On Dealing with Minima at the Border of a Simplicial Feasible Area in Simplicial Branch and Bound

• Published in: Journal of optimization theory and applications Vol. 203; no. 2; pp. 1880 - 1909
• Main Authors: G.-Tóth, Boglárka, Hendrix, Eligius M. T., Casado, Leocadio G., Messine, Frédéric
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-11-2024
• Subjects: Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Engineering; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Theory of Computation; Simplex; Directional derivative; Branch and bound; Monotonic; Face
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-024-02480-9

We consider a simplicial branch and bound Global Optimization algorithm, where the search region is a simplex. Apart from using longest edge bisection, a simplicial partition set can be reduced due to monotonicity of the objective function. If there is a direction in which the objective function is monotone over a simplex, depending on whether the facets that may contain the minimum are at the border of the search region, we can remove the simplex completely, or reduce it to some of its border facets. Our research question deals with finding monotone directions and labeling facets of a simplex as border after longest edge bisection and reduction due to monotonicity. Experimental results are shown over a set of global optimization problems where the feasible set is defined as a simplex, and a global minimum point is located at a face of the simplicial feasible area.