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Convex underestimation for posynomial functions of positive variables. (English) Zbl 1152.90610
Summary: The approximation of the convex envelope of nonconvex functions is an essential part in deterministic global optimization techniques [C.A. Floudas, Deterministic Global Optimization: Theory, Methods and Application, Kluwer, Boston (2000)]. Current convex underestimation algorithms for multilinear terms, based on arithmetic intervals or recursive arithmetic intervals [A.S.C. Hamed, Calculation of bounds on variables and underestimating convex functions for nonconvex functions, PhD thesis, The George Washington University (1991); C.D. Maranas and C.A. Floudas, J. Glob. Optim. 7, No.2, 143–182 (1995; Zbl 0841.90115); H.S. Ryoo and N.V. Sahinidis, J. Glob. Optim. 19, No.4, 403–424 (2001; Zbl 0982.90054)], introduce a large number of linear cuts. C.A. Meyer and C.A. Floudas [in: Floudas, Christodoulos A. (ed.) et al., Frontiers in global optimization. Boston, MA: Kluwer Academic Publishers. Nonconvex Optim. Appl. 74, 327–352 (2004; Zbl 1176.90469); J. Glob. Optim. 29; No.2; 125–155, (2004; Zbl 1085.90047)] introduced the complete set of explicit facets for the convex and concave envelopes of trilinear monomials with general bounds. This study proposes a novel method to underestimate posynomial functions of strictly positive variables.

MSC:
90C30 Nonlinear programming
90C25 Convex programming
Software:
BARON
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