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

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.


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