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Modern Methods of Optimization pp 1–23Cite as

Computability and Complexity of Polynomial Optimization Problems

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Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 378))

Abstract

The paper is devoted to such general questions as the computability, decidability and complexity of polynomial optimization problems even for integer variables. The first part relates polynomial optimization to Tarski algebra, algorithmical semialgebraic geometry and Matija, sevich’s result on diophantine sets. The second part deals with the particular case of integer polynomial optimization, where all polynomials involved are quasiconvex with integer coefficients. In particular, we show that the polynomial inequality system f 1 ≤ 0,..., f s ≤ 0 admits an integer solution if and only if it admits such a solution in a ball B(0, R) with log R = (sd)0 (n) ℓ, where d ≥ 2 is a degree bound for all polynomials f l,..., f s , ∈ ℤ[x 1,..., x n ] and ℓ is a bound for the binary length of all coefficients.

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Author information

Authors and Affiliations

  1. FB Mathematik, Humboldt-Universität, PSF 1297, D-0 1086, Berlin, Germany

    Bernd Bank & Reinhard Mandel

  2. Working Group Noaïota Fitchas, Universidad de Buenos Aires, Fac. de Ciencias Exactas, Depto de Matemática, Ciudad Universitaria, 1428, Buenos Aires, Argentina

    Joos Heintz, Teresa Krick & Pablo Solernó

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  1. Bernd Bank
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  2. Joos Heintz
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  3. Teresa Krick
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  4. Reinhard Mandel
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  5. Pablo Solernó
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Editors and Affiliations

  1. Department of Mathematics, Technische Hochschule Darmstadt, Schloßgartenstr. 7, W-6100, Darmstadt, Germany

    Werner Krabs

  2. Department of Mathematics, University of Bayreuth, Universitätsstr. 30, W-8580, Bayreuth, Germany

    Jochem Zowe

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© 1992 Springer-Verlag Berlin Heidelberg

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Bank, B., Heintz, J., Krick, T., Mandel, R., Solernó, P. (1992). Computability and Complexity of Polynomial Optimization Problems. In: Krabs, W., Zowe, J. (eds) Modern Methods of Optimization. Lecture Notes in Economics and Mathematical Systems, vol 378. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-02851-3_1

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  • DOI: https://doi.org/10.1007/978-3-662-02851-3_1

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