New methods for calculating $$\alpha $$ BB-type underestimators

Anders Skjäl and Tapio Westerlund ()

Journal of Global Optimization, 2014, vol. 58, issue 3, 427 pages

Abstract: Most branch-and-bound algorithms in global optimization depend on convex underestimators to calculate lower bounds of a minimization objective function. The $$\alpha $$ BB methodology produces such underestimators for sufficiently smooth functions by analyzing interval Hessian approximations. Several methods to rigorously determine the $$\alpha $$ BB parameters have been proposed, varying in tightness and computational complexity. We present new polynomial-time methods and compare their properties to existing approaches. The new methods are based on classical eigenvalue bounds from linear algebra and a more recent result on interval matrices. We show how parameters can be optimized with respect to the average underestimation error, in addition to the maximum error commonly used in $$\alpha $$ BB methods. Numerical comparisons are made, based on test functions and a set of randomly generated interval Hessians. The paper shows the relative strengths of the methods, and proves exact results where one method dominates another. Copyright Springer Science+Business Media New York 2014

Keywords: Global optimization; $$\alpha $$ BB; Convex relaxation; Nonconvex optimization; 90C26 (search for similar items in EconPapers)
Date: 2014
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Citations: View citations in EconPapers (7)

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DOI: 10.1007/s10898-013-0057-y

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