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.

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.

The Markowitz model for single period portfolio optimization quantifies the problem by means of only two criteria: the mean, representing the expected outcome, and the risk, a scalar measure of the variability of outcomes. The classical Markowitz model uses the variance as the risk measure, thus resulting in a quadratic optimization problem. Following Sharpe's work on linear approximation to the mean‐variance model, many attempts have been made to linearize the portfolio optimization problem. There were introduced several alternative risk measures which are computationally attractive as (for discrete random variables) they result in solving Linear Programming (LP) problems. The LP solvability is very important for applications to real‐life financial decisions where the constructed portfolios have to meet numerous side constraints and take into account transaction costs. This paper provides a systematic overview of the LP solvable models with a wide discussion of their properties.

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.