New methods for calculating α BB-type underestimators

Anders Skjäl, Tapio Westerlund

    Research output: Contribution to journalArticleScientificpeer-review

    18 Citations (Scopus)


    Most branch-and-bound algorithms in global optimization depend on convex underestimators to calculate lower bounds of a minimization objective function. The BB methodology produces such underestimators for sufficiently smooth functions by analyzing interval Hessian approximations. Several methods to rigorously determine the 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 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.
    Original languageUndefined/Unknown
    Pages (from-to)411–427
    Number of pages17
    JournalJournal of Global Optimization
    Issue number3
    Publication statusPublished - 2014
    MoE publication typeA1 Journal article-refereed


    • alpha BB
    • Convex relaxation
    • Global optimization
    • Nonconvex optimization

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