Comparing Discrete and Continuous Genotypes on the Constrained Portfolio Selection Problem

Abstract

Abstract. In financial engineering the problem of portfolio selection has drawn much attention in the last decades. But still unsolved problems remain, while on the one hand the type of model to use is still debated, even the most common models cannot be solved efficiently, if real world constraints are added. This is not only because the portfolio selection problem is multi-objective, but also because constraints may turn a formerly continuous problem into a discrete one. Therefore, we suggest to use a Multi-Objective Evolutionary Algorithm and compare discrete and continuous representations. To meet constraints we apply a repair mechanism and examine the impact of Lamarckism and the Baldwin Effect on several instances of the portfolio selection problem.

Evolutionary multi-objective portfolio optimization in practical context

“…Such a scheme facilitates the removal and adding of assets to portfolios, resulting in smaller portfolios generally. This representation has been popular in [5,14,15,17]. Alternatively, the weight vector can just comprise a few assets that are randomly chosen prior to the algorithmic run [16,24] .…”

“…While floor constraint has been actively studied in [5,10,[12][13][14][15][16][17], the general floor and ceiling constraint has been less explored. Cardinality constraint specifies the maximum and minimum number of assets that a portfolio can hold due to monitoring, diversification or transaction cost control reasons.…”

“…This constraint has been simplified in several related works, where either the inequality restriction in (6) is replaced by an equality restriction instead, i.e., portfolios are restricted to a particular fixed value of cardinality [10][11][12][13][14][15][16]18] or only the maximum cardinality constraint is considered [5,17,19] . Round-lot constraint requires the number of any asset included in the portfolio to be in exact multiples of the normal trading lots [5,20] .…”

Natural Computing in Finance – A Review

“…The formulation of the portfolio problem includes cardinality and buy-in constraints, and a repair mechanism was applied in order to ensure that generated solutions were feasible. The utility of differing crossover operators and differing genotypic representations for the portfolio selection problem was examined by [103,104], and the application of EC hybrids was examined by [106] and [105]. The impact of cardinality constraints was examined in [46] and [82] (the latter also adopted an EC hybrid approach).…”

Hybrid Metaheuristic for Portfolio Selection: Comparison with an exact solver and search space analysis

“…Every object have to be instantiated by a choice, and the combination of these choices leads to a specific formulation (model) of the problem, hence to different optimisation results. For instance, as stated by di Tollo and Roli [8], two main choices are possible for variables: continuous [15], [29], [31], [28] and integer [30], [21]. Choosing continuous variables is quite 'natural' and its representation is independent of the actual budget, while integer values (ranging between zero and the maximum available budget, or equal to the number of 'rounds') allow us to add constraints taking into account actual budget, minimum lots and to tackle other objective functions to better explain the problem at hand.…”