. In the case of one--dimensional random element the stability results obtained on the base of the Wasserstein metric are very suitable for numerical investigation. The aim has been to present numerical studies for independent and some types of dependent random samples.

#### Stochastic Data Envelopment Analysis

Standard model for evaluating Pareto-Koopmans technical efficiency of production units - Data Envelopment Analysis, assumes the exact data. Nonetheless, in reality, the uncertainty in data occurs, and any defence is necessary. We have analyzed several approaches to this problem, and we have taken into account $\alpha$-stochastic concept. We have questioned the conditions for production unit to be $\alpha$-stochastically inefficient, and we have introduced new criteria for general distribution of errors, and stronger for the special case, for the normal distribution. In addition, we have asked the opposite question, the conditions for efficiency. Also in this paper, new conditions for general and for normal distribution were revealed.

#### Mean-Variance Optimality in Markov Decision Processes

Mean variance selection rules, originally introduced by Markowitz for the portfolio selection problem, may very well capture risk sensitive behaviour of a decision maker. For this reason mean variance optimality was also studied stochastic dynamic models, in particular for discrete- and continuous time Markov chains and for semi-Markov reward processes. On the base of the results for the growth rate of the variance of total reward for discrete- and continuous-time Markov reward chains, we were able to estimate the growth rate of the variance for the more general semi-Markov reward processes. Additional attention was paid to the mean-variance optimality in Markov decision processes with respect to various mean-variance optimality criteria. In particular, on employing algorithmic procedures and using a computer program created in the SAS software we were able to solve medium size problems (up to 100 states and 100 admissible actions in each state) within 10 minutes on a standard PC computer.

### People

• Vlasta Kaòková
stochastic programming problems, stochastic decision procedures.
Economic dynamics, dynamic programming, stochastic systems.
• Martin Šmíd
approximation methods, multistage stochastic programming.
• Michal Houda
stochastic programming, stability, empirical estimates, approximations.
• Milan Sitaø
stochastic dynamic programming.
• Petr Chovanec
stochastic programming, applications to social problems.
• Jakub Jeøábek
stochastic dynamic programming.

### Selected publications

• Chovanec P.: New criteria for stochastic DEA. In: Proceedings of the 23rd International Conference Mathematical Methods in Economics 2005 (Hana Skalská, ed.). Gaudeamus, Hradec Králové 2005, pp. 164-170.
• Chovanec P.: Production possibility frontier and stochastic programming In: Proceedings of the 14th Annual Conference of Doctoral Students -- WDS 2005 (Jana Šafránková ed.), MATFYZPRESS, Prague 2005, pp. 108--113. ISBN 80-86732-59-2.
• van Dijk N.M., Sladký K.: Total reward variance in discrete and continuous time Markov chains. In: Operations Research Proceedings 2004 (Selected Papers of the International Conference on Operations Research 2004, Tilburg, September 1-3, 2004, H. Fleuren, D. den Hertog, P. Kort, eds.) Springer, Berlin- Heidelberg 2005, pp. 319-326.
• Kaòková V.: On stability of stochastic programming problems with linear recourse. In: Proceedings of the 23rd International Conference Mathematical Methods in Economics 2005 (Hana Skalská, ed.). Gaudeamus, Hradec Králové 2005, pp. 188--195.
• Houda M.: Using metrics in stability of stochastic programming problems. Acta Oeconomica Pragensia, 13 (2005), 1, 128-134.
• Houda M.: Estimation in chance-constrained problem. In: Proceedings of the 23rd International Conference Mathematical Methods in Economics 2005. (Skalská H. ed.). Gaudeamus, Hradec Králové 2005, pp. 134-139.
• Sladký K.: On mean reward variance in semi-Markov processes. Mathematical Methods of Operations Research 62 (2005), No. 3, pp. 387-397.
• Sladký K., Sitaø M.: Optimal solutions for undiscounted variance penalized Markov decision chains. Lecture Notes in Economics and Mathematical Systems 532 (IFIP/IIASA/GAMM Workshop "Dynamic Stochastic Optimization", Y. Ermoliev, K. Marti and G. Pflug, eds.) Springer, Berlin 2004, pp. 43-66.
• Sladký K., Sitaø M.: Mean variance optimality in Markov decision chains. In: Proceedings of the 23rd International Conference Mathematical Methods in Economics 2005 (H. Skalská, ed.). University of Hradec Králové, Hradec Králové 2005, pp. 350--357.
• Sladký K., Sitaø M.: Algorithmic procedures for mean variance optimality in Markov decision chains. Operations Research Proceedings 2005 (Selected Papers of the International Conference on Operations Research 2005, Bremen, September 7--9). Springer, Berlin (accepted for publication).

### Grants and projects

• Stochastic Decision Approaches in Nonlinear Economic Models
Vlasta Kaòková, grant No. 402/04/1294 of the Czech Science Foundation.
• Validation of Economic Decision Models and Results
Karel Sladký, grant No. 402/05/0115 of the Czech Science Foundation, with Charles University of Prague.