Optimal algorithms for global optimization in case of unknown Lipschitz constant

We consider the global optimization problem for d-variate Lipschitz functions which, in a certain sense, do not increase too slowly in a neighborhood of the global minimizer(s). On these functions, we apply optimization algorithms which use only function values. We propose two adaptive deterministic methods. The first one applies in a situation when the Lipschitz constant L is known. The second one applies if L is unknown. We show that for an optimal method, adaptiveness is necessary and that randomization (Monte-Carlo) yields no further advantage. Both algorithms presented have the optimal rate of convergence. Key words: Global optimization, Lipschitz functions, complexity, optimal rate of convergence MSC: primary 90C60, 90C56; secondary 68Q25, 26B35

• Type: Conference
• Year: 2004
• Where: DAGSTUHL
• Authors: Matthias U. Horn
• Keywords: DAGSTUHL 2004, DAGSTUHL 2007, Global Optimization, Global Optimization Problem, Lipschitz Functions