A Direct Multisearch Filter Method for Biobjective Optimization
Published in French-German-Portuguese Conference on Optimization, 2022
French-German-Portuguese Conference on Optimization
This is the 20th edition of the series of French-German conferences which initiated in Oberwolfach, Germany, in 1980. Since 1998, the conference has been organized under the participation of a third invited European country. This time it will be jointly organized with Portugal and will take place at the University of Porto. Initially planned to 2021, the French German Portuguese conference in optimization was postponed to 2022.
The purpose of this conference series has been to bring together researchers working in all different areas of Optimization and particularly to encourage young researchers to present their work. Theoretical aspects of Optimization, in addition to applications and algorithms, will be covered. The conference is held every two years and usually brings together up to 150 mathematicians. Communications will be in the form of invited plenary talks, minisymposia, and contributed talks.
A Direct Multisearch Filter Method for Biobjective Optimization
Abstract: In practical applications, it is common to have several conflicting objective functions to optimize. Frequently, these functions are of black-box type, preventing the use of multiobjective derivative-based optimization techniques. Direct Multisearch (DMS) is a multiobjective derivative-free optimization class of methods, with a well-established convergence analysis and competitive computational implementations, often successfully used for benchmark of new algorithms and in practical applications.
From the theoretical point of view, DMS is developed for continuous optimization with general constraints, making use of an extreme barrier approach, only evaluating feasible points. In this work, we propose the integration of a filter approach in DMS, to address biobjective optimization problems with linear and nonlinear constraints. The linear constraints are explicitly treated by the algorithm, by adapting the positive generating sets considered at each iteration to the geometry of the nearby constraints. The violations of the nonlinear constraints are aggregated in a third objective function and are treated as an additional objective to be minimized.
We will describe the proposed algorithmic structure in detail, provide results on the theoretical properties of the method, and report numerical experiments that state the good performance of this approach to address nonlinear constraints, both in an academic test set and in a real application.
Key words: Derivative-free Optimization, Multiobjective Optimization, Constrained Optimization, Direct Multisearch, Filter methods
Joint work with Ana L. Custódio
Acknowledgments: This research was financially supported by Fundação para a Ciência e a Tecnologia (FCT) (Portuguese Foundation for Science and Technology) through projects UIDB/00297/2020,UIDP/00297/2020, UI/BD/151246/2021 (Centro de Matemática e Aplicações - NOVA Math), and PTDC/MAT-APL/28400/2017 (NOVA Math).