Bit complexity for multi-homogeneous polynomial system solving - Application to polynomial minimization

Multi-homogeneous polynomial systems arise in many applications. We provide bit complexity estimates for solving them which, up to a few extra other factors, are quadratic in the number of solutions and linear in the height of the input system under some genericity assumptions. The assumptions essentially imply that the Jacobian matrix of the system under study has maximal rank at the solution set and that this solution set if finite. The algorithm is probabilistic and a probability analysis is provided. Next, we apply these results to the problem of optimizing a linear map on the real trace of an algebraic set. Under some genericity assumptions, we provide bit complexity estimates for solving this polynomial minimization problem.

[1]  Marc Giusti,et al.  A Gröbner Free Alternative for Polynomial System Solving , 2001, J. Complex..

[2]  Daniel Bienstock,et al.  Polynomial Solvability of Variants of the Trust-Region Subproblem , 2014, SODA.

[3]  Simone Naldi.  Solving Rank-Constrained Semidefinite Programs in Exact Arithmetic , 2016, ISSAC.

[4]  François Le Gall,et al.  Powers of tensors and fast matrix multiplication , 2014, ISSAC.

[5]  Ya-Xiang Yuan,et al.  On a subproblem of trust region algorithms for constrained optimization , 1990, Math. Program..

[6]  M. Laurent.  Sums of Squares, Moment Matrices and Optimization Over Polynomials , 2009 .

[7]  Dima Grigoriev,et al.  Polynomial-time computing over quadratic maps i: sampling in real algebraic sets , 2004, computational complexity.

[8]  Ariel Waissbein,et al.  Deformation Techniques for Sparse Systems , 2006, Found. Comput. Math..

[9]  Jean B. Lasserre,et al.  Global Optimization with Polynomials and the Problem of Moments , 2000, SIAM J. Optim..

[10]  Angelos Mantzaflaris,et al.  On the Bit Complexity of Solving Bilinear Polynomial Systems , 2016, ISSAC.

[11]  B. Bank,et al.  Polar varieties and efficient real elimination , 2000 .

[12]  I. Shafarevich.  Basic algebraic geometry , 1974 .

[13]  Journal für die reine und angewandte Mathematik , 1893 .

[14]  Juan Sabia,et al.  Computing multihomogeneous resultants using straight-line programs , 2007, J. Symb. Comput..

[15]  Pablo A. Parrilo,et al.  Semidefinite programming relaxations for semialgebraic problems , 2003, Math. Program..

[16]  Sally Morrison.  The Differential Ideal [P]: Minfty , 1999, J. Symb. Comput..

[17]  Lihong Zhi,et al.  Computing rational solutions of linear matrix inequalities , 2013, ISSAC '13.

[18]  Mohab Safey El Din,et al.  Probabilistic Algorithm for Polynomial Optimization over a Real Algebraic Set , 2013, SIAM J. Optim..

[19]  J. E. Morais,et al.  When Polynomial Equation Systems Can Be "Solved" Fast? , 1995, AAECC.

[20]  Fabrice Rouillier,et al.  Solving Zero-Dimensional Systems Through the Rational Univariate Representation , 1999, Applicable Algebra in Engineering, Communication and Computing.

[21]  Marc Moreno Maza,et al.  Lifting techniques for triangular decompositions , 2005, ISSAC.

[22]  Pierre-Jean Spaenlehauer,et al.  On the Complexity of Computing Critical Points with Gröbner Bases , 2013, SIAM J. Optim..

[23]  Éric Schost,et al.  Polar varieties and computation of one point in each connected component of a smooth real algebraic set , 2003, ISSAC '03.

[24]  Angelos Mantzaflaris,et al.  Multihomogeneous resultant formulae for systems with scaled support , 2009, ISSAC '09.

[25]  Bruno Grenet,et al.  On the complexity of the multivariate resultant , 2013, J. Complex..

[26]  Marc Giusti,et al.  Le rôle des structures de données dans les problèmes d'élimination , 1997 .

[27]  Ya-Xiang Yuan,et al.  Optimality Conditions for the Minimization of a Quadratic with Two Quadratic Constraints , 1997, SIAM J. Optim..

[28]  Marc Giusti,et al.  Intrinsic complexity estimates in polynomial optimization , 2013, J. Complex..

[29]  A. Morgan,et al.  A homotopy for solving general polynomial systems that respects m-homogeneous structures , 1987 .

[30]  Alexander I. Barvinok,et al.  Feasibility testing for systems of real quadratic equations , 1992, STOC '92.

[31]  Michael L. Overton,et al.  Narrowing the difficulty gap for the Celis–Dennis–Tapia problem , 2015, Math. Program..

[32]  Richard A. Tapia,et al.  A trust region strategy for nonlinear equality constrained op-timization , 1984 .

[33]  Teresa Krick,et al.  Sharp estimates for the arithmetic Nullstellensatz , 1999, math/9911094.

[34]  Elias P. Tsigaridas,et al.  On the Minimum of a Polynomial Function on a Basic Closed Semialgebraic Set and Applications , 2013, SIAM J. Optim..

[35]  Palaiseau Cedex,et al.  Computing Parametric Geometric Resolutions , 2001 .

[36]  Shuzhong Zhang,et al.  Strong Duality for the CDT Subproblem: A Necessary and Sufficient Condition , 2008, SIAM J. Optim..

[37]  Jean-Charles Faugère,et al.  Critical points and Gröbner bases: the unmixed case , 2012, ISSAC.

[38]  K. B. O’Keefe,et al.  The differential ideal $[uv]$ , 1966 .

[39]  Andrew J. Sommese,et al.  The numerical solution of systems of polynomials - arising in engineering and science , 2005 .

[40]  Patrizia M. Gianni,et al.  Algebraic Solution of Systems of Polynomial Equations Using Groebner Bases , 1987, AAECC.

[41]  Marc Giusti,et al.  Generalized polar varieties: geometry and algorithms , 2005, J. Complex..

[42]  D. Mumford.  Algebraic Geometry I: Complex Projective Varieties , 1981 .

[43]  D. Eisenbud.  Commutative Algebra: with a View Toward Algebraic Geometry , 1995 .

[44]  Mohab Safey El Din,et al.  Deciding reachability of the infimum of a multivariate polynomial , 2011, ISSAC '11.

[45]  Simone Naldi,et al.  Solving Rank-Constrained Semidefinite Programs in Exact Arithmetic , 2016, J. Symb. Comput..

[46]  J. E. Morais,et al.  Straight--Line Programs in Geometric Elimination Theory , 1996, alg-geom/9609005.

[47]  F. J. Rayner,et al.  An algebraically closed field , 1968, Glasgow Mathematical Journal.

[48]  Marie-Françoise Roy,et al.  Computing roadmaps of semi-algebraic sets (extended abstract) , 1996, STOC '96.

[49]  Stephen Melczer,et al.  Symbolic-Numeric Tools for Analytic Combinatorics in Several Variables , 2016, ISSAC.

[50]  Mohab Safey El Din,et al.  Exact algorithms for linear matrix inequalities , 2015, SIAM J. Optim..

[51]  R. Gregory Taylor,et al.  Modern computer algebra , 2002, SIGA.

[52]  S. Basu,et al.  Algorithms in real algebraic geometry , 2003 .

[53]  B. Bank,et al.  Polar Varieties and Efficient Real Equation Solving: The Hypersurface Case , 1996 .

[54]  Stéphan Thomassé,et al.  On the complexity of partial derivatives , 2016, STACS.

[55]  Marie-Françoise Roy,et al.  Zeros, multiplicities, and idempotents for zero-dimensional systems , 1996 .

[56]  F. S. Macaulay,et al.  The Algebraic Theory of Modular Systems , 1972 .

[57]  Grégoire Lecerf.  Computing an equidimensional decomposition of an algebraic variety by means of geometric resolutions , 2000, ISSAC.

[58]  Lihong Zhi,et al.  Computing Rational Points in Convex Semialgebraic Sets and Sum of Squares Decompositions , 2010, SIAM J. Optim..

[59]  Carlos D'Andrea,et al.  Heights of varieties in multiprojective spaces and arithmetic Nullstellensatze , 2011, 1103.4561.

[60]  Éric Schost,et al.  A Nearly Optimal Algorithm for Deciding Connectivity Queries in Smooth and Bounded Real Algebraic Sets , 2013, J. ACM.

[61]  Tomás Recio,et al.  Algorithms in Algebraic Geometry and Applications , 2011 .

[62]  Sally Morrison.  The Differential Ideal [ P ] : M ∞ , 1999 .

[63]  Craig Huneke,et al.  Commutative Algebra I , 2012 .

[64]  Daniel Bienstock,et al.  A Note on Polynomial Solvability of the CDT Problem , 2014, SIAM J. Optim..

[65]  Daniel Perrucci,et al.  A Probabilistic Symbolic Algorithm to Find the Minimum of a Polynomial Function on a Basic Closed Semialgebraic Set , 2013, Discret. Comput. Geom..

[66]  Fabrice Rouillier,et al.  Finding at Least One Point in Each Connected Component of a Real Algebraic Set Defined by a Single Equation , 2000, J. Complex..

[67]  L. Kronecker.  Grundzüge einer arithmetischen Theorie der algebraische Grössen. , 2022 .