Convexification, concavification and monotonization in global optimization

Convexification, concavification and monotonization in global optimization

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Article ID: iaor20023009
Country: Netherlands
Volume: 105
Issue: 1
Start Page Number: 213
End Page Number: 226
Publication Date: Jul 2001
Journal: Annals of Operations Research
Authors: , , ,
Keywords: programming: convex
Abstract:

We show in this paper that via certain convexification, concavification and monotonization schemes a nonconvex optimization problem over a simplex can be always converted into an equivalent better-structured nonconvex optimization problem, e.g., a concave optimization problem or a programming problem applied to the difference of two convex functions (a D.C. problem), thus facilitating the search of a global optimum by using the existing methods in concave minimization and D.C. programming. We first prove that a monotone optimization problem (with a monotone objective function and monotone constraints) can be transformed into a concave minimization problem over a convex set or a D.C. programming problem via pth power transformation. We then prove that a class of nonconvex minimization problems can be always reduced to a monotone optimization problem, thus a concave minimization problem or a D.C. programming problem.

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