Journal ArticleDOI
Outer Approximation by Polyhedral Convex Sets
TLDR
In this article, a general theorem on convergence is presented and new classes of outer approximation methods using polyhedral convex sets are derived, related to the cut map-separator theory of Eaves and Zangwill.Abstract:
This paper deals with outer approximation methods for solving possibly multiextremal global optimization problems. A general theorem on convergence is presented and new classes of outer approximation methods using polyhedral convex sets are derived. The underlying theory is then related to the cut map-separator theory of Eaves and Zangwill. Two constraint dropping strategies are deduced.read more
Citations
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Journal ArticleDOI
DC programming: overview
Reiner Horst,Nguyen Van Thoai +1 more
TL;DR: The purpose of this overview is to discuss main theoretical results, some applications, and solution methods for this interesting and important class of programming problems.
Book ChapterDOI
Concave Minimization: Theory, Applications and Algorithms
TL;DR: The purpose of this chapter is to present the essential elements of the theory, applications, and solution algorithms of concave minimization, including three fundamental classes of solution approaches that use deterministic (rather than stochastic) methods.
Journal ArticleDOI
Linear multiplicative programming
Hiroshi Konno,Takahito Kuno +1 more
TL;DR: It is shown that LMP can be solved efficiently by the combination of the parametric simplex method and any standard convex minimization procedure, and can be extended to a convex multiplicative programming problem (CMP), which minimizes the product of two convex functions under convex constraints.
Journal ArticleDOI
Global minimization of a generalized convex multiplicative function
TL;DR: An algorithm for generalized convex multiplicative programming problems, a special class of nonconvex minimization problems in which the objective function is expressed as a sum ofp products of two convex functions, is discussed.
Journal ArticleDOI
On solving a D.C. programming problem by a sequence of linear programs
TL;DR: 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.
References
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Journal ArticleDOI
Newton's method for convex programming and Tchebycheff approximation
E. W. Cheney,A. A. Goldstein +1 more
TL;DR: The rationale of Newton's method is exploited here in order to develop effective algorithms for solving the following general problem: given a convex continuous function F defined on a closed convex subset K of E,~, obtain a point x of K such that F(x)_<_F(y) for all y in K.
Journal ArticleDOI
A Successive Underestimation Method for Concave Minimization Problems
James E. Falk,Karla R. Hoffman +1 more
TL;DR: A new method designed to globally minimize concave functions over linear polyhedra is described, and an example problem is solved, and computational considerations are discussed.
Journal ArticleDOI
A method for globally minimizing concave functions over convex sets
TL;DR: A method is described for globally minimizing concave functions over convex sets whose defining constraints may be nonlinear that allows the objective function to be lower semicontinuous and nonseparable, and is guaranteed to converge to the global solution.