Nonlinear Integer Programming

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms.We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.

[1]  Ralph E. Gomory,et al.  An algorithm for integer solutions to linear programs , 1958 .

[2]  J. E. Kelley,et al.  The Cutting-Plane Method for Solving Convex Programs , 1960 .

[3]  Robert G. Jeroslow,et al.  There Cannot be any Algorithm for Integer Programming with Quadratic Constraints , 1973, Oper. Res..

[4]  Jack E. Graver,et al.  On the foundations of linear and integer linear programming I , 1975, Math. Program..

[5]  E. Balas.  Nonconvex Quadratic Programming via Generalized Polars , 1975 .

[6]  A. Buckley,et al.  An alternate implementation of Goldfarb's minimization algorithm , 1975, Math. Program..

[7]  Garth P. McCormick,et al.  Computability of global solutions to factorable nonconvex programs: Part I — Convex underestimating problems , 1976, Math. Program..

[8]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[9]  M. Garey.  Johnson: computers and intractability: a guide to the theory of np- completeness (freeman , 1979 .

[10]  Mihalis Yannakakis,et al.  The complexity of restricted spanning tree problems , 1982, JACM.

[11]  James P. Jones.  Universal Diophantine Equation , 1982, J. Symb. Log..

[12]  Kenneth McAloon.  On the Complexity of Models of Arithmetic , 1982, J. Symb. Log..

[13]  James E. Falk,et al.  Jointly Constrained Biconvex Programming , 1983, Math. Oper. Res..

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

[15]  Omprakash K. Gupta,et al.  Branch and Bound Experiments in Convex Nonlinear Integer Programming , 1985 .

[16]  James E. Goehring,et al.  THE ENGLISH TRANSLATION , 1986 .

[17]  Ignacio E. Grossmann,et al.  An outer-approximation algorithm for a class of mixed-integer nonlinear programs , 1986, Math. Program..

[18]  Vijay V. Vazirani,et al.  Matching is as easy as matrix inversion , 1987, STOC.

[19]  N. Z. Shor.  An approach to obtaining global extremums in polynomial mathematical programming problems , 1987 .

[20]  M. Brion.  Points entiers dans les polyèdres convexes , 1988 .

[21]  J. Renegar,et al.  On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part I , 1989 .

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

[23]  Mark Evan Hartmann,et al.  Cutting planes and the complexity of the integer hull , 1989 .

[24]  András Sebö,et al.  Hilbert Bases, Caratheodory's Theorem and Combinatorial Optimization , 1990, IPCO.

[25]  Teresa Krick,et al.  A Gemometrical Bound for Integer Programming with Polynomial Constraints , 1991, FCT.

[26]  Donald Goldfarb,et al.  A Logarithmic Barrier Function Algorithm for Quadratically Constrained Convex Quadratic Programming , 1991, SIAM J. Optim..

[27]  James Renegar,et al.  On the Computational Complexity of Approximating Solutions for Real Algebraic Formulae , 1992, SIAM J. Comput..

[28]  James Renegar,et al.  On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part III: Quantifier Elimination , 1992, J. Symb. Comput..

[29]  I. Grossmann,et al.  An LP/NLP based branch and bound algorithm for convex MINLP optimization problems , 1992 .

[30]  William J. Cook,et al.  On integer points in polyhedra , 1992, Comb..

[31]  László Lovász,et al.  The Generalized Basis Reduction Algorithm , 1990, Math. Oper. Res..

[32]  Joos Heintz,et al.  Une borne optimale pour la programmation entière quasi-convexe , 1993 .

[33]  Alexander I. Barvinok,et al.  A Polynomial Time Algorithm for Counting Integral Points in Polyhedra when the Dimension Is Fixed , 1993, FOCS.

[34]  Ravindra K. Ahuja,et al.  Network Flows: Theory, Algorithms, and Applications , 1993 .

[35]  Panos M. Pardalos,et al.  The Quadratic Assignment Problem: A Survey and Recent Developments , 1993, Quadratic Assignment and Related Problems.

[36]  B. Sturmfels.  Gröbner bases and convex polytopes , 1995 .

[37]  B. Reznick,et al.  Sums of squares of real polynomials , 1995 .

[38]  T. Westerlund,et al.  An extended cutting plane method for solving convex MINLP problems , 1995 .

[39]  David P. Williamson,et al.  Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming , 1995, JACM.

[40]  Nikolaos V. Sahinidis,et al.  BARON: A general purpose global optimization software package , 1996, J. Glob. Optim..

[41]  Edward M. B. Smith,et al.  On the optimal design of continuous processes , 1996 .

[42]  M. Brion,et al.  Residue formulae, vector partition functions and lattice points in rational polytopes , 1997 .

[43]  Egon Balas,et al.  programming: Properties of the convex hull of feasible points * , 1998 .

[44]  A. Barvinok,et al.  An Algorithmic Theory of Lattice Points in Polyhedra , 1999 .

[45]  Edward M. B. Smith,et al.  A symbolic reformulation/spatial branch-and-bound algorithm for the global optimisation of nonconvex MINLPs , 1999 .

[46]  Sanjay Mehrotra,et al.  A branch-and-cut method for 0-1 mixed convex programming , 1999, Math. Program..

[47]  Sebastián Ceria,et al.  Convex programming for disjunctive convex optimization , 1999, Math. Program..

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

[49]  Arkadi Nemirovski,et al.  Lectures on modern convex optimization - analysis, algorithms, and engineering applications , 2001, MPS-SIAM series on optimization.

[50]  A. Prestel,et al.  Distinguished representations of strictly positive polynomials , 2001 .

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

[52]  Johan Håstad,et al.  Some optimal inapproximability results , 2001, JACM.

[53]  Nikolaos V. Sahinidis,et al.  Semidefinite Relaxations of Fractional Programs via Novel Convexification Techniques , 2001, J. Glob. Optim..

[54]  Nikolaos V. Sahinidis,et al.  Convex extensions and envelopes of lower semi-continuous functions , 2002, Math. Program..

[55]  Endre Boros,et al.  Pseudo-Boolean optimization , 2002, Discret. Appl. Math..

[56]  T. Westerlund,et al.  Solving Pseudo-Convex Mixed Integer Optimization Problems by Cutting Plane Techniques , 2002 .

[57]  Nikolaos V. Sahinidis,et al.  Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming , 2002 .

[58]  Raymond Hemmecke,et al.  On the positive sum property and the computation of Graver test sets , 2003, Math. Program..

[59]  Monique Laurent,et al.  A Comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre Relaxations for 0-1 Programming , 2003, Math. Oper. Res..

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

[61]  Leo Liberti,et al.  Convex Envelopes of Monomials of Odd Degree , 2003, J. Glob. Optim..

[62]  Ju. V. Matijasevic,et al.  ENUMERABLE SETS ARE DIOPHANTINE , 2003 .

[63]  Bernd Sturmfels,et al.  Higher Lawrence configurations , 2003, J. Comb. Theory, Ser. A.

[64]  L. Liberti.  Comparison of Convex Relaxations for Monomials of Odd Degree , 2003 .

[65]  Christodoulos A. Floudas,et al.  Trilinear Monomials with Mixed Sign Domains: Facets of the Convex and Concave Envelopes , 2004, J. Glob. Optim..

[66]  Nikolaos V. Sahinidis,et al.  Global optimization of mixed-integer nonlinear programs: A theoretical and computational study , 2004, Math. Program..

[67]  Uriel G. Rothblum,et al.  Convex Combinatorial Optimization , 2003, Discret. Comput. Geom..

[68]  Jesús A. De Loera,et al.  The Complexity of Three-Way Statistical Tables , 2002, SIAM J. Comput..

[69]  Jeff T. Linderoth.  A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs , 2005, Math. Program..

[70]  Mehmet Tolga Çezik,et al.  Cuts for mixed 0-1 conic programming , 2005, Math. Program..

[71]  George L. Nemhauser,et al.  A polyhedral study of nonconvex quadratic programs with box constraints , 2005, Math. Program..

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

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

[74]  George L. Nemhauser,et al.  A branch-and-cut algorithm for nonconvex quadratic programs with box constraints , 2005, Math. Program..

[75]  Jesús A. De Loera,et al.  All Linear and Integer Programs Are Slim 3-Way Transportation Programs , 2006, SIAM J. Optim..

[76]  Jesús A. De Loera,et al.  FPTAS for mixed-integer polynomial optimization with a fixed number of variables , 2006, SODA '06.

[77]  Claudio Gentile,et al.  Perspective cuts for a class of convex 0–1 mixed integer programs , 2006, Math. Program..

[78]  Jesús A. De Loera,et al.  Markov bases of three-way tables are arbitrarily complicated , 2006, J. Symb. Comput..

[79]  Jesús A. De Loera,et al.  Integer Polynomial Optimization in Fixed Dimension , 2006, Math. Oper. Res..

[80]  John J. H. Forrest,et al.  Rapid development of an open-source MINLP solver with COIN-OR , 2007 .

[81]  Matthias Köppe,et al.  A Primal Barvinok Algorithm Based on Irrational Decompositions , 2006, SIAM J. Discret. Math..

[82]  Seth Sullivant,et al.  A finiteness theorem for Markov bases of hierarchical models , 2007, J. Comb. Theory, Ser. A.

[83]  Jon Lee,et al.  In situ column generation for a cutting-stock problem , 2007, Comput. Oper. Res..

[84]  Franz Rendl,et al.  A Branch and Bound Algorithm for Max-Cut Based on Combining Semidefinite and Polyhedral Relaxations , 2007, IPCO.

[85]  Matthias Ko¨ppe.  A Primal Barvinok Algorithm Based on Irrational Decompositions , 2007 .

[86]  Giovanni Rinaldi,et al.  Efficient Reduction of Polynomial Zero-One Optimization to the Quadratic Case , 2007, SIAM J. Optim..

[87]  Samuel Burer,et al.  A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations , 2008, Math. Program..

[88]  Shmuel Onn,et al.  Nonlinear bipartite matching , 2008, Discret. Optim..

[89]  Jon Lee,et al.  Disjunctive Cuts for Non-convex Mixed Integer Quadratically Constrained Programs , 2008, IPCO.

[90]  Jesús A. De Loera,et al.  N-fold integer programming , 2006, Discret. Optim..

[91]  Kaj-Mikael Björk,et al.  Global solution of optimization problems with signomial parts , 2008, Discret. Optim..

[92]  Jesús A. De Loera,et al.  FPTAS for optimizing polynomials over the mixed-integer points of polytopes in fixed dimension , 2008, Math. Program..

[93]  R. Weismantel,et al.  Nonlinear optimization for matroid intersection and extensions , 2008, 0807.3907.

[94]  Andries E. Brouwer,et al.  Small Integral Trees , 2008, Electron. J. Comb..

[95]  Jeff T. Linderoth,et al.  FilMINT: An Outer-Approximation-Based Solver for Nonlinear Mixed Integer Programs , 2008 .

[96]  Robert Weismantel,et al.  The Convex Envelope of (n--1)-Convex Functions , 2008, SIAM J. Optim..

[97]  S. Ulbrich,et al.  MIXED INTEGER SECOND ORDER CONE PROGRAMMING , 2008 .

[98]  Oktay Günlük,et al.  Perspective Relaxation of Mixed Integer Nonlinear Programs with Indicator Variables , 2008, IPCO.

[99]  Matthias Köppe,et al.  Computing Parametric Rational Generating Functions with a Primal Barvinok Algorithm , 2008, Electron. J. Comb..

[100]  Gérard Cornuéjols,et al.  An algorithmic framework for convex mixed integer nonlinear programs , 2008, Discret. Optim..

[101]  Eva Riccomagno,et al.  Nonlinear Matroid Optimization and Experimental Design , 2007, SIAM J. Discret. Math..

[102]  Jon Lee,et al.  On test sets for nonlinear integer maximization , 2008, Oper. Res. Lett..

[103]  Gérard Cornuéjols,et al.  A Feasibility Pump for mixed integer nonlinear programs , 2009, Math. Program..

[104]  Claudio Gentile,et al.  A computational comparison of reformulations of the perspective relaxation: SOCP vs. cutting planes , 2009, Oper. Res. Lett..

[105]  Sinan Gürel,et al.  A strong conic quadratic reformulation for machine-job assignment with controllable processing times , 2009, Oper. Res. Lett..

[106]  Miguel A. Lejeune,et al.  An Exact Solution Approach for Portfolio Optimization Problems Under Stochastic and Integer Constraints , 2009, Oper. Res..

[107]  Christodoulos A. Floudas.  Generalized Benders Decomposition , 2009, Encyclopedia of Optimization.

[108]  Panos M. Pardalos,et al.  Quadratic Assignment Problem , 1997, Encyclopedia of Optimization.

[109]  Leo Liberti,et al.  Branching and bounds tighteningtechniques for non-convex MINLP , 2009, Optim. Methods Softw..

[110]  P. Bonami,et al.  An Exact Solution Approach for Integer Constrained Portfolio Optimization Problems Under Stochastic Constraints , 2009 .

[111]  Jon Lee,et al.  Nonlinear Optimization over a Weighted Independence System , 2009, AAIM.

[112]  Ralph E. Gomory,et al.  Outline of an Algorithm for Integer Solutions to Linear Programs and An Algorithm for the Mixed Integer Problem , 2010, 50 Years of Integer Programming.

[113]  Oktay Günlük,et al.  Perspective reformulations of mixed integer nonlinear programs with indicator variables , 2010, Math. Program..

[114]  Franz Rendl,et al.  Solving Max-Cut to optimality by intersecting semidefinite and polyhedral relaxations , 2009, Math. Program..

[115]  Samuel Burer,et al.  Computable representations for convex hulls of low-dimensional quadratic forms , 2010, Math. Program..

[116]  Jon Lee,et al.  Convex relaxations of non-convex mixed integer quadratically constrained programs: extended formulations , 2010, Math. Program..

[117]  Jon Lee,et al.  Convex relaxations of non-convex mixed integer quadratically constrained programs: projected formulations , 2011, Math. Program..

[118]  Raymond Hemmecke,et al.  A polynomial oracle-time algorithm for convex integer minimization , 2007, Math. Program..

[119]  Erik Massop.  Hilbert's tenth problem , 2012 .

[120]  Lisa Turner,et al.  Applications of Second Order Cone Programming , 2012 .