Convex envelopes of monomials of odd degree

Convex envelopes of nonconvex functions are widely used to calculate lower bounds to solutions of nonlinear programming problems (NLP), 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 ε 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.