On Solving a D.C. Programming Problem by a Sequence of Linear Programs R. HORST’, T.Q. PHONG’, Ng. V. THOAI’, and J. de VRIES’ ‘Fachbereich ZV - Mathematik, Universitiit Trier, Trier, West Germany; ‘Institute of Technology, Danang, Vietnam; ‘Institute of Mathematics, Hanoi, Vietnam (Received: 28 May 1990; accepted: 4 December 1990) Abstract. We are dealing with a numerical method for solving the problem of minimizing a difference of two convex functions (a d.c. function) over a closed convex set in R”. This algorithm combines a new prismatic branch and bound technique with polyhedral outer approximation in such a way that only linear programming problems have to be solved. Key words. Nonlinear programming, global optimization, d.c. programming, branch and bound, outer approximation, prismatic partition 1. Introduction In this paper we consider the multiextremal global optimization problem glob mint f(x) - g(4) s.t. hi(x)<0 (j=l,. . . ,J) 07 where f, g, hj (j=l,. . . , J) are finite convex functions on IR”. Problem (P) is frequently called a d.c. optimization problem, where d.c. is an abbreviation for the difference of two convex functions. The formulation of (P) indicates that we are interested in finding a global minimum of the objective function (f(x) - g(x)) over the feasible set ‘D:={XER”:hj(x)sO (j=l,..,J)}. We assume that D is compact (which implies of course that a global solution of (P) exists whenever D is nonempty), and that a feasible point is known in advance. Problem (P) is of interest from a practical as well as from a theoretical viewpoint. From the theoretical viewpoint we should like to mention that the class of d.c. functions, i.e., functions that can be represented as difference of two convex functions, enjoys a remarkable stability with respect to operations fre- quently encountered in optimization. For example, the class of d.c. functions is closed under operations such as sum, multiplication, multiplication with a scalar, forming maximum and minimum of a finite number of functions, etc. Moreover, we known that every locally d.c. function, i.e., every function that is d.c. in a Parts of this research were accomplished while the third author was visiting the University of Trier, Germany, as a fellow of the Alexander von Humboldt foundation. Journal of Global Optimization 1: 183-203, 1991 @ 1991 Kluwer Academic Publishers. Printed in the Netherlands.