A Sequential Descent Method for Global Optimization

TL;DR

Numerical examples show that the global minimum can be sought by the proposed method of sequential descent, which has the descent property and the convergence is monotonic.

Abstract

In this paper, a sequential search method for finding the global minimum of an objective function is presented, The descent gradient search is repeated until the global minimum is obtained. The global minimum is located by a process of finding progressively better local minima. We determine the set of points of intersection between the curve of the function and the horizontal plane which contains the local minima previously found. Then, a point in this set with the greatest descent slope is chosen to be a initial point for a new descent gradient search. The method has the descent property and the convergence is monotonic. To demonstrate the effectiveness of the proposed sequential descent method, several non-convex multidimensional optimization problems are solved. Numerical examples show that the global minimum can be sought by the proposed method of sequential descent.

Figures (8)

• Figure 1: First local optimal solution $x^{1,*}$
• Figure 2: Intersection between the curve of the function and the horizontal plan containing $f(x^{1,*})$
• Figure 3: Second local optimal solution $x^{2,*}$
• Figure 4: Intersection between the curve of the function and the horizontal plan containing $f(x^{2,*})$
• Figure 5: First local optimal solution $x^{1,*}$

Theorems & Definitions (1)

• Definition 2.1