Pareto Front Reconstruction of Multi-Objective Optimization Problems

The thesis is concerned with multi-objective optimization (MOO) problems under various constraint types. In particular, we consider the unconstrained, the box constrained, the general convex constrained and the cardinality constrained settings. As mathematical tool, MOO has received much attention over the years, being suitable both in operation research applications and in many real-world problems where contrasting goals have to be taken into account. In the dissertation, a general overview of the main MOO concepts is given, focusing on the notions typically employed in the gradient-based MOO approaches. In particular, we present a general formulation for the search direction problem and a general framework for single-point methodologies, i.e., approaches designed to generate a single solution to the problem at hand. The features of each of the two concepts are discussed and analyzed; finally, we show how they can be reduced to well-know schemes from the MOO literature. Then, we propose novel gradient-based methodologies aimed to reconstruct the Pareto front for the considered problem classes. In some of the mentioned settings it is the first attempt to define this type of approaches generating multiple solutions rather than one, while in the others the topic has been almost unexplored yet. However, in the MOO context, returning an approximation of the Pareto front rather than a single solution can be much more useful for the final user, so as they can choose among multiple tradeoffs of the objectives the one that is the most suitable for themselves. For each methodology, detailed descriptions of the algorithmic scheme and characteristic features are reported; moreover, each methodology is theoretically analyzed and its convergence properties are stated and proved. Finally, the proposed methods are numerically tested with wide benchmarks of test problems, comparing each of them with state-of-the-art approaches from the MOO literature. The results show the efficiency and the effectiveness of the proposals w.r.t. the competitors in diverse experimental settings.