Bouchet, P.-Y., Audet, C., & Bourdin, L. (2024). Convergence towards a local minimum by direct search methods with a covering step. arXiv preprint arXiv:2401.07097v3.
This paper aims to enhance the convergence analysis of Direct Search Methods (DSMs) for optimization problems with potentially discontinuous objective functions by introducing a novel "covering step."
The authors propose the covering DSM (cDSM), which incorporates a covering step designed to ensure the density of evaluated trial points in a neighborhood of any refined point. They provide a theoretical analysis of the cDSM's convergence properties, proving that under specific assumptions, all refined points generated by the algorithm are local solutions to the optimization problem.
The covering step significantly strengthens the convergence analysis of DSMs, guaranteeing the local optimality of refined points under weaker assumptions than previous work. This enhancement makes cDSMs a powerful tool for optimizing potentially discontinuous objective functions.
This research significantly advances the theoretical understanding and practical applicability of DSMs for optimization problems involving discontinuous objective functions, which are common in various real-world applications.
The paper focuses on a specific set of assumptions regarding the objective function. Further research could explore the applicability of the covering step to broader classes of functions or investigate its performance with different covering oracle implementations.
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