Geodesic
Stochastic optimization problems with second order stochastic dominance constraints via Wasserstein metric
Kybernetika, Tome 54 (2018) no. 6, pp. 1231-1246
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

Optimization problems with stochastic dominance constraints are helpful to many real-life applications. We can recall e. g., problems of portfolio selection or problems connected with energy production. The above mentioned constraints are very suitable because they guarantee a solution fulfilling partial order between utility functions in a given subsystem $ {\cal U} $ of the utility functions. Especially, considering $ {\cal U} := {\cal U}_{1} $ (where ${\cal U}_{1} $ is a system of non decreasing concave nonnegative utility functions) we obtain second order stochastic dominance constraints. Unfortunately it is also well known that these problems are rather complicated from the theoretical and the numerical point of view. Moreover, these problems goes to semi-infinite optimization problems for which Slater's condition is not necessary fulfilled. Consequently it is suitable to modify the constraints. A question arises how to do it. The aim of the paper is to suggest one of the possibilities how to modify the original problem with an "estimation" of a gap between the original and a modified problem. To this end the stability results obtained on the base of the Wasserstein metric corresponding to ${\cal L}_{1}$ norm are employed. Moreover, we mention a scenario generation and an investigation of empirical estimates. At the end attention will be paid to heavy tailed distributions.
Optimization problems with stochastic dominance constraints are helpful to many real-life applications. We can recall e. g., problems of portfolio selection or problems connected with energy production. The above mentioned constraints are very suitable because they guarantee a solution fulfilling partial order between utility functions in a given subsystem $ {\cal U} $ of the utility functions. Especially, considering $ {\cal U} := {\cal U}_{1} $ (where ${\cal U}_{1} $ is a system of non decreasing concave nonnegative utility functions) we obtain second order stochastic dominance constraints. Unfortunately it is also well known that these problems are rather complicated from the theoretical and the numerical point of view. Moreover, these problems goes to semi-infinite optimization problems for which Slater's condition is not necessary fulfilled. Consequently it is suitable to modify the constraints. A question arises how to do it. The aim of the paper is to suggest one of the possibilities how to modify the original problem with an "estimation" of a gap between the original and a modified problem. To this end the stability results obtained on the base of the Wasserstein metric corresponding to ${\cal L}_{1}$ norm are employed. Moreover, we mention a scenario generation and an investigation of empirical estimates. At the end attention will be paid to heavy tailed distributions.
DOI : 10.14736/kyb-2018-6-1231
Classification : 90C15
Keywords: stochastic programming problems; second order stochastic dominance constraints; stability; Wasserstein metric; relaxation; scenario generation; empirical estimates; light- and heavy-tailed distributions; crossing 
@article{10_14736_kyb_2018_6_1231,
     author = {Ka\v{n}kov\'a, Vlasta and Omel\v{c}enko, Vadim},
     title = {Stochastic optimization problems with second order stochastic dominance constraints via {Wasserstein} metric},
     journal = {Kybernetika},
     pages = {1231--1246},
     year = {2018},
     volume = {54},
     number = {6},
     doi = {10.14736/kyb-2018-6-1231},
     mrnumber = {3902631},
     zbl = {07031771},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.14736/kyb-2018-6-1231/}
}
TY  - JOUR
AU  - Kaňková, Vlasta
AU  - Omelčenko, Vadim
TI  - Stochastic optimization problems with second order stochastic dominance constraints via Wasserstein metric
JO  - Kybernetika
PY  - 2018
SP  - 1231
EP  - 1246
VL  - 54
IS  - 6
UR  - http://geodesic.mathdoc.fr/articles/10.14736/kyb-2018-6-1231/
DO  - 10.14736/kyb-2018-6-1231
LA  - en
ID  - 10_14736_kyb_2018_6_1231
ER  - 
%0 Journal Article
%A Kaňková, Vlasta
%A Omelčenko, Vadim
%T Stochastic optimization problems with second order stochastic dominance constraints via Wasserstein metric
%J Kybernetika
%D 2018
%P 1231-1246
%V 54
%N 6
%U http://geodesic.mathdoc.fr/articles/10.14736/kyb-2018-6-1231/
%R 10.14736/kyb-2018-6-1231
%G en
%F 10_14736_kyb_2018_6_1231
Kaňková, Vlasta; Omelčenko, Vadim. Stochastic optimization problems with second order stochastic dominance constraints via Wasserstein metric. Kybernetika, Tome 54 (2018) no. 6, pp. 1231-1246. doi: 10.14736/kyb-2018-6-1231

[1] Dentcheva, D., Ruszczyński, A.: Optimization with stochastic dominance constraints. SIAM J. Optim. 14 (2003), 548-566. | DOI | MR

[2] Kaňková, V., Houda, M.: Thin and heavy tails in stochastic programming. Kybernetika 51 (2015), 3, 433-456. | DOI | MR

[3] Kaňková, V., Omelchenko, V.: Empirical estimates in stochastic programs with probability and second order stochastic dominance constraints. Acta Math. Universitatis Comenianae 84 (2015), 2, 267-281. | MR

[4] Kaňková, V.: Stability, empirical estimates and scenario generation in stochastic optimization-applications in finance. Kybernetika 53 (2017), 6, 1026-1046. | DOI | MR

[5] Klebanov, L. B.: Heavy Tailed Distributions. MATFYZPRESS, Prague 2003. | DOI

[6] Kotz, S., Balakrishnan, N., Johnson, N. J.: Continuous Multivariate Distributions (Volume 1: Models and Applications.). John Wiley and Sons, New York 2000. | MR

[7] Odintsov, K.: Decision Problems and Empirical Data; Applicationns to New Types of Problems (in Czech). Diploma Thesis, Faculty of Mathematics and Physics, Charles University Prague, Prague 2012.

[8] Ogryczak, W., Ruszczyński, A.: From stochastic dominance to mean-risk models: Semideviations as risk measures. Europ. J. Oper. Res. 116 (1999), 33-50. | DOI

[9] Omelčenko, V.: Problems of Stochastic Optimization under Uncertainty, Quantitative Methods, Simulations, Applications in Gas Storage Valuation. Doctoral Thesis, Faculty of Mathematics and Physics, Charles University Prague, Prague 2016.

[10] Rockafellar, R., Wets, R. J. B.: Variational Analysis. Springer, Berlin 1983. | DOI | Zbl

[11] Samarodnitsky, G., Taqqu, M.: Stable Non-Gaussian Random Processes. Chapman and Hall, New York 1994. | DOI | MR

Cité par Sources :