Granular sieving algorithm for selecting best n$$ n $$ parameters

摘要

A common type problem of optimization is to find simultaneously n$$ n $$ parameters that globally minimize an objective function of n$$ n $$ variables. Such problems are seen in signal and image processing and in various applications of mathematical analysis of several complex variables and Clifford algebras. Objective functions are usually assumed to be Lipschitzian with maybe unknown Lipschitz constants. A number of methods have been established to discard the sets called “bad sets” in a partition that is impossible to contain any optimal point, as well as to treat the unknown Lipschitz bound problem along with the algorithm. In the present paper, a simple criterion of eliminating bad sets is proposed for the first time. The elimination method leads to a concise and rigorous proof of convergence. The algorithm, on the range space side, converges to the global minimum with an exponential rate, while on the domain space side, converges with equal accuracy to the set of all the global minimizers. To treat the unknown Lipschitz constant dilemma, we propose a practical pseudo‐Lipshitz bound process. The methodology is of fundamental nature with straightforward mathematical formulation applicable to multivariate objective functions defined on any compactly connected manifolds in higher dimensions. The method is tested against an extensive number of benchmark functions in the literature. The experimental results exhibit considerable effectiveness and applicability of the algorithm.