Global optimization via α-dense curves

Purpose - To study constrained or unconstrained global optimization problems in a cube of R-d where d is a positive integer. Design/methodology/approach - alpha-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 a-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 a-dense curves.