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Constrained Multiobjective Derivative-free Optimization
Publication . Silva, Everton José da; Custódio, Ana
The present work focus on multiobjective derivative-free optimization, proposing strate-
gies to address constrained problems.
For that, a direct multisearch method that combines a filter-based strategy with an
inexact restoration phase, DMS-FILTER-IR, was developed. In this approach, feasibility
is treated as an additional component of the objective function that must be minimized.
The inexact restoration step attempts to generate new feasible points, thereby prioritizing
feasibility, a requirement for the strong performance of any filter approach. Theoretical
results are provided, analyzing the different types of sequences of points generated by
the new algorithm, and numerical experiments on a set of nonlinearly constrained biob-
jective problems are reported, stating the good algorithmic performance of the proposed
approach.
The filter approach reformulates the original problem by aggregating the constraint
violations into an additional objective function component, thereby increasing the number
of objectives by one. From a theoretical point of view, DMS-FILTER-IR was developed for
continuous constrained multiobjective optimization with an arbitrary number of objectives.
However, when the original problem has three or more objectives, the increase of this
number caused by the use of the filter, originates a many-objective optimization problem.
Thus, strategies to address many-objective optimization problems are also investigated.
Based on reduction approaches, employing correlation or sketching techniques, we
propose a new variant of DMS, namely DMS-Reduction. This reduction method aims to
tackle large problems by decreasing both the number of objective function components
and the number of problem variables. Reducing the number of components of the
objective function to be optimized at each iteration, has the additional benefit of potentially
conducting to a reduction in the number of variables to be optimized, since there could
be the case that not all variables are related to the objective function components selected.
We detail the proposed algorithm and report a large set of numerical experiments that
demonstrate the potential of this approach in addressing many-objective optimization
problems.
A different way of addressing constraints is resorting to penalization techniques. We investigate the use of a logarithmic barrier technique, both in single and multiobjective
derivative-free optimization. For single-objective optimization, we propose the joint use
of a mixed penalty-logarithmic barrier method and direct search. A merit function is
employed in which the set of inequality constraints is divided into two groups: one is
treated with a logarithmic barrier approach, while the other, together with the equality
constraints, is addressed using a penalization term. This strategy is incorporated into a
generalized pattern search method, enabling the effective handling of general nonlinear
constraints. Convergence to KKT-stationary points is established under continuous differ-
entiability assumptions, without requiring any form of convexity. Numerical experiments
demonstrate the robustness, efficiency, and overall effectiveness of the proposed method,
when compared with state-of-the-art solvers. The logarithmic barrier approach is then ex-
tended to the multiobjective setting via LOG-DMS, demonstrating improved exploration
of the Pareto front under inequality constraints. Comparative experiments indicate that
LOG-DMS outperforms traditional DMS and DMS-FILTER-IR in terms of hypervolume
metrics.
Overall, the contributions of this thesis not only advance the theoretical understanding
of constrained derivative-free optimization methods but also provide practical, competitive
algorithms for solving optimization problems.
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Entidade financiadora
Fundação para a Ciência e a Tecnologia
Programa de financiamento
Concurso para Financiamento de Projetos de Investigação Científica e Desenvolvimento Tecnológico em Todos os Domínios Científicos - 2017
Número da atribuição
PTDC/MAT-APL/28400/2017