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On the complexity of quasiconvex integer minimization problem

Author

Listed:
  • A. Yu. Chirkov

    (Lobachevsky State University of Nizhny Novgorod)

  • D. V. Gribanov

    (Lobachevsky State University of Nizhny Novgorod
    National Research University Higher School of Economics)

  • D. S. Malyshev

    (National Research University Higher School of Economics)

  • P. M. Pardalos

    (National Research University Higher School of Economics
    University of Florida)

  • S. I. Veselov

    (Lobachevsky State University of Nizhny Novgorod)

  • N. Yu. Zolotykh

    (Lobachevsky State University of Nizhny Novgorod)

Abstract

In this paper, we consider the class of quasiconvex functions and its proper subclass of conic functions. The integer minimization problem of these functions is considered, assuming that the optimized function is defined by the comparison oracle. We will show that there is no a polynomial algorithm on $$\log R$$ log R to optimize quasiconvex functions in the ball of radius R using only the comparison oracle. On the other hand, if the optimized function is conic, then we show that there is a polynomial on $$\log R$$ log R algorithm (the dimension is fixed). We also present an exponential on the dimension lower bound for the oracle complexity of the conic function integer optimization problem. Additionally, we give examples of known problems that can be polynomially reduced to the minimization problem of functions in our classes.

Suggested Citation

  • A. Yu. Chirkov & D. V. Gribanov & D. S. Malyshev & P. M. Pardalos & S. I. Veselov & N. Yu. Zolotykh, 2019. "On the complexity of quasiconvex integer minimization problem," Journal of Global Optimization, Springer, vol. 73(4), pages 761-788, April.
    • Handle: RePEc:spr:jglopt:v:73:y:2019:i:4:d:10.1007_s10898-018-0729-8
    DOI: 10.1007/s10898-018-0729-8
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    References listed on IDEAS

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    1. Ravi Kannan, 1987. "Minkowski's Convex Body Theorem and Integer Programming," Mathematics of Operations Research, INFORMS, vol. 12(3), pages 415-440, August.
    2. Wojciech Banaszczyk & Alexander E. Litvak & Alain Pajor & Stanislaw J. Szarek, 1999. "The Flatness Theorem for Nonsymmetric Convex Bodies via the Local Theory of Banach Spaces," Mathematics of Operations Research, INFORMS, vol. 24(3), pages 728-750, August.
    3. H. W. Lenstra, 1983. "Integer Programming with a Fixed Number of Variables," Mathematics of Operations Research, INFORMS, vol. 8(4), pages 538-548, November.
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