Local Tuning in Nested Scheme of Global Optimization

Numerical methods for global optimization of the multidimensional multiextremal functions in the framework of the approach oriented at dimensionality reduction by means of the nested optimization scheme are considered. This scheme reduces initial multidimensional problem to a set of univariate subproblems connected recursively. That enables to apply efficient univariate algorithms for solving the multidimensional problems. The nested optimization scheme served as the source of many methods for optimization of Lipschitzian function. However, in all of them there is the problem of estimating the Lipschitz constant as the parameter of the function optimized and, as a consequence, of tuning to it the optimization method. In the methods proposed earlier, as a rule, a global estimate (related to whole search domain) is used whereas local Lipschitz constants in some subdomains can differ significantly from the global constant. It can slow down the optimization process considerably. To overcome this drawback in the article the finer estimates of a priori unknown Lipschitz constants taking into account local properties of the objective function are considered and used in the nested optimization scheme. The results of numerical experiments presented demonstrate the advantages of methods with mixed (local and global) estimates of Lipschitz constants in comparison with the use of the global ones only.