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Several real-world applications could be modeled as Mixed-Integer Non-LinearProgramming (MINLP) problems, and some prominent examples include portfoliooptimization, remote sensing technology, and so on. Most of the models forthese applications are non-convex and always involve some conflictingobjectives. The mathematical and heuristic methods have their advantages insolving this category of problems. In this work, we turn to Multi-ObjectiveEvolutionary Algorithms (MOEAs) for finding elegant solutions for suchproblems. In this framework, we investigate a multi-objective constrainedportfolio optimization problem, which can be cast as a classical financialproblem and can also be naturally modeled as an MINLP problem. Consequently, wepoint out one challenge, faced by a direct coding scheme for MOEAs, to thisproblem. It is that the dependence among variables, like the selection andweights for one same asset, will likely make the search difficult. We thus,propose a Compressed Coding Scheme (CCS), compressing the two dependentvariables into one variable to utilize the dependence and thereby meeting thischallenge. Subsequently, we carry out a detailed empirical study on two sets ofinstances. The first part consists of 5 instances from OR-Library, which issolvable for the general mathematical optimizer, like CPLEX, while theremaining 15 instances from NGINX are addressed only by MOEAs. The twobenchmarks, involving the number of assets from 31 to 2235, consistentlyindicate that CCS is not only efficient but also robust for dealing with theconstrained multi-objective portfolio optimization.

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