Informatica

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.

We consider a problem of nonlinear stochastic optimization with linear constraints. The method of ɛ‐feasible solution by series of Monte‐Carlo estimators has been developed for solving this problem avoiding “jamming” or “zigzagging”. Our approach is distinguished by two peculiarities: the optimality of solution is tested in a statistical manner and the Monte‐Carlo sample size is adjusted so as to decrease the total amount of Monte‐Carlo trials and, at the same time, to guarantee the estimation of the objective function with an admissible accuracy. Under some general conditions we prove by the martingale approach that the proposed method converges a.s. to the stationary point of the problem solved. As a counterexample the maximization of the probability of portfolio desired return is given, too.

Stochastic programming problems with simple recourse belong to problems depending on a random element only through the corresponding probability measure. Consequently, this probability measure can be treated as a parameter of the problem.

It is well known that many practical optimization problems with random elements lead from the mathematical point of view to deterministic optimization problems depending on the random elements through probability laws only. Further, it is also well known that these probability laws are known very seldom. Consequently, statistical estimates of the unknown probability measure, if they exist, must be employed to obtain some estimates of the optimal value and the optimal solution, at least.

Error bounds are developed for a class of quadratic programming problems. The absolute error between an approximate feasible solution, generated via a dual formulation, and the true optimal solution is measured. Furthermore, these error bounds involve considerably less work computationally than existing estimates.