Find out how to access previewonly content
Computation with Polynomial Equations and Inequalities Arising in Combinatorial Optimization
 Jesus A. De Loera,
 Peter N. Malkin,
 Pablo A. Parrilo
 … show all 3 hide
Abstract
This is a survey of a recent methodology to solve systems of polynomial equations and inequalities for problems arising in combinatorial optimization. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create largescale linear algebra or semidefinite programming relaxations of many kinds of feasibility or optimization questions.
 E. Balas, S. Ceria, and G. Cornu´ejols, A liftandproject cutting plane algorithm for mixed 0–1 programs, Mathematical Programming, 58 (1993), pp. 295–324.
 J. Bochnak, M. Coste, and M.F. Roy, Real algebraic geometry, Springer, 1998.
 M. Clegg, J. Edmonds, and R. Impagliazzo, Using the Groebner basis algorithm to find proofs of unsatisfiability, in STOC ’96: Proceedings of the twentyeighth annual ACM symposium on Theory of computing, New York, NY, USA, 1996, ACM, pp. 174–183.
 N. Courtois, A. Klimov, J. Patarin, and A. Shamir, Efficient algorithms for solving overdefined systems of multivariate polynomial equations, in EUROCRYPT, 2000, pp. 392–407.
 D. Cox, J. Little, and D. O’Shea, Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer Verlag, 1992.
 , Using Algebraic Geometry, Vol. 185 of Graduate Texts in Mathematics, Springer, 2nd ed., 2005.
 J. De Loera, C. Hillar, P. Malkin, and M. Omar, Recognizing graph theoretic properties with polynomial ideals. http://arxiv.org/abs/1002.4435, 2010.
 J. De Loera, J. Lee, P. Malkin, and S. Margulies, Hilbert’s Nullstellensatz and an algorithm for proving combinatorial infeasibility, in Proceedings of the Twentyfirst International Symposium on Symbolic and Algebraic Computation (ISSAC 2008), 2008.
 J. De Loera, J. Lee, S. Margulies, and S. Onn, Expressing combinatorial optimization problems by systems of polynomial equations and the nullstellensatz, to appear in the Journal of Combinatorics, Probability and Computing (2008).
 A. Dickenstein and I. Emiris, eds., Solving Polynomial Equations: Foundations, Algorithms, and Applications, Vol. 14 of Algorithms and Computation in Mathematics, Springer Verlag, Heidelberg, 2005.
 W. Eberly and M. Giesbrecht, Efficient decomposition of associative algebras over finite fields, Journal of Symbolic Computation, 29 (2000), pp. 441–458. CrossRef
 A.V. Gelder, Another look at graph coloring via propositional satisfiability, Discrete Appl. Math., 156 (2008), pp. 230–243. CrossRef
 E. Gilbert, Random graphs, Annals of Mathematical Statistics, 30 (1959), pp. 1141–1144. CrossRef
 J. Gouveia, M. Laurent, P.A. Parrilo, and R.R. Thomas, A new semidefinite programming relaxation for cycles in binary matroids and cuts in graphs. http://arxiv.org/abs/0907.4518, 2009.
 J. Gouveia, P.A. Parrilo, and R.R. Thomas, Theta bodies for polynomial ideals, SIAM Journal on Optimization, 20 (2010), pp. 2097–2118. CrossRef
 D. Grigoriev and N. Vorobjov, Complexity of Nullstellensatz and Positivstellensatz proofs, Annals of Pure and Applied Logic, 113 (2002), pp. 153–160. CrossRef
 D. Henrion and J.B. Lasserre, GloptiPoly: Global optimization over polynomials with MATLAB and SeDuMi, ACM Trans. Math. Softw., 29 (2003), pp. 165–194. CrossRef
 , Detecting global optimality and extracting solutions in GloptiPoly, in Positive polynomials in control, Vol. 312 of Lecture Notes in Control and Inform. Sci., Springer, Berlin, 2005, pp. 293–310.
 T. Hogg and C. Williams, The hardest constraint problems: a double phase transition, Artif. Intell., 69 (1994), pp. 359–377. CrossRef
 A. Kehrein and M. Kreuzer, Characterizations of border bases, Journal of Pure and Applied Algebra, 196 (2005), pp. 251 – 270. CrossRef
 A. Kehrein, M. Kreuzer, and L. Robbiano, An algebraist’s view on border bases, in Solving Polynomial Equations: Foundations, Algorithms, and Applications, A. Dickenstein and I. Emiris, eds., Vol. 14 of Algorithms and Computation in Mathematics, Springer Verlag, Heidelberg, 2005, ch. 4, pp. 160–202.
 J. Koll´ar, Sharp effective Nullstellensatz, Journal of the AMS, 1 (1988), pp. 963–975.
 J. Lasserre, Global optimization with polynomials and the problem of moments, SIAM J. on Optimization, 11 (2001), pp. 796–817. CrossRef
 J. Lasserre, M. Laurent, and P. Rostalski, Semidefinite characterization and computation of zerodimensional real radical ideals, Found. Comput. Math., 8 (2008), pp. 607–647. CrossRef
 , A unified approach to computing real and complex zeros of zerodimensional ideals, in Emerging Applications of Algebraic Geometry, M. Putinar and S. Sullivant, eds., vol. 149 of IMA Volumes in Mathematics and its Applications, Springer, 2009, pp. 125–155.
 J.B. Lasserre, An explicit equivalent positive semidefinite program for nonlinear 0–1 programs, SIAM J. on Optimization, 12 (2002), pp. 756–769. CrossRef
 M. Laurent, A comparison of the SheraliAdams, Lov´aszSchrijver, and Lasserre relaxations for 0–1 programming, Math. Oper. Res., 28 (2003), pp. 470–496. CrossRef
 , Semidefinite relaxations for maxcut, in The Sharpest Cut: The Impact of Manfred Padberg and His Work, M. Gr¨otschel, ed., Vol. 4 of MPSSIAM Series in Optimization, SIAM, 2004, pp. 257–290.
 , Semidefinite representations for finite varieties, Mathematical Programming, 109 (2007), pp. 1–26.
 , Sums of squares, moment matrices and optimization over polynomials, in Emerging Applications of Algebraic Geometry, M. Putinar and S. Sullivant, eds., Vol. 149 of IMA Volumes in Mathematics and its Applications, Springer, 2009, pp. 157–270.
 J. L¨ofberg, YALMIP: A toolbox for modeling and optimization in MATLAB, in Proceedings of the CACSD Conference, Taipei, Taiwan, 2004.
 L. Lov´asz, Stable sets and polynomials, Discrete Math., 124 (1994), pp. 137–153.
 , Semidefinite programs and combinatorial optimization, in Recent advances in algorithms and combinatorics, B. Reed and C. Sales, eds., Vol. 11 of CMS Books in Mathematics, Spring, New York, 2003, pp. 137–194.
 L. Lov´asz and A. Schrijver, Cones of matrices and setfunctions and 0–1 optimization, SIAM J. Optim., 1 (1991), pp. 166–190.
 S. Margulies, Computer Algebra, Combinatorics, and Complexity: Hilbert’s Nullstellensatz and NPComplete Problems, PhD thesis, UC Davis, 2008.
 M. Marshall, Positive polynomials and sums of squares., Mathematical Surveys and Monographs, 146. Providence, RI: American Mathematical Society (AMS). xii, p. 187, 2008.
 B. Mourrain, A new criterion for normal form algorithms, in Proc. AAECC, Vol. 1719 of LNCS, Springer, 1999, pp. 430–443.
 B. Mourrain and P. Tr´ebuchet, Stable normal forms for polynomial system solving, Theoretical Computer Science, 409 (2008), pp. 229 – 240. Symbolic Numerical Computations.
 Y. Nesterov, Squared functional systems and optimization problems, in High Performance Optimization, J.F. et al., eds., ed., Kluwer Academic, 2000, pp. 405–440.
 P.A. Parrilo, Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization, PhD thesis, California Institute of Technology, May 2000.
 , Semidefinite programming relaxations for semialgebraic problems, Mathematical Programming, 96 (2003), pp. 293–320.
 P.A. Parrilo and B. Sturmfels, Minimizing polynomial functions, in Proceedings of the DIMACS Workshop on Algorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science (March 2001), S. Basu and L. GonzalezVega, eds., American Mathematical Society, Providence RI, 2003, pp. 83–100.
 S. Prajna, A. Papachristodoulou, P. Seiler, and P.A. Parrilo, SOSTOOLS: Sum of squares optimization toolbox for MATLAB, 2004.
 G. Reid and L. Zhi, Solving polynomial systems via symbolicnumeric reduction to geometric involutive form, Journal of Symbolic Computation, 44 (2009), pp. 280–291. CrossRef
 S. Roman, Advanced Linear Algebra, Vol. 135 of Graduate Texts in Mathematics, Springer New York, third ed., 2008.
 A. Schrijver, Theory of linear and integer programming, Wiley, 1986.
 H. Sherali and W. Adams, A hierarchy of relaxations between the continuous and convex hull representations for zero–one programming problems, SIAM Journal on Discrete Mathematics, 3 (1990), pp. 411–430. CrossRef
 N.Z. Shor, Class of global minimum bounds of polynomial functions, Cybernetics, 23 (1987), pp. 731–734. CrossRef
 G. Stengle, A Nullstellensatz and a Positivstellensatz in semialgebraic geometry, Mathematische Annalen, 207 (1973), pp. 87–97. CrossRef
 H. Stetter, Numerical Polynomial Algebra, SIAM, 2004.
 L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Review, 38 (1996), pp. 49–95. CrossRef
 L. Zhang, zchaff v2007.3.12. Available at http://www.princeton.edu/_chaff/zchaff.html, 2007
 Title
 Computation with Polynomial Equations and Inequalities Arising in Combinatorial Optimization
 Book Title
 Mixed Integer Nonlinear Programming
 Pages
 pp 447481
 Copyright
 2012
 DOI
 10.1007/9781461419273_16
 Print ISBN
 9781461419266
 Online ISBN
 9781461419273
 Series Title
 The IMA Volumes in Mathematics and its Applications
 Series Volume
 154
 Series ISSN
 09406573
 Publisher
 Springer New York
 Copyright Holder
 Springer Science+Business Media, LLC
 Additional Links
 Topics
 Keywords

 Polynomial equations and inequalities
 combinatorial optimization
 Nullstellensatz
 Positivstellensatz
 graph colorability
 maxcut
 stable sets
 semidefinite programming
 largescale linear algebra
 semialgebraic sets
 real algebra
 Industry Sectors
 eBook Packages
 Editors

 Jon Lee ^{(ID1)}
 Sven Leyffer ^{(ID2)}
 Editor Affiliations

 ID1. , College of Engineering, University of Michigan
 ID2. , Mathematics and Computer Science, Argonne National Laboratory
 Authors

 Jesus A. De Loera ^{(1)}
 Peter N. Malkin ^{(1)}
 Pablo A. Parrilo ^{(2)}
 Author Affiliations

 1. Department of Mathematics, University of California at Davis, Davis, CA, 95616, USA
 2. Laboratory for Information and Decision Systems, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA
Continue reading...
To view the rest of this content please follow the download PDF link above.