Modeling and Optimizing the Multi-Objective Portfolio Optimization Problem with Trapezoidal Fuzzy Parameters

Abstract

A common issue in the Multi-Objective Portfolio Optimization Problem (MOPOP) is the presence of uncertainty that affects individual decisions, e.g., variations on resources or benefits of projects. Fuzzy numbers are successful in dealing with imprecise numerical quantities, and they found numerous applications in optimization. However, so far, they have not been used to tackle uncertainty in MOPOP. Hence, this work proposes to tackle MOPOP’s uncertainty with a new optimization model based on fuzzy trapezoidal parameters. Additionally, it proposes three novel steady-state algorithms as the model’s solution process. One approach integrates the Fuzzy Adaptive Multi-objective Evolutionary (FAME) methodology; the other two apply the Non-Dominated Genetic Algorithm (NSGA-II) methodology. One steady-state algorithm uses the Spatial Spread Deviation as a density estimator to improve the Pareto fronts’ distribution. This research work’s final contribution is developing a new defuzzification mapping that allows measuring algorithms’ performance using widely known metrics. The results show a significant difference in performance favoring the proposed steady-state algorithm based on the FAME methodology.

The Hypervolume Newton Method for Constrained Multi-objective Optimization Problems

“…As an example, in many portfolio problems, one is interested in maximizing the expected return, and social responsibility or sustainability, while minimizing the risk of a financial portfolio ( [5], [6]). In multi-objective optimization we distinguish between the decision space, which contains the vectors of decision variables, and the objective space, which is the k dimensional real vectors and comprises the images of the vector-valued objective function.…”

Numerical and Evolutionary Optimization 2020

“…In [8], Estrada-Padilla et al propose a new methodology to deal with uncertainties in multi-objective portfolio optimization problems by using fuzzy numbers. The results show a significant difference in performance favoring the proposed steady-state algorithm based on the fuzzy adaptive multi-objective evolutionary (FAME) methodology.…”