Subset Selection Algorithms in Multiobjective Optimisation”

Typically, multiobjective optimization problems are solved according to the Pareto principle of optimality: A solution is called Pareto-optimal if no other feasible solution exists which is better in all objectives. Each of these Pareto-optimal solutions corresponds to a different compromise among the objective functions and each of them is potentially relevant for a decision maker. Therefore, the goal of multiobjective optimization is to compute the Pareto-optimal set or, since this set is usually very large in practice, a good approximation of it. The quality of the approximation (or representation) of the Pareto-optimal set can be determined in several manners: cardinality, accuracy, representation error or cluster density.
The goal of this project is to develop efficient algorithms that select i) a subset of a point set that is close as possible from another point set; ii) a representative subset of a point set. In this work, it is assumed that the point sets contain only nondominated elements and the cardinality of the subset to be chosen is fixed a priori.
The first task is related to subset selection procedures within heuristic approaches to multiobjective optimization. The goal is to choose a subset of solutions at each iteration that is as close as possible from a reference set, for instance, the Pareto-optimal set. Some work is already done for the case of the epsilon-indicator as a measure of closeness between nondominated point sets (see Section 5 of article A. Ponte, L. Paquete, J.R. Figueira, On beam search for multicriteria combinatorial optimization problems). The goal is to implement the approach described in the article and extend it to more than two dimensions.
The second task is a topic of a bilateral research project RepSys - Representation Systems with Quality Guarantees for Multi-Objective Optimization Problems, with Kaiserslautern University of Technology and University of Wuppertal, in Germany. The goal is to develop algorithms that select the representative subset of the Pareto-optimal set that maximizes a given measure, such as epsilon-indicator, hypervolume, uniformity and coverage, or combination thereof. Some part of this work is already described for uniformity and coverage measures (L. Paquete, CM Fonseca, K. Klamroth, M. Stiglmayr, Concise representation of nondominated sets in discrete multicriteria optimization).
Note: This is a draft of the abstract, considering the goals of the thesis. Given that the thesis is far from over, an accurate abstract is still hard to write, and it will probably change for the final version of the thesis.