Browsing by Author "Custódio, A. L."
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- Direct multisearch for multiobjective optimizationPublication . Custódio, A. L.; Madeira, JFA; Vaz, A. I. F.; Vicente, L. N.In practical applications of optimization it is common to have several conflicting objective functions to optimize. Frequently, these functions are subject to noise or can be of black-box type, preventing the use of derivative-based techniques. We propose a novel multiobjective derivative-free methodology, calling it direct multisearch (DMS), which does not aggregate any of the objective functions. Our framework is inspired by the search/poll paradigm of direct-search methods of directional type and uses the concept of Pareto dominance to maintain a list of nondominated points (from which the new iterates or poll centers are chosen). The aim of our method is to generate as many points in the Pareto front as possible from the polling procedure itself, while keeping the whole framework general enough to accommodate other disseminating strategies, in particular, when using the (here also) optional search step. DMS generalizes to multiobjective optimization (MOO) all direct-search methods of directional type. We prove under the common assumptions used in direct search for single objective optimization that at least one limit point of the sequence of iterates generated by DMS lies in (a stationary form of) the Pareto front. However, extensive computational experience has shown that our methodology has an impressive capability of generating the whole Pareto front, even without using a search step. Two by-products of this paper are (i) the development of a collection of test problems for MOO and (ii) the extension of performance and data profiles to MOO, allowing a comparison of several solvers on a large set of test problems, in terms of their efficiency and robustness to determine Pareto fronts.
- GLODS: global and local optimization using direct searchPublication . Custódio, A. L.; Madeira, JFALocating and identifying points as global minimizers is, in general, a hard and time-consuming task. Difficulties increase in the impossibility of using the derivatives of the functions defining the problem. In this work, we propose a new class of methods suited for global derivative-free constrained optimization. Using direct search of directional type, the algorithm alternates between a search step, where potentially good regions are located, and a poll step where the previously located promising regions are explored. This exploitation is made through the launching of several instances of directional direct searches, one in each of the regions of interest. Differently from a simple multistart strategy, direct searches will merge when sufficiently close. The goal is to end with as many direct searches as the number of local minimizers, which would easily allow locating the global extreme value. We describe the algorithmic structure considered, present the corresponding convergence analysis and report numerical results, showing that the proposed method is competitive with currently commonly used global derivative-free optimization solvers.
- MultiGLODS: global and local multiobjective optimization using direct searchPublication . Custódio, A. L.; F. Aguillar Madeira, JoséThe optimization ofmultimodal functions is a challenging task, in particular when derivatives are not available for use. Recently, in a directional direct search framework, a clever multistart strategy was proposed for global derivative-free optimization of single objective functions. The goal of the current work is to generalize this approach to the computation of global Pareto fronts for multiobjective multimodal derivative-free optimization problems. The proposed algorithm alternates between initializing new searches, using a multistart strategy, and exploring promising subregions, resorting to directional direct search. Components of the objective function are not aggregated and new points are accepted using the concept of Pareto dominance. The initialized searches are not all conducted until the end, merging when they start to be close to each other. The convergence of the method is analyzed under the common assumptions of directional direct search. Numerical experiments show its ability to generate approximations to the different Pareto fronts of a given problem.