Evolutionary Algorithms (EAs) have received a lot of attention regarding their potential to solve complex optimization problems using problem-specific variation operators. A search directed by a population of candidate solutions is quite robust with respect to a moderate noise and multi-modality of the optimized function, in contrast to some classical optimization methods such as quasi-Newton methods. The main limitation of EAs, the large number of function evaluations required, prevents from using EAs on computationally expensive problems, where one evaluation takes much longer than 1 second.
The present thesis focuses on an evolutionary algorithm, Covariance Matrix Adaptation Evolution Strategy (CMA-ES), which has become a standard powerful tool for continuous black-box optimization. We present several state-of-the-art algorithms, derived from CMA-ES, for solving single- and multi-objective black-box optimization problems.
First, in order to deal with expensive optimization, we propose to use comparison-based surrogate (approximation) models of the optimized function, which do not exploit function values of candidate solutions, but only their quality-based ranking. The resulting self-adaptive surrogate-assisted CMA-ES represents a tight coupling of statistical machine learning and CMA-ES, where a surrogate model is build, taking advantage of the function topology given by the covariance matrix adapted by CMA-ES. This allows to preserve two key invariance properties of CMA-ES: invariance with respect to i). monotonous transformation of the function, and ii). orthogonal transformation of the search space. For multi-objective optimization we propose two mono-surrogate approaches: i). a mixed variant of One Class Support Vector Machine (SVM) for dominated points and Regression SVM for non-dominated points; ii). Ranking SVM for preference learning of candidate solutions in the multi-objective space. We further integrate these two approaches into multi-objective CMA-ES (MO-CMA-ES) and discuss aspects of surrogate-model exploitation.
Second, we introduce and discuss various algorithms, developed to understand, explore and expand frontiers of the Evolutionary Computation domain, and CMA-ES in particular. We introduce linear time Adaptive Coordinate Descent method for non-linear optimization, which inherits a CMA-like procedure of adaptation of an appropriate coordinate system without losing the initial simplicity of Coordinate Descent. For multi-modal optimization we propose to adaptively select the most suitable regime of restarts of CMA-ES and introduce corresponding alternative restart strategies. For multi-objective optimization we analyze case studies, where original parent selection procedures of MO-CMA-ES are inefficient, and introduce reward-based parent selection strategies, focused on a comparative success of generated solutions.
The performance of the HCMA algorithm
compared to 50 selected classical (BFGS/fminunc, fmincon, Nelder–Mead, NEWUOA, DIRECT, GLOBAL) and evolutionary (CMA-ES, PSO, DE, GA) algorithms on 24 different (uni-modal / multi-modal, separable / non-separable, moderate / ill-conditioned, with / without global structure) 20-dimensional continuous optimization problems of the Black-Box Optimization Benchmarking (BBOB) workshop 2009, 2010, 2012, 2013
. My other algorithms are excluded for the sake of clarity.
x-axis: the number of objective function calls/evaluations divided by problem dimension, log10 scale.
y-axis: the proportion of function-target pairs solved using a given number of objective function evaluations.
2nd Microsoft Research Prize for the Best Poster Presentation,
ISSPR: International Summer School on Pattern Recognition, 2011, Plymouth, UK.