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A Generalization of the Classical αBB Convex Underestimation via Diagonal and Nondiagonal Quadratic Terms

Abstract

The classical αBB method determines univariate quadratic perturbations that convexify twice continuously differentiable functions. This paper generalizes αBB to additionally consider nondiagonal elements in the perturbation Hessian matrix. These correspond to bilinear terms in the underestimators, where previously all nonlinear terms were separable quadratic terms. An interval extension of Gerschgorin’s circle theorem guarantees convexity of the underestimator. It is shown that underestimation parameters which are optimal, in the sense that the maximal underestimation error is minimized, can be obtained by solving a linear optimization model.

Theoretical results are presented regarding the instantiation of the nondiagonal underestimator that minimizes the maximum error. Two special cases are analyzed to convey an intuitive understanding of that optimally-selected convexifier. Illustrative examples that convey the practical advantage of these new αBB underestimators are presented.

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Acknowledgements

A.S. gratefully acknowledges financial support from the Center of Excellence in Optimization and Systems Engineering at Åbo Akademi University. C.A.F and R.M. are thankful for support from the National Science Foundation (CBET – 0827907). This material is based upon work supported by the National Science Foundation Graduate Research Fellowship to R.M. under Grant No. DGE-0646086.

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Skjäl, A., Westerlund, T., Misener, R. et al. A Generalization of the Classical αBB Convex Underestimation via Diagonal and Nondiagonal Quadratic Terms. J Optim Theory Appl 154, 462–490 (2012). https://doi.org/10.1007/s10957-012-0033-6

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Keywords

  • Global optimization
  • αBB
  • Convex relaxations
  • Nonconvex optimization