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 D.C. programming 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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Li, D., Sun, X., Biswal, M. et al. Convexification, Concavification and Monotonization in Global Optimization. Annals of Operations Research 105, 213–226 (2001). https://doi.org/10.1023/A:1013313901854
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DOI: https://doi.org/10.1023/A:1013313901854