Journal:Informatica
Volume 15, Issue 2 (2004), pp. 271–282
Abstract
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.
Journal:Informatica
Volume 14, Issue 1 (2003), pp. 37–62
Abstract
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.