Nonlinear Stochastic Programming Involving CVaR in the Objective and Constraints

Article type: Research Article

Authors: Dumskis, Valerijonas¹ | Sakalauskas, Leonidas^{2; *}

Affiliations: [1] Šiauliai University, Šiauliai, Lithuania | [2] Institute of Mathematics and Informatics, Vilnius University, Lithuania, e-mail: valius.du@svajone.su.lt, leonidas.sakalauskas@mii.vu.lt

Correspondence: [*] Corresponding author.

Abstract: The nonlinear stochastic programming problem involving CVaR in the objective and constraints is considered. Solving the latter problem in a framework of bi-level stochastic programming, the extended Lagrangian is introduced and the related KKT conditions are derived. Next, the sequential simulation-based approach has been developed to solve stochastic problems with CVaR by finite sequences of Monte Carlo samples. The approach considered is grounded by the rule for iterative regulation of the Monte Carlo sample size and the stochastic termination procedure, taking into account the stochastic model risk. The rule is introduced to regulate the size of the Monte Carlo sample inversely proportionally to the square of the stochastic gradient norm allows us to solve stochastic nonlinear problems in a rational way and ensures the convergence. The proposed termination procedure enables us to test the KKT conditions in a statistical way and to evaluate the confidence intervals of the objective and constraint functions in a statistical way as well. The results of the Monte Carlo simulation with test functions and solution of the practice sample of trade-offs of gas purchases, storage and service reliability, illustrate the convergence of the approach considered as well as the ability to solve in a rational way the nonlinear stochastic programming problems handling CVaR in the objective and constraints, with an admissible accuracy, treated in a statistical manner.

Keywords: stochastic programming, Monte Carlo method, stochastic gradient, CVAR

Journal: Informatica, vol. 26, no. 4, pp. 569-591, 2015