Analyse de sensibilité pour la réduction de dimension en optimisation sans dérivée
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- In today’s industry, the look for highest productivity at smallest costs naturally creates optimization problems. Precise models often create complex problems along with the need for derivative free optimization methods. Those are methods which can handle non-linear, non-differentiable or noisy objective functions. MADS algorithms are well-known black box optimization methods which solve this type of problem through calls to a black box, i.e. some kind of computer code. When little is known about the problem, the exploration of the search space requires a large number of black box evaluations. However, in the context of black box optimization, problems take the form of expensive-to-evaluate functions. The total number of evaluations is therefore very limited and this constitutes the main challenge of the field. When considering black box problems in large dimensions, the limited budget of evaluations is even more constraining. Standard black box algorithms need to be adapted, for example through dimension reduction scheme. STATS-MADS is a Mads-based algorithm which applies an analysis of variance to rank most influential input variables. Then the method alternates between optimizing the problem in a smaller dimension, where least influential variables were fixed, and the problem in its original large dimension. Most of the improvement of the objective value was done during the optimization in the small dimension. We propose a new MADS algorithm conceived to handle large-scale black box problems. This method applies a principal component analysis to identify most influential directions in the search space and is called PCA-MADS. Similarly to STATS-MADS, PCA-MADS alternates between an optimization in a smaller dimension, where the input can only evolve in the few most influential directions, and a poll in the large dimension. However, its structure allows to skip the poll in the large dimension as long as the optimization in the smaller dimension generates new improving solutions. A MATLAB implementation of the PCA-MADS method, based on the LTMADS instance was run on problems of up to 1500 variables. Its performances are compared to other derivative free methods such as CMA-ES, MADS and mainly STATS-MADS. The results of these tests clearly indicate the value of the approach developed for PCA-MADS.
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Vanden Bulcke, R. (2020). Analyse de sensibilité pour la réduction de dimension en optimisation sans dérivée. https://dial-mem.test.bib.ucl.ac.be/handle/2078.2/24355