PhD defense of Gaëtan SERRÉ

Gaëtan SERRÉ

Methodological advances and formalization in global optimization

Abstract

Global optimization algorithms are numerous and diverse, yet beneath their surface lie shared mathematical structures that can be identified and exploited. This thesis advances the field by recognizing these underlying structures, organizing algorithms into families, developing methodological advances rooted in this perspective, and establishing formal foundations through rigorous mathematical frameworks.

Among global optimization algorithms, two families stand out and deserve distinguishing: decision-based approaches and McKean-Vlasov-based approaches. These families share deep mathematical structures that enable reasoning and result transfer across algorithms. Within this family framework, this thesis develops two major methodological contributions: i), the introduction of Stein Boltzmann Sampling, a novel McKean-Vlasov-based algorithm; ii) a modular approach to adding common noise in McKean-Vlasov-based algorithm, improving exploration while preserving mean-field stochasticity. The common noise approach exemplifies the power of family reasoning: it is a single module applicable uniformly across an entire algorithmic family. This thesis also validates this family-centric perspective through a Python/C++ library that exploits shared structure to maximize code reuse and deploy plugins applicable to all algorithms within a family.

This family-centric reasoning naturally motivates investigating what unifies optimization algorithms at a larger scale. The second part develops an abstract mathematical framework unifying all optimization algorithms, including those from the two distinguished families, within a common language. This unified framework enables reasoning about all algorithms simultaneously. One crucial application of this framework is establishing rigorous foundations for global optimization research through formal methods. Specifically, this thesis develops contributions in L∀∃N that formalize key results on Markov kernels, and implements both the unifying framework and concrete optimization algorithms, building a practical formal library. This enables both rigorous reasoning about optimization algorithm theory and reusable formal developments, opening possibilities for mechanized verification and integration with emerging tools like large language models.

PhD surpervisors

• Nicolas VAYATIS
• Argyris KALOGERATOS

Jury members

• Gérard BIAU, Full professor, Sorbonne Université, Rapporteur
• Massimiliano PONTIL, Professor, Italian Institute of Technology, Rapporteur
• Gabriel PEYRE, Directeur de recherche, École Normale Supérieure, Examinateur
• Alain TROUVE, Professeure des universités, École Normale Supérieure Paris-Saclay, Examinateur
• Sylvie BOLDO, Directrice de recherche, Inria Saclay, Examinateur
• Anna KORBA, Professeure, ENSAE, Examinateur

Invited jury member

• Rémy Degenne, researcher, Inria Lill

Keywords

Global optimization, Frugal optimization, Formalization,

Publications