Convex Envelopes of Monomials of Odd Degree

  • Authors:

  • Leo Liberti;Constantinos C. Pantelides

  • Affiliations:

  • Centre for Process Systems Engineering, Imperial College of Science, Technology and Medicine, London SW7 2BY, UK;Centre for Process Systems Engineering, Imperial College of Science, Technology and Medicine, London SW7 2BY, UKCorresponding author. e-mail: c.pantelides@ic.ac.uk

  • Venue:
  • Journal of Global Optimization
  • Year:
  • 2003

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Abstract

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, x2k+1, where k ∈ 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.