Utilizing Dependence among Variables in Evolutionary Algorithms for Mixed-Integer Programming: A Case Study on Multi-Objective Constrained Portfolio Optimization

Several real-world applications could be modeled as Mixed-Integer Non-Linear Programming (MINLP) problems, and some prominent examples include portfolio optimization, remote sensing technology, and so on. Most of the models for these applications are non-convex and always involve some conflicting objectives. The mathematical and heuristic methods have their advantages in solving this category of problems. In this work, we turn to Multi-Objective Evolutionary Algorithms (MOEAs) for finding elegant solutions for such problems. In this framework, we investigate a multi-objective constrained portfolio optimization problem, which can be cast as a classical financial problem and can also be naturally modeled as an MINLP problem. Consequently, we point out one challenge, faced by a direct coding scheme for MOEAs, to this problem. It is that the dependence among variables, like the selection and weights for one same asset, will likely make the search difficult. We thus, propose a Compressed Coding Scheme (CCS), compressing the two dependent variables into one variable to utilize the dependence and thereby meeting this challenge. Subsequently, we carry out a detailed empirical study on two sets of instances. The first part consists of 5 instances from OR-Library, which is solvable for the general mathematical optimizer, like CPLEX, while the remaining 15 instances from NGINX are addressed only by MOEAs. The two benchmarks, involving the number of assets from 31 to 2235, consistently indicate that CCS is not only efficient but also robust for dealing with the constrained multi-objective portfolio optimization.