| Abstract [eng] |
Global optimization problems arise in practice whenever there is a need to select a collection of variables corresponding to the best value of some objective funcion, e.~g. the lowest price. In a typical "black box" situation, when an analytic expression of the function is unavailable, algorithms based on statistical or Lipschitz obejctive function models can be applied. Statistical models are especially useful when objective function evaluations are expensive, therefore the efficiency of the algorithms has to be increased in terms of the number of trials. To this end, two approaches combining the statistical global search with local search techniques were suggested and their efficiency solving difficult multimodal problems was experimentally demonstrated. The optimization efficiency is influenced by the selected statistical model as well. Model selection problem was investigated experimentally and respective guidlines were formulated based on a priori information about the objective function complexity. In order to ensure not only efficient, but also theoretically justified search for optimal solutions, theoretical investigation of two algorithms was performed. First, asymptotic properties of a simplicial statistical model were investigated, allowing to relate a heuristic simplex selection criterion to the probability of improvement. Second, an optimal algorithm was derived, justifying the use of a certain trisection procedure in a set of Lipchitz optimization algorithms. |
