Global optimization of Hölder functions

Abstract

Abstract. We propose a branch-and-bound framework for the global optimization of unconstrained HOlder functions. The general framework is used to derive two algorithms. The first one is a generalization of Piyavskii's algorithm for univariate Lipschitz functions. The second algorithm, using a piecewise constant upper-bounding function, is designed for multivariate HOlder functions. A proof of convergence is provided for both algorithms. Computational experience is reported on several test functions from the literature.

New Underestimator for Multivariate Global Optimization With Box Constraints

“…The solution found by our method is (1.7936, 1.3772) and f (1.7936, 1.3772) = 0.16905, with practically the same precision as in [3], the number of function…”

“…with practically the same precision as in [3], the number of function evaluations is 889 which is less than that of [3] which is equal to 1013.…”

Index information algorithm with local tuning for solving multidimensional global optimization problems with multiextremal constraints

“…On the one hand, the information global optimization algorithm with local tuning proposed in [31] for solving the problem (7) and, consequently, the problem (8)- (9). On the other hand, the index algorithm with local tuning proposed in [35] for solving the one-dimensional problem…”

“…It uses Peano type space-filling curves (see [2,5,26,38,42] for examples of usage of space-filling curves in mathematical programming) to reduce the original Lipschitz multi-dimensional problem to a Hölder univariate one (a comprehensive presentation of this approach can be found in [42]). Global optimization of Hölder functions (see [9,19,45]) has given new tools for solving the reduced one-dimensional problem. Peano curves avoid constructions of support (or auxiliary) functions usually used in the multi-dimensional Lipschitz optimization (see, for example, [15,17,18,25,27,42] and references given therein).…”

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“…They developed methods generating sequences converging to the optimum. Other authors, Gourdin E. Jaumard B. and Ellaia R. [11], Lera D. and Sergeyev Ya. D. [12], Rahal M. and Ziadi A.…”