Thèse d'habilitation à diriger des recherches "Analysis of Comparison-based Stochastic Continuous Black-Box Optimization Algorithms". (Thèse d'habilitation à diriger des recherches "Analysis of Comparison-based Stochastic Continuous Black-Box Optimization Algorithms")

This manuscript presents a large part of my research since the end of my PhD. Most of my work is related to numerical (also referred to as continuous) optimization, at the exception of one contribution done during my postdoc in Zurich introducing a new stochastic algorithm to simulate chemical or biochemical systems [23]. The optimization algorithms at the core of my work are adaptive derivative-free stochastic (or randomized) optimization methods. The algorithms are tailored to tackle dificult numerical optimization problems in a so-called black-box context where the objective function to be optimized is seen as a black-box. For a given input solution, the black-box returns solely the objective function value but no gradient or higher order derivatives are assumed. The optimization algorithm can use the information returned by the black-box, i.e. the history of function values associated to the queried search points, but no other knowledge that could be within the black-box (parameters describing the class of functions the function belongs to, ...). This black-box context is very natural in industrial settings where the function to be optimized can be given by an executable file for which the source code is not provided. It is also natural in situations where the function is given by a large simulation code from which it is hard to extract any useful information for the optimization. This context is also called derivative-free optimization (DFO) in the mathematical optimization community. Well-known DFO methods are the Nelder-Mead algorithm [79, 77], pattern search methods [54, 90, 6] or more recently the NEW Unconstraint Optimization Algorithm (NEWUOA) developed by Powell [82, 81]. In this context, I have been focusing on DFO methods in the literal sense. However the methods my research is centered on have a large stochastic component and originate from the community of bio-inspired algorithms mainly composed of computer scientists and engineers. The methods were introduced at the end of the 70's. A parallel with Darwin's theory of the evolution of species based on blind variation and natural selection was recognized and served as source of inspiration for those methods. Nowadays this field of bio-inspired methods is referred to as evolutionary computation (EC) and a generic term for the methods is evolutionary algorithms. The probably most famous examples of bio-inspired methods are genetic algorithms (GAs). However today GAs are known to be not competitive for numerical optimization. Evolution Strategies (ES) introduced in the end of the 70's [83] have emerged as the main sub-branch of EC devoted to continuous optimization. One important feature of ES is that they are comparison-based algorithms. The present most advanced ES algorithm, the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) [50] is a variable metric method recognized as the state-of-the-art method for stochastic numerical optimization. It is used in many applications in industry and academy. Because of historical reasons, the developments and work on Evolution Strategies are mainly carried out in the EC field where practice and effectiveness is definitely as (or more) important as having a theorem proven about an algorithm. However ES algorithms are simply adaptive stochastic iterative methods and they need to be studied from a mathematical perspective as well as any other iterative method in optimization or other domain in order to understand the methods better and convince a broader class of people about their soundness. Questions like their convergence and speed of convergence central in optimization need to be addressed. My research is encompassed within this general context: I am particularly interested by the mathematical aspects of adaptive stochastic methods like ES (and of course CMA-ES) or more generally adaptive stochastic optimization algorithms. Evolution strategies have this attractive facet that while introduced in the bio-inspired and engineering context, they turn out to be methods with deep theoretical foundations related to invariance, information geometry, stochastic approximation and strongly connected to Markov chain Monte Carlo (MCMC) algorithms. Those foundations and connections are relatively new and to a small (for some topics) or large (for others) extent partly related to some of my contributions. They will be explained within the manuscript. I particularly care that the theory I am working on relates to practical algorithms or has an impact on (new) algorithm designs. I attempt to illustrate this within the manuscript. While optimization is the central theme of my research, I have been tackling various aspect of optimization. Although most of my work is devoted to single-objective optimization, I have also been working on multi-objective optimization where the goal is to optimize simultaneously several conflicting objectives and where instead of a single solution, a set of solutions, the so-called Pareto set composed of the best compromises is searched. In the field of single-objective optimization, I have been tackling diverse contexts like noisy optimization where for a given point in a search space we do not observe one deterministic value but a distribution of possible function values, large-scale optimization where one is interested in tackling problems of the order of 104 (medium large-scale) to 106 variables (large-scale) and to a smaller extent constrained optimization. In addition to investigating theoretical questions, I have been also working on designing new algorithms that calls for theory complemented with numerical simulations. Last I have tackled some applications mainly in the context of the PhD of Mohamed Jebalia with an application in chromatography and of the PhD of Zyed Bouzarkouna (PhD financed by the French Institute for petrol) on the placement of oil wells. Furthermore, a non neglect-able part of my research those past years has been devoted to benchmarking of algorithms. Benchmarking complements theory as it is difficult to assess theoretically the performance of algorithms on all typical functions one is interested. The main motivation has then been to improve the standards on how benchmarking is done. Those contributions were done along with the development of the Comparing COntinuous Optimizers platform (COCO). My work is articulated around three main complementary axis, namely theory / algorithm design and applications. An overview of the contributions presented within this habilitation organized along those axes is given in Figure 3.1.