Derivative-free optimization: worst-case complexity for line-search methods and a mixed penalty-barrier approach

This thesis investigates new advances in Derivative-Free Optimization (DFO), focusing on complexity analysis for line-search methods and proposing a novel mixed penalty-barrier approach for handling constraints in black-box optimization problems. Classical optimization approaches often rely on gradient information; however, in practical applications, derivatives may be unavailable, unreliable, or costly to compute. DFO addresses this gap by developing methods that depend solely on function evaluations, which are crucial for applications with noisy, expensive, or simulation-based objective functions. Chapter 2 presents a theoretical analysis of line-search based DFO algorithms, concentrating on their worst-case complexity bounds in both unconstrained and bound-constrained contexts. For unconstrained problems, a worst-case complexity bound is established that matches the iteration requirements of existing direct search methods, marking a significant theoretical contribution to the field. Additionally, for the bound-constrained case, a criticality measure is introduced to evaluate solution quality, with complexity bounds developed to ensure efficient progress. An active-set identification property is also demonstrated, showing that the method can recognize and exploit bounds that are active at the solution, enhancing computational efficiency for bound-constrained problems. Chapter 3 introduces a sequential mixed-penalty approach for solving general nonlinear constrained black-box optimization problems, where traditional derivative-based constraint-handling techniques are unsuitable. This approach combines two distinct penalty mechanisms: a logarithmic barrier for handling inequality constraints and an exterior penalty for equality constraints. This dual strategy leverages a line-search framework to enforce constraint satisfaction while accommodating variable bounds, demonstrating improved feasibility attainment in constrained black-box settings. Furthermore, a direct search variant of the algorithm is developed, incorporating the mixed penalty strategy to manage unrelaxable constraints effectively. Together, these contributions advance the field of DFO by providing rigorous complexity bounds for line-search methods and a robust mixed penalty-barrier framework for constrained optimization. Empirical tests validate the efficacy of these algorithms across a diverse range of constrained optimization problems, underscoring their applicability in real-world black-box settings.