Convex underestimators of polynomials

Jean Lasserre () and Tung Thanh ()

Journal of Global Optimization, 2013, vol. 56, issue 1, 25 pages

Abstract: Convex underestimators of a polynomial on a box. Given a non convex polynomial $${f\in \mathbb{R}[{\rm x}]}$$ and a box $${{\rm B}\subset \mathbb{R}^n}$$ , we construct a sequence of convex polynomials $${(f_{dk})\subset \mathbb{R}[{\rm x}]}$$ , which converges in a strong sense to the “best” (convex and degree-d) polynomial underestimator $${f^{*}_{d}}$$ of f. Indeed, $${f^{*}_{d}}$$ minimizes the L 1 -norm $${\Vert f-g\Vert_1}$$ on B, over all convex degree-d polynomial underestimators g of f. On a sample of problems with non convex f, we then compare the lower bounds obtained by minimizing the convex underestimator of f computed as above and computed via the popular α BB method and some of its other refinements. In most of all examples we obtain significantly better results even with the smallest value of k. Copyright Springer Science+Business Media, LLC. 2013

Keywords: Convex underestimators; Polynomials; Semidefinite programming (search for similar items in EconPapers)
Date: 2013
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Citations: View citations in EconPapers (4)

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DOI: 10.1007/s10898-012-9974-4

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