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 Kaňková
stochastic programming problems, stochastic decision procedures.
- Karel Sladký
Economic dynamics, dynamic programming, stochastic systems.
- Martin Šmíd
approximation methods, multistage stochastic programming.
- Michal Houda
stochastic programming, stability, empirical estimates, approximations.
- Milan Sitař
stochastic dynamic programming.
- Petr Chovanec
stochastic programming, applications to social problems.
- Jakub Jeřábek
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.
- Kaňková 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., Sitař 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., Sitař 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., Sitař 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 Kaňková, 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.
