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English version: Journal of Applied and Industrial Mathematics, 2018, 12:3, 587–594 |
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Volume 25, No 3, 2018, P. 23-35 UDC 519.854
Keywords: quasiconvex function, oracle, integer lattice. DOI: 10.17377/daio.2018.25.585 Sergey I. Veselov 1 Received 6 July 2017 References[1] I. M. Vinogradov, Osnovy teorii chisel, Lan’, Moscow, 2009. Translated under the title Elements of Number Theory, Dover, Mineola, NY, 2016.[2] N. Yu. Zolotykh and A. Yu. Chirkov, Lower bound for complexity of minimization of quasi-convex function on integer lattice, Vestn. Nizhegorod. Univ. N. I. Lobachevskogo, No. 5 (2), 93–96, 2012. [3] A. G. Sukharev, A. V. Timokhov, and V. V. Fyodorov, Kurs metodov optimizatsii (A Course in Optimization Methods), Nauka, Moscow, 1986. [4] A. Yu. Chirkov, Minimization of quasi-convex function on two-dimensional integer lattice, Vestn. Nizhegorod. Univ. N. I. Lobachevskogo, Mat. Model. Optim. Upr., No. 1, 227–238, 2003. [5] M. Avriel and D. J. Wilde, Optimality proof for the symmetric Fibonacci search technique, Fibonacci Q., 4, No. 3, 265–269, 1966. [6] A. Basu and T. Oertel, Centerpoints: A link between optimization and convex geometry, SIAM J. Optim., 27, No. 2, 866–889, 2017. [7] S. Heinz, Complexity of integer quasiconvex polynomial optimization, J. Complexity, 21, No. 4, 543–556, 2005. [8] S. Heinz, Quasiconvex functions can be approximated by quasiconvex polynomials, ESAIM, Control Optim. Calc. Var., 14, No. 4, 795–801, 2008. [9] R. Hildebrand and M. Köppe, A new Lenstra-type algorithm for quasiconvex polynomial integer minimization with complexity $2^{O(n log n)}$, Discrete Optim., 10, No. 1, 69–84, 2013. [10] J. Kiefer, Sequential minimax search for a maximum, Proc. AMS, 4, No. 3, 502–506, 1953. [11] T. Oertel, Integer convex minimization in low dimensions, PhD Dissertation, ETH Zurich, 2014. [12] T. Oertel, C. Wagner, and R. Weismantel, Integer convex minimization by mixed integer linear optimization, Oper. Res. Lett., 42, No. 6, 424–428, 2014. [13] A. Schrijver, Theory of Linear and Integer Programming, John Wiley & Sons, Chichester, GB, 1998. [14] W. Sun and Y. Yuan, Optimization Theory and Methods: Nonlinear Programming, Springer, New York, 2006 (Springer Optim. Its Appl., Vol. 1). |
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