Section: Scientific Foundations

Continuous Optimization

Participants : Ouassim Ait ElHara, Yohei Akimoto, Anne Auger, Zyed Bouzarkouna, Alexandre Chotard, Nikolaus Hansen, Ilya Loshchilov, Verena Heidrich-Meisner, Yann Ollivier, Marc Schoenauer, Michèle Sebag, Olivier Teytaud, Mouadh Yagoubi.

Our main expertise in continuous optimization is on stochastic search algorithms. We address theory, algorithm design and applications. The methods we investigate are adaptive techniques able to learn iteratively parameters of the distribution used to sample solutions. The Covariance Matrix Adaptation Evolution Strategy (CMA-ES) is nowadays one of the most powerful methods for derivative-free continuous optimization. We work on different variants of the CMA-ES to improve it in various contexts as described below. We have proven the convergence of simplified variants of the CMA-ES algorithm using the theory of stochastic approximation providing the first proofs of convergence on composite of twice continuously differentiable functions and used Markov chain analysis for analyzing the step-size adaptation rule of the CMA-ES algorithm.

New algorithms based on surrogates and for constrained optimization.

A new variant of CMA-ES to address constraint optimization has been designed [22] . In the Zyed Bouzarkouna's PhD, defended in April 2012, a CIFRE PhD in cooperation with IFP-EN (Institut Français du Pétrole - Energies Nouvelles), new variants of CMA-ES coupled with local-meta-models for expensive optimization have been proposed. They have been applied to well placement problem in oil industry [2] . In the context of his PhD thesis (to be defended in January 2013), Ilya Loshchilov has proposed different surrogate variants of CMA-ES based on ranking-SVM that preserve the invariance to monotonic transformation of the CMA-ES algorithm [51] . He has also explored new restart mechanisms for CMA-ES [47] .

Benchmarking.

We have continued our effort for improving standards in benchmarking and pursued the development of the COCO (COmparing Continuous Optimizers) platform. We have organized the ACM GECCO 2012 workshop on Black-Box-Optimization Benchmarking (see http://coco.gforge.inria.fr/doku.php?id=bbob-2012 ) and benchmarked different variants of the CMA-ES algorithms [27] [30] [29] [28] [48] [49] [50] . Our new starting ANR project NumBBO, centered on the COCO platform, aims at extending it for large-scale, expensive, constrained and multi-objective optimization.

Theoretical proofs of convergence.

We have defined and analyzed a variant of CMA-ES being able to prove that its covariance matrix converges to the inverse Hessian on convex-quadratic functions [18] . We have analyzed the convergence of continuous time trajectories associated to step-size adaptive Evolution Strategies on monotonic $C^2$-composite Functions and proved the local convergence towards local minima [19] . This work has been the starting point for analyzing convergence of step-size adaptive ESs using the framework of stochastic approximation (work about to be submitted). In the context of his thesis, Alexandre Chotard has analyzed the step-size adaptation algorithm of CMA-ES on linear function using the theory of Markov Chains [35] .

Multi-objective optimization.

Mouadh Yagoubi has completed his PhD [3] , a CIFRE cooperation with PSA (Peugeot-Citroen automotive industry). His work addressed the multi-disciplinary multi-objective optimization of a diesel engine, that led to propose some asynchronous parallelization of expensive objective functions (more than 2 days per evaluation for the complete 3D model) [57] .