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Positivity and Optimization: Beyond Polynomials

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Book cover Handbook on Semidefinite, Conic and Polynomial Optimization

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 166))

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Abstract

The recent progress optimization theory, due to a novel synthesis of real algebraic geometry and moment problems techniques, can naturally be reduced to positivity certificates for polynomial functions defined on basic semi-algebraic sets. There are however classical problems of applied mathematics which require exact positivity criteria for non-polynomial functions, such as splines, wavelets, periodic or almost periodic functions. While we do not lack fine analysis results referring to the positivity of such functions, traditionally stated in terms of Fourier-Laplace transforms type, the algebraic machinery of modern optimization theory based on polynomial algebra fails when applied to this more general context. A notorious example being the stability problem of differential equations with delays in the argument. In all these cases, the exact algebraic certificates must be complemented by approximation theory results. Without aiming at completeness, the present chapter offers a glimpse at a series of specific non-polynomial optimization problems, by identifying in every instance the specific results needed to run a robust algebraic relaxation scheme.

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Notes

  1. 1.

    With g : ℝn → ℝ and f : ℝ → ℝ, x↦(f ∘ g)(x) : = f(g(x)).

  2. 2.

    The authors wish to thank D. Plaumann for pointing out this fact.

  3. 3.

    The semidefinite relaxation was solved with the GloptiPoly software [17] dedicated to solving the generalized problem of moments as defined in [32].

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Acknowledgements

The authors thank Daniel Plaumann for a careful reading and constructive comments on an earlier version of the manuscript. Partially supported by the National Science Foundation DMS 10-01071, U.S.A.

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Authors and Affiliations

  1. LAAS-CNRS and Institute of Mathematics, University of Toulouse, 7 Avenue du Colonel Roche, 31077, Toulouse Cédex 4, France

    Jean B. Lasserre

  2. Department of Mathematics, University of California at Santa Barbara, Santa Barbara, CA, 93106-3080, USA

    Mihai Putinar

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  1. Jean B. Lasserre
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Correspondence to Jean B. Lasserre .

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Editors and Affiliations

  1. Department of Mathematics and Industrial, Ecole Polytechnique Montreal, Édouard-Montpetit Boulevard 2350, Montreal, QC, Canada, H3C 3A7

    Miguel F. Anjos

  2. LAAS-CNRS, Avenue du Colonel Roche 7, 31077, Toulouse Cedex 4, France

    Jean B. Lasserre

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Lasserre, J.B., Putinar, M. (2012). Positivity and Optimization: Beyond Polynomials. In: Anjos, M.F., Lasserre, J.B. (eds) Handbook on Semidefinite, Conic and Polynomial Optimization. International Series in Operations Research & Management Science, vol 166. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0769-0_14

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