Convex Underestimation of Twice Continuously Differentiable Functions by Piecewise Quadratic Perturbation: Spline αBB Underestimators

  • Authors:

  • Clifford A. Meyer;Christodoulos A. Floudas

  • Affiliations:

  • Department of Chemical Engineering, Princeton University, Princeton, USA 08544;Department of Chemical Engineering, Princeton University, Princeton, USA 08544

  • Venue:
  • Journal of Global Optimization
  • Year:
  • 2005

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Abstract

This paper describes the construction of convex underestimators for twice continuously differentiable functions over box domains through piecewise quadratic perturbation functions. A refinement of the classical 驴 BB convex underestimator, the underestimators derived through this approach may be significantly tighter than the classical 驴BB underestimator. The convex underestimator is the difference of the nonconvex function f and a smooth, piecewise quadratic, perturbation function, q. The convexity of the underestimator is guaranteed through an analysis of the eigenvalues of the Hessian of f over all subdomains of a partition of the original box domain. Smoothness properties of the piecewise quadratic perturbation function are derived in a manner analogous to that of spline construction.