Informatica

We present an algorithm to solve multistage stochastic convex problems, whose objective function and constraints are nonlinear. It is based on the twin-node-family concept involved in the Branch-and-Fix Coordination method. These problems have 0–1 mixed-integer and continuous variables in all the stages. The non-anticipativity constraints are satisfied by means of the twin-node-family strategy.

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.