What is it about?

A gradient-free deterministic method is developed to solve global optimization problems for Lipschitz continuous functions defined in arbitrary path-wise connected compact sets in Euclidean spaces. The method can be regarded as granular sieving with simultaneous analysis in both the domain and range of the objective function. With straightforward mathematical formulation that is uniformly applicable to both univariate and multivariate functions, all the global minimizers and the global minimum of the objective function are located through two decreasing sequences of compact sets in, respectively, the domain and range spaces. The algorithm is easy to implement and has moderate computational cost. Furthermore, the method is tested against extensive benchmark functions in the literature, and the results show remarkable effectiveness and applicability.

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Why is it important?

In many fields of science, economics, engineering and medicine, practical applications can be formulated as global optimization problems, such as various cost functions to be minimized. Global optimization algorithms are generally divided into the deterministic and the stochastic types. The differences lie on whether the probability method is involved. In the literature, stochastic methods offer satisfactory accuracy with acceptable efficiency than deterministic ones in many cases. Deterministic methods, on the other hand, provide mathematical guarantees on the precision of the solutions. Here, we introduce a practical and efficient deterministic method for general global optimization. It provides an exhaustive search method that is applicable to objective functions of all dimensions, and theoretically ensures that the global minimum together with the entire set of minimizers can be found. The method is tested against an extensive number of benchmark functions and compared with 12 other popular methods on the top 10 hardest functions listed in the test functions index. The results show remarkable effectiveness and applicability, and in many tested examples surpass the popular stochastic algorithms. We anticipate that the method can be practically used in many fields and will arouse attention and interest in optimization studies.

Perspectives

This article looks the deterministic global optimization problem in another angle, which is different from other related methods in the literature. It can be easily implemented like stochastic methods but with theoretically guaranteed global minimum and complete set of minimizers. This article could lead to the idea of deterministic evolutionary algorithms, which is a breakthrough in the field, as the current algorithms in this field are stochastic.

Liming Zhang
University of Macau

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This page is a summary of: Granular sieving algorithm for selecting best n$$ n $$ parameters, Mathematical Methods in the Applied Sciences, March 2022, Wiley, DOI: 10.1002/mma.8254.
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