Abstract
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 nondecreasing functions is no longer finite dimensional. In this paper we consider a specific nonconvex optimization problem min {f(x):h j (x)⩽g(x),j=1, ⋯,m, x∈X}, wheref(·),h j (·) andg(·) are finite convex functions andX 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.
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This paper is dedicated to Phil Wolfe on the occasion of his 65th birthday.
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Thach, P.T., Konno, H. A generalized Dantzig—Wolfe decomposition principle for a class of nonconvex programming problems. Mathematical Programming 62, 239–260 (1993). https://doi.org/10.1007/BF01585169
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DOI: https://doi.org/10.1007/BF01585169