A GENERALIZED DANTZIG-WOLFE DECOMPOSITION PRINCIPLE FOR A CLASS OF NONCONVEX PROGRAMMING-PROBLEMS

Since Dantzig-Wolfe's pioneering contribution, the decomposition approach using a pricing mechanism has been developed for a wide class of mathematical programs. For convex programs a linear space of Lagrangean multipliers is enough to define price functions. For general mathematical programs the price functions could be defined by using a subclass of nondecreasing functions. However the space of non-decreasing functions is no longer finite dimensional. In this paper we consider a specific nonconvex optimization problem min{f(x):h(j)(x) less-than-or-equal-to g(x), j = 1,. . . , m, x is-an-element-of x}, where f(.), h(j)(.) and g(.) are finite convex functions and X is a closed convex set. We generalize optimal price functions for this problem in such a way that the parameters of generalized price functions are defined in a finite dimensional space. Combining convex duality and a nonconvex duality we can develop a decomposition method to find a globally optimal solution.