## The Directed Search Method for Constrained Multi-objective Optimization Problems

### Erick Ignacio Mejía Estrada

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Abstract

Optimization is to find the best solution to a given problem. In different situations of real life, we can usually find problems where we have to solve several objectives simultaneously which are subject to several constraints. Such problems are known as constrained multi-objective optimization problems (MOPs). The mathematical model of a MOP can be stated as $F : Q \subset \Rn \rightarrow \Rk$, where $F = \{ f_1, \ldots, f_k \}$ and $f_i : Q \rightarrow \R$. The subset Q is defined, in the constrained case, by l inequality and m equality constraints. One characteristic of the MOPs is that, since the objectives are in conflict with each other, there is no unique solution, but rather a set of trade-off solutions. Nowadays, there exist several approaches to solve MOPs; nevertheless, there is no general method to find the solution of every kind of existing problems. One class of methods are the gradient-based techniques. In such cases, the gradient information is used to compute a descent direction, that is, a direction where (ideally) all the objective values are improved. On the other hand, in recent decades, researchers have focused their attention on continuation methods. Such methods search along the set of solutions if at least one solution is at hand. Recently, a novel idea has been proposed to guide the search process along a given direction $\alpha\in\Rk$ defined in objective space. Based on this, a descent and a continuation method were proposed. In both cases, it is necessary to solve a sub-problem which is formulated as an initial value problem (IVP). The solution of that problem is a point   $x^*$ whose image $F(x^*)$ is at the boundary of the image of the problem F(Q). Nevertheless, its characteristics have not been fully explored and exploited yet. Moreover, it is stated just for unconstrained MOPs. In this thesis project we state a predictor corrector (PC) method as a solution of the IVP. We focused our attention on finding quickly a solution and keeping the search direction. In order to make more efficient the optimization process, we propose some techniques to calculate step sizes for a suitable computation of the predictor point, enabling an efficient corrector step. We also adressed the case, in unconstrained MOPs, where a boundary point is reached but it is not optimal. Our proposal is a minimization problem to approximate the search direction $\alpha$. Relevant in this work is the extension of the descent and the continuation method to constrained MOPs. We state the movement toward constrained Pareto sets and, the movement along active constraints until an optimal point is reached, in case a dominated point is achieved.\newline. We present numerical results of the proposed techniques applied on some benchmark problems, with and without restrictions, and problems with more than two objectives..