Global optimization via α‐dense curves

Abstract

Purpose

To study constrained or unconstrained global optimization problems in a cube of R^d where d is a positive integer.

Design/methodology/approach

α‐dense curves are initially used to transform this problem into a global optimization problem of a single variable. The optimization of the one variable is then treated by means of the Legendre‐Fenchel Transform. This discrete convex envelope of the one variable function obtained previously, can then be computed.

Findings

Global optimization problems of this nature have already been extensively studied by the authors. In this paper they have coupled the Alienor method with Legendre‐Fenchel Tranform to compute a discrete convex envelope of the function to minimize. A fast algorithm was successfully used to do this.

Research limitations/implications

This approach to global optimization is based on α‐dense curves and numerical tests performed on a Pentium IV (1,700 MHz) computer used with Mathematica 4 software.

Practical implications

Gives the solutions illustrated in the numerous examples provided that show the practicality of the methodology.

Originality/value

A new approach based on extensive research into global optimization via α‐dense curves.

Citation

Benabidallah, A. and Cherruault, Y. (2005), "Global optimization via α‐dense curves", Kybernetes, Vol. 34 No. 7/8, pp. 999-1011. https://doi.org/10.1108/03684920510605821