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Integer Conic Function Minimization Based on the Comparison Oracle

  • Dmitriy V. GribanovEmail author
  • Dmitriy S. Malyshev
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11548)

Abstract

Let \(f : \mathbb {R}^n \rightarrow \mathbb {R}\) be a conic function and \(x_0 \in \mathbb {R}^n\). In this note, we show that the shallow separation oracle for the set \(K = \{x \in \mathbb {R}^n : f(x) \le f(x_0)\}\) can be polynomially reduced to the comparison oracle of the function f. Combining these results with known results of D. Dadush et al., we give an algorithm with \((O(n))^n \log R\) calls to the comparison oracle for checking the non-emptiness of the set \(K \cap \mathbb {Z}^n\), where K is included to the Euclidean ball of a radius R. Additionally, we give a randomized algorithm with the expected oracle complexity \((O(n))^n \log R\) for the problem to find an integral vector that minimizes values of f on an Euclidean ball of a radius R. It is known that the classes of convex, strictly quasiconvex functions, and quasiconvex polynomials are included into the class of conic functions. Since any system of conic functions can be represented by a single conic function, the last facts give us an opportunity to check the feasibility of any system of convex, strictly quasiconvex functions, and quasiconvex polynomials by an algorithm with \((O(n))^n \log R\) calls to the comparison oracle of the functions. It is also possible to solve a constraint minimization problem with the considered classes of functions by a randomized algorithm with \((O(n))^n \log R\) expected oracle calls.

Keywords

Nonlinear integer programming Conic function Convex function Quasiconvex function Comparison oracle Separation oracle Membership oracle Convex set Integral lattice 

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Lobachevsky State University of Nizhny NovgorodNizhny NovgorodRussian Federation
  2. 2.National Research University Higher School of EconomicsNizhny NovgorodRussian Federation

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