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# MOBES: A Multiobjective Evolution Strategy for Constrained Optimization Problems

Authors:
• Environmental sensing and Driver Monitoring

## Abstract

. In this paper a new MultiOBjective Evolution Strategy (MOBES) for solving multiobjective optimization problems subject to linear and nonlinear constraints is presented. MOBES is based on the new concept of C-, F - and N -fitness, which allows systematically to handle constraints and (in)feasible individuals. The existence of niche infeasible individuals in every population enables to explore new areas of the feasible region and new feasible pareto-optimal solutions. Moreover, MOBES proposed a new selection algorithm for searching, maintaining a set of feasible pareto-optimal solutions in every generation. The performance of the MOBES can be successfully evaluated on two selected test problems. 1 Introduction 1.1 Multiobjective optimization problem The general multiobjective optimization problem with linear and nonlinear constraints can be formally stated as below: min x f(x) = min x (f 1 (x); f 2 (x); Delta Delta Delta ; fN (x)) where x = (x 1 ; x 2 ; Delta Delta...
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... For SOO, this corresponds to a single 1 = 1, but for MOO, this corresponds to a random selection of 's, and we obtain as per Eq. (8). With probability, we sample the search space using the combination of aleatoric and epistemic uncertainties, 1 · + 2 ·ˆ, to find a point where the performance estimate is the most uncertain (line 9). ...
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Researchers constantly strive to explore larger and more complex search spaces in various scientific studies and physical experiments. However, such investigations often involve sophisticated simulators or time-consuming experiments that make exploring and observing new design samples challenging. Previous works that target such applications are typically sample-inefficient and restricted to vector search spaces. To address these limitations, this work proposes a constrained multi-objective optimization (MOO) framework, called BREATHE, that searches not only traditional vector-based design spaces but also graph-based design spaces to obtain best-performing graphs. It leverages second-order gradients and actively trains a heteroscedastic surrogate model for sample-efficient optimization. In a single-objective vector optimization application, it leads to 64.1% higher performance than the next-best baseline, random forest regression. In graph-based search, BREATHE outperforms the next-best baseline, i.e., a graphical version of Gaussian-process-based Bayesian optimization, with up to 64.9% higher performance. In a MOO task, it achieves up to 21.9$\times$ higher hypervolume than the state-of-the-art method, multi-objective Bayesian optimization (MOBOpt). BREATHE also outperforms the baseline methods on most standard MOO benchmark applications.
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... BR1, BR2, BR3 and BR4 are real-world constraint problems. Also, BR6 (Binh and Korn 1997) is a constraint problem. Table 5 specifies the characteristics of these problems. ...
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... .} and ζ ∈ (0, 1). (45) Similarly, from line 6 of Algorithm 2, we have λ = ζ l |λ min | for some l ∈ {2, 3, 4, . . .} and ζ ∈ (0, 1). ...
... In the weight scalarization method, all of the objective functions are consolidated into a single function that appears as a linear function. Many studies have adopted the scalarization approach to optimize different mathematical functions such as a quadratic functions [38] and logarithmic function [39]. By employing the scalarization approach to minimise (14) and (15) under (16) constraint, the following expression is formulated ...
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