A Faster Algorithm for Quasi-convex Integer Polynomial Optimization

We present a faster exponential-time algorithm for integer optimization over quasi-convex polynomials. We study the minimization of a quasiconvex polynomial subject to s quasi-convex polynomial constraints and integrality constraints for all variables. The new algorithm is an improvement upon the best known algorithm due to Heinz (Journal of Complexity, 2005). A lower time complexity is reached through applying a stronger ellipsoid rounding method and applying a recent advancement in the shortest vector problem to give a smaller exponential-time complexity of a Lenstra-type algorithm. For the bounded case, our algorithm attains a time-complexity of s(rlMd) O(1) 2 2n log2(n)+O(n) when M is a bound on the number of monomials in each polynomial and r is the binary encoding length of a bound on the feasible region. In the general case, sl O(1) d O(n) 2 2n log2(n) . In each we assume d 2 is a bound on the total degree of the polynomials and l bounds the maximum binary encoding size of the input.

[1]  Michael J. Todd,et al.  On Khachiyan's algorithm for the computation of minimum-volume enclosing ellipsoids , 2007, Discret. Appl. Math..

[2]  M. Kochol Constructive approximation of a ball by polytopes , 1994 .

[3]  Leonid Khachiyan,et al.  Integer Optimization on Convex Semialgebraic Sets , 2000, Discret. Comput. Geom..

[4]  Ravi Kannan,et al.  Polynomial Algorithms for Computing the Smith and Hermite Normal Forms of an Integer Matrix , 1979, SIAM J. Comput..

[5]  Kurt M. Anstreicher,et al.  Ellipsoidal Approximations of Convex Sets Based on the Volumetric Barrier , 1999, Math. Oper. Res..

[6]  J. Miller Numerical Analysis , 1966, Nature.

[7]  Sebastian Heinz,et al.  Complexity of integer quasiconvex polynomial optimization , 2005, J. Complex..

[8]  Leonid Khachiyan,et al.  Rounding of Polytopes in the Real Number Model of Computation , 1996, Math. Oper. Res..

[9]  Phong Q. Nguyen,et al.  Sieve algorithms for the shortest vector problem are practical , 2008, J. Math. Cryptol..

[10]  Daniele Micciancio,et al.  Faster exponential time algorithms for the shortest vector problem , 2010, SODA '10.

[11]  Daniele Micciancio,et al.  A Deterministic Single Exponential Time Algorithm for Most Lattice Problems based on Voronoi Cell Computations ( Extended Abstract ) , 2009 .

[12]  László Lovász,et al.  Covering Minima and Lattice Point Free Convex Bodies , 1986, FSTTCS.

[13]  Alexander E. Litvak,et al.  The Flatness Theorem for Nonsymmetric Convex Bodies via the Local Theory of Banach Spaces , 1999, Math. Oper. Res..

[14]  Kurt M. Anstreicher,et al.  Improved Complexity for Maximum Volume Inscribed Ellipsoids , 2002, SIAM J. Optim..

[15]  Hendrik W. Lenstra,et al.  Integer Programming with a Fixed Number of Variables , 1983, Math. Oper. Res..

[16]  Piyush Kumar,et al.  Minimum-Volume Enclosing Ellipsoids and Core Sets , 2005 .

[17]  J. Douglas Faires,et al.  Numerical Analysis , 1981 .

[18]  George Labahn,et al.  Asymptotically fast computation of Hermite normal forms of integer matrices , 1996, ISSAC '96.

[19]  W. Banaszczyk New bounds in some transference theorems in the geometry of numbers , 1993 .

[20]  Shirley Dex,et al.  JR 旅客販売総合システム(マルス)における運用及び管理について , 1991 .

[21]  Yurii Nesterov,et al.  Rounding of convex sets and efficient gradient methods for linear programming problems , 2004, Optim. Methods Softw..

[22]  Daniele Micciancio The Shortest Vector in a Lattice is Hard to Approximate to within Some Constant , 2000, SIAM J. Comput..

[23]  B. Bank,et al.  Parametric Integer Optimization , 1988 .

[24]  Martin Henk Note on Shortest and Nearest Lattice Vectors , 1997, Inf. Process. Lett..

[25]  Ravi Kumar,et al.  A sieve algorithm for the shortest lattice vector problem , 2001, STOC '01.

[26]  Friedrich Eisenbrand,et al.  Integer Programming and Algorithmic Geometry of Numbers - A tutorial , 2010, 50 Years of Integer Programming.

[27]  Ravi Kannan,et al.  Minkowski's Convex Body Theorem and Integer Programming , 1987, Math. Oper. Res..

[28]  Ravi Kannan,et al.  Improved algorithms for integer programming and related lattice problems , 1983, STOC.

[29]  J. G. Pierce,et al.  Geometric Algorithms and Combinatorial Optimization , 2016 .

[30]  Martin Kochol,et al.  A note on approximation of a ball by polytopes , 2004, Discret. Optim..

[31]  Dimitris Bertsimas,et al.  Optimization over integers , 2005 .