On the use of polynomial models in multiobjective directional direct search
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On the use of polynomial models in multiobjective directional direct search C. P. Brás1 · A. L. Custódio1 Received: 2 August 2019 / Accepted: 29 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Polynomial interpolation or regression models are an important tool in Derivativefree Optimization, acting as surrogates of the real function. In this work, we propose the use of these models in the multiobjective framework of directional direct search, namely the one of Direct Multisearch. Previously evaluated points are used to build quadratic polynomial models, which are minimized in an attempt of generating nondominated points of the true function, defining a search step for the algorithm. Numerical results state the competitiveness of the proposed approach. Keywords Multiobjective optimization · Derivative-free optimization · Direct search methods · Quadratic polynomial interpolation and regression · Minimum Frobenius norm models Mathematics Subject Classification 90C29 · 90C56 · 65D05 · 90C30
1 Introduction Multiobjective optimization is a common topic in practical applications, when several objectives need to be optimized and are conflicting among each other [7, 34]. Applications appear on different domains such as engineering, finance, or medicine [1, 8, 28, 35, 37]. In this work we address the Multiobjective Derivative-free Optimization problem
* A. L. Custódio [email protected] C. P. Brás [email protected] 1
Department of Mathematics, FCT-UNL-CMA, Campus de Caparica, 2829‑516 Caparica, Portugal
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C. P. Brás, A. L. Custódio
min
( )⊤ F(x) ≡ f1 (x), … , fm (x)
s.t.
x ∈ Ω ⊆ ℝn ,
(1.1)
where m ≥ 2 , Ω ⊆ ℝn represents the feasible region and each fi ∶ Ω ⊆ ℝn → ℝ ∪ {+∞}, i = 1, 2, … , m denotes a component of the objective function, for which derivatives are not available, neither can be numerically approximated. Often the objective function is nonsmooth or the corresponding evaluation is expensive and/or unreliable, justifying the derivative-free approach. A comprehensive review of single objective derivative-free optimization methods can be found in [3, 15]. Several methods were already proposed for this class of problems. Some, like [6, 39], rely on the aggregation of the different components of the objective function, addressing the multiobjective derivative-free optimization problem through a series of single objective minimizations. In [6], the authors use a directional direct search method for solving the single objective optimization problems, whereas [39] resorts to a derivative-free trust-region approach. The works [13, 17, 33] follow a different strategy, making use of the concept of Pareto dominance, without considering any aggregation techniques. Directional direct search methods are generalized in [17] to multiobjective optimization, while [33] consists in a linesearch-based method and [13] can be regarded as a multiobjective version of implicit filtering. Being a directional direct search method, the algorithmic structure o
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