تکامل در درختان: در طراحی یک استراتژی تکامل برای سناریو بر اساس چند دوره بهینه سازی سبد سرمایه گذاری تحت هزینه های معاملاتی

Scenario-based optimization is a problem class often occurring in finance, planning and control. While the standard approach is usually based on linear stochastic programming, this paper develops an Evolution Strategy (ES) that can be used to treat nonlinear planning problems arising from Value at Risk (VaR)-constraints and not necessarily proportional transaction costs. Due to the VaR-constraints the optimization problem is generally of non-convex type and its decision version is already NP-complete. The developed ES is the first algorithm in the field of evolutionary and swarm intelligence that tackles this kind of optimization problem. The algorithm design is based on the covariance matrix self-adaptation ES (CMSA-ES). The optimization is performed on scenario trees where in each node specific constraints (balance equations) must be fulfilled. In order to evaluate the performance of the ES proposed, instances of increasing problem hardness are considered. The application to the general case with nonlinear node constraints shows not only the potential of the ES designed, but also its limitations. The latter are basically determined by the high dimensionalities of the search spaces defined by the scenario trees.

This paper designs evolution strategies (ESs) for discrete-time multi-period multi-asset portfolio optimization problems with Value at Risk (VaR)-constraints and not necessarily proportional transaction costs. It is a follow-up of [1], which analyzed the design of Evolution Strategies (ES) for one-period portfolio optimization problems under VaR-constraints without transaction costs, where it turned out that the algorithm design was challenging due to the combination of seemingly simple constraints. The problem considered in this paper is even more complex since in the multi-period case, node constraints (in general non-linear balance equations) must be fulfilled forcing the ES to evolve on a non-linear manifold. The considered problem is a stochastic control problem whose information and decision structure is defined on scenario trees. In finance and operations research literature, problems of this kind are approached by linear stochastic programming [2] and [3]. However, due to the nonlinearities in the problem class considered here, the problem must be linearized to allow for the application of this standard approach. In [4] it is stated that metaheuristics might be successfully applied to such types of problems. However, up to now this problem class has not been tackled by swarm or evolutionary methods. Since Evolutionary Algorithms (EAs) allow for a great flexibility they might be well-suited for the treatment of nonlinearities arising from VaR-constraints and non-linear transaction costs. In contrast to the problem considered in this paper, classical portfolio optimization in the framework of the Markowitz model is intrinsically a multiobjective optimization problem. In [5] recent trends for EAs applied to such portfolio optimization problems are discussed and in [6] a particle swarm approach has been proposed and compared to other algorithms to tackle that problem class. However, virtually all the featured algorithms found in the literature deal only with the one-period problem and the multi-period problem is not considered. In cases where an EA was applied to the multi-period portfolio problem [7], [8] and [9], the problem itself is different from (and to some extent simpler) the one considered here. This paper is the first that provides a design methodology and an ES implementation for evolutionary optimization on scenario trees encountered in non-linear multi-period stochastic problems. Section 2 introduces the problem in its general form. Section 3 deals with the problem in the special case of vanishing transaction costs and uses the standard CMSA-ES as an algorithmic skeleton to design an ES, the multi-period (mp) CMSA-ES, that operates on the tree structure of the optimization problem. Section 4 presents an experimental evaluation of the novel mpCMSA-ES. While the development of Section 3 is based on linear balance equations in the tree nodes, Section 5 extends the approach to the problem in its general form with nonlinear constraints arising for example from non-linear transaction costs. Section 6 summarizes the paper and provides an outlook.