A mapping-based constraint-handling technique for evolutionary algorithms with its applications to portfolio optimization problems

A novel Constraint-Handling Technique (CHT) for Evolutionary Algorithms (EAs) applied to constrained optimization problems is proposed. It is assumed that the feasible region of the constrained optimization problem is defined by a convex-hull of multiple vertices. On the other hand, without loss of generality, the search space of EA is given by a hyper-cube. The proposed CHT called Convex-Hull Mapping (CHM) transforms the real vector in the search space of EA into the solution in the feasible region. It is also proven that CHM performs a surjective mapping from the search space of EA to the feasible region. Although the proposed CHM can be applied to any EAs, one of the latest EAs, or Adaptive Differential Evolution (ADE), is used in this paper. By using ADE, CHM is compared with conventional CHTs in a real-world optimization problem in the field of finance, namely the portfolio optimization problem. Portfolio optimization is the process of determining the best proportion of investment in different assets according to some objective. Specifically, to reveal the characteristic of CHM depending on the number of the above vertices, three different formulations of the portfolio optimization problem are employed to evaluate the performance of ADE using CHM. Numerical experiments show that CHM is better than conventional CHTs in most cases. Moreover, the hybrid method combining CHM with a conventional CHT outperforms the original CHT.