Convex envelopes of monomials of odd degree. (English) Zbl 1030.90117

Liberti, Leo; Pantelides, Constantinos C.

J. Glob. Optim. 25, No. 2, 157-168 (2003).

Summary: Convex envelopes of nonconvex functions are widely used to calculate lower bounds to solutions of nonlinear programming problems, particularly within the context of spatial branch-and-bound methods for global optimization. This paper proposes a nonlinear continuous and differentiable convex envelope for monomial terms of odd degree, \(x^{2k+1}\), where \(k\in \mathbb N\) and the range of \(x\) includes zero. We prove that this envelope is the tightest possible. We also derive a linear relaxation from the proposed envelope, and compare both the nonlinear and linear formulations with relaxations obtained using other approaches.

MSC:

90C29	Multi-objective and goal programming
90C57	Polyhedral combinatorics, branch-and-bound, branch-and-cut

Keywords:

nonconvex functions; branch-and-bound methods; linear relaxation; Convex relaxation; Global optimization