Modeling disjunctive constraints with a logarithmic number of binary variables and constraints

Many combinatorial constraints over continuous variables such as SOS1 and SOS2 constraints can be interpreted as disjunctive constraints that restrict the variables to lie in the union of a finite number of specially structured polyhedra. Known mixed integer binary formulations for these constraints have a number of binary variables and extra constraints linear in the number of polyhedra. We give sufficient conditions for constructing formulations for these constraints with a number of binary variables and extra constraints logarithmic in the number of polyhedra. Using these conditions we introduce mixed integer binary formulations for SOS1 and SOS2 constraints that have a number of binary variables and extra constraints logarithmic in the number of continuous variables. We also introduce the first mixed integer binary formulations for piecewise linear functions of one and two variables that use a number of binary variables and extra constraints logarithmic in the number of linear pieces of the functions. We prove that the new formulations for piecewise linear functions have favorable tightness properties and present computational results showing that they can significantly outperform other mixed integer binary formulations.

[1]  Jon Lee.  All-Different Polytopes , 2002, J. Comb. Optim..

[2]  Herbert S. Wilf,et al.  Combinatorial Algorithms: An Update , 1987 .

[3]  Hanif D. Sherali,et al.  On mixed-integer zero-one representations for separable lower-semicontinuous piecewise-linear functions , 2001, Oper. Res. Lett..

[4]  G. Dantzig.  Discrete-Variable Extremum Problems , 1957 .

[5]  Don Coppersmith,et al.  Parsimonious binary-encoding in integer programming , 2005, Discret. Optim..

[6]  Toshimde Ibaraki.  Integer programming formulation of combinatorial optimization problems , 1976, Discret. Math..

[7]  Jon Lee,et al.  Polyhedral methods for piecewise-linear functions I: the lambda method , 2001, Discret. Appl. Math..

[8]  George L. Nemhauser,et al.  Models for representing piecewise linear cost functions , 2004, Oper. Res. Lett..

[9]  George L. Nemhauser,et al.  Mixed-Integer Models for Nonseparable Piecewise-Linear Optimization: Unifying Framework and Extensions , 2010, Oper. Res..

[10]  Charles E. Blair.  Representation for multiple right-hand sides , 1991, Math. Program..

[11]  Robert G. Jeroslow,et al.  Representability in mixed integer programmiing, I: Characterization results , 1987, Discret. Appl. Math..

[12]  R. R. Meyer.  A theoretical and computational comparison of “equivalent” mixed‐integer formulations , 1981 .

[13]  George L. Nemhauser,et al.  A Branch-and-Cut Algorithm Without Binary Variables for Nonconvex Piecewise Linear Optimization , 2006, Oper. Res..

[14]  M. Todd.  Union Jack Triangulations , 1977 .

[15]  George L. Nemhauser,et al.  Nonconvex, lower semicontinuous piecewise linear optimization , 2008, Discret. Optim..

[16]  Charles Eugene Blair.  Two Rules for Deducing Valid Inequalities for 0-1 Problems , 1976 .

[17]  G. Dantzig.  ON THE SIGNIFICANCE OF SOLVING LINEAR PROGRAMMING PROBLEMS WITH SOME INTEGER VARIABLES , 1960 .

[18]  Manfred W. Padberg,et al.  Approximating Separable Nonlinear Functions Via Mixed Zero-One Programs , 1998, Oper. Res. Lett..

[19]  Robert R. Meyer,et al.  On the existence of optimal solutions to integer and mixed-integer programming problems , 1974, Math. Program..

[20]  Hanif D. Sherali,et al.  Optimization with disjunctive constraints , 1980 .

[21]  Robert G. Jeroslow.  Representability of functions , 1989, Discret. Appl. Math..

[22]  Robert G. Jeroslow,et al.  Cutting-Plane Theory: Disjunctive Methods , 1977 .

[23]  R. Kevin Wood,et al.  Explicit-Constraint Branching for Solving Mixed-Integer Programs , 2000 .

[24]  R. G. Jeroslow,et al.  Experimental Results on the New Techniques for Integer Programming Formulations , 1985 .

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

[26]  E. Balas.  Disjunctive programming and a hierarchy of relaxations for discrete optimization problems , 1985 .

[27]  George L. Nemhauser,et al.  Branch-and-cut for combinatorial optimization problems without auxiliary binary variables , 2001, The Knowledge Engineering Review.

[28]  Jesús M. Carnicer,et al.  Piecewise linear interpolants to Lagrange and Hermite convex scattered data , 1996, Numerical Algorithms.

[29]  R. Meyer.  Integer and mixed-integer programming models: General properties , 1975 .

[30]  George B. Dantzig,et al.  Linear programming and extensions , 1965 .

[31]  J. K. Lowe.  Modelling with Integer Variables. , 1984 .

[32]  R. R. Meyer,et al.  Mixed integer minimization models for piecewise-linear functions of a single variable , 1976, Discret. Math..

[33]  J. Tomlin.  A Suggested Extension of Special Ordered Sets to Non-Separable Non-Convex Programming Problems* , 1981 .

[34]  Stephen C. Graves,et al.  A composite algorithm for a concave-cost network flow problem , 1989, Networks.

[35]  Lawrence J. Watters.  Letter to the Editor - Reduction of Integer Polynomial Programming Problems to Zero-One Linear Programming Problems , 1967, Oper. Res..

[36]  François Margot,et al.  On a Binary-Encoded ILP Coloring Formulation , 2007, INFORMS J. Comput..

[37]  Thomas L. Magnanti,et al.  A Comparison of Mixed - Integer Programming Models for Nonconvex Piecewise Linear Cost Minimization Problems , 2003, Manag. Sci..

[38]  Egon Balas.  On the convex hull of the union of certain polyhedra , 1988 .

[39]  A. S. Manne,et al.  On the Solution of Discrete Programming Problems , 1956 .

[40]  Thomas L. Magnanti,et al.  Separable Concave Optimization Approximately Equals Piecewise Linear Optimization , 2004, IPCO.

[41]  Robert G. Jeroslow.  A simplification for some disjunctive formulations , 1988 .

[42]  Egon Balas,et al.  Projection, Lifting and Extended Formulation in Integer and Combinatorial Optimization , 2005, Ann. Oper. Res..

[43]  Alexander Martin,et al.  Mixed Integer Models for the Stationary Case of Gas Network Optimization , 2006, Math. Program..