A Broyden-based algorithm for multi-objective local-search optimization

Abstract

In multi-objective optimization, the direction vectors in the objective space that improve the current non-dominated set are named improvement directions. The Improvement Direction Mapping (IDM) methods apply a spatial transformation to map improvement directions in the objective space to search for directions in the variable space. Jacobian based transformations can perform the mapping, however, they require the analytic expressions of the objectives and their derivatives. Hence, they serve as a reference in academic problems but are impractical in real-world applications. Furthermore, they are commonly ill-conditioned and under-determined; consequently, they deviate the search as the current non-dominated set approaches to the Pareto set. This work proposes a solution via an iterative updating based on the Broyden method. The proposal promotes parallelism between the displacement of the non-dominated solutions in the objective space and improvement directions delivered by Chebyshev scalarizing functions. Compared to other transformations reported in the specialized literature, the proposed method demonstrates a better conditioning that provokes to err at a lesser extent in providing successful search directions. This impacts the algorithm performance, mainly in locations close to the Pareto set. These advantages are demonstrated using benchmark functions and metrics.