chapter 10 disjunctive programming - lara:...

Chapter 10Disjunctive Programming

Egon Balas

Introduction by Egon Balas

In April 1967 I and my family arrived into the US as fresh immigrants from be-hind the Iron Curtain. After a fruitful semester spent with George Dantzig’s groupin Stanford, I started working at CMU. My debut in integer programming and entryticket into Academia was the additive algorithm for 0-1 programming [B65], an im-plicit enumeration procedure based on logical tests akin to what today goes underthe name of constraint propagation. As it used only additions and comparisons, itwas easy to implement and was highly popular for a while. However, I was awareof its limitations and soon after I joined CMU I started investigating cutting planeprocedures, trying to use for this purpose the tools of convex analysis: support func-tions and their level sets, maximal convex extensions, polarity, etc. During the fiveyears starting in 1969, I proposed a number of procedures based on the central ideaof intersection cuts [2] (numbered references are to those at the end of the paper,whereas mnemonicized ones are to the ones listed at the end of this introduction):Given any convex set S containing the LP optimum of a mixed integer program(MIP) but containing no feasible integer point in its interior, one can generate avalid cut by intersecting the boundary of S with the extreme rays of the cone definedby the optimal solution to the linear programming relaxation of the MIP and tak-ing the hyperplane defined by the intersection points as the cut. The search for themost appropriate sets S in this role has led to the concept of outer polars and relatedconstructs [3, 6]. In our days, the idea of intersection cuts has been revived in theform of cutting planes from convex sets with lattice-free interiors, and is the objectof numerous investigations (e.g., [ALWW07], [BC07], [CM07], [DW07]).

It was this line of research that has led to the idea of disjunctive programming,through a process outlined in section 1 of the paper below. Optimizing a function

Egon BalasCarnegie Mellon University, Pittsburgh, USAe-mail: [email protected]

283 M. Jünger et al. (eds.), 50 Years of Integer Programming 1958-2008, DOI 10.1007/978-3-540-68279-0_10, © Springer-Verlag Berlin Heidelberg 2010

284 Egon Balas

subject to a set of linear inequalities connected by conjunction or disjunction is aspecial type of nonconvex programming problem called disjunctive programming.Mixed 0-1 programming is its most important special case. More broadly speak-ing, disjunctive programming is optimization over a union of polyhedra. The basicdocument on disjunctive programming is the July 1974 technical report “Disjunc-tive Programming: Properties of the convex hull of feasible points,” MSRR #348,referenced as [10] in the survey below, which however has not appeared in printuntil 24 years later, when it was published as an invited paper with a preface byGerard Cornuejols and Bill Pulleyblank [B98]. The reasons for this situation arecomplex. The 1974 paper was not rejected, but the report I received at the end ofa very long refereeing process was asking for a revision that would have involveda major rewriting effort at a time when I was engaged in other, complementary re-search projects. As a result, when shortly thereafter I was invited by the late PeterHammer to prepare an introductory survey of disjunctive programming for the up-coming 1977 ARIDAM conference in Vancouver, I decided to incorporate into thatsurvey the main results of MSSR #348 and forego its publication as a stand alonepaper. This is how this survey came about.

Looking back on that decision, I find that not much of substance was lost. Sec-tions 2–6 of this survey adequately summarize the main findings of the 1974 pa-per, with one important exception, which I will now explain. MSRR #348 exploredthe basic properties of disjunctive programs from a polyhedral point of view. Itgave two compact characterizations of the convex hull of a union P of polyhe-dra Pi = {x : Aix ≥ bi}, i ∈ Q, in a higher dimensional space, a procedure we calltoday extended formulation. The first one describes convP as the set of points xsatisfying x = ∑(xi : i ∈ Q) for some (xi,xi

0), i ∈ Q, such that Aixi ≥ bixi0, i ∈ Q,

∑(xi0 : i ∈ Q) = 1, xi

0 ≥ 0, i ∈ Q. The second one describes the facets of convPas the vertices of the reverse polar of P, defined as P# := {y : yx ≥ 1 ∀x ∈ P},shown to be the set of points y satisfying y ≥ uiAi, i ∈ Q for some ui ≥ 0 such thatuibi ≥ 1, i ∈ Q. Both characterizations are linear in |Q|, the number of polyhedra inthe union. In a certain sense the two characterizations are equivalent: projecting thehigher-dimensional polyhedron of the convex hull characterization onto the x-spaceyields the set of all valid inequalities and conversely, taking the reverse polar of thereverse polar yields the convex hull. Thus in the survey paper I only included thesecond characterization, so that the first one did not appear in print until 1985 [B85].But it is precisely this first characterization which has served as a prototype for themany extended formulations that have proved to be such prolific tools for polyhedralanalysis of combinatorial problems, starting with the early studies of this type in the1980’s [BP83], [BP89], [BLP89], and continuing with a plethora of results, the morerecent ones being exemplified by [PW06], [A06], [CW08]. It is also this first char-acterization that was generalized to nonlinear disjunctive programming [SM99] andwas extensively used in the modeling of a variety of practical situations in industry.

Habent sua fata libelli, goes the Latin saying: books have their own fate. Thisapparently also applies to papers or theorems or discoveries. While the work ondisjunctive programming, including the cutting planes that it entailed, stirred littleif any enthusiasm at the time of its inception, about 15 years later when Sebas-

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tian Ceria, Gerard Cornuejols and myself recast essentially the same results in anew framework which we called lift-and-project [BCC93], the reaction was quitedifferent. This time our work was focused on algorithmic aspects, with the cuttingplanes generated in rounds and embedded into an enumerative framework (branchand cut), and was accompanied by the development of an efficient computer code(MIPO, developed by Sebastian) that was able to solve many problem instances thathad been impervious to solution by branch-and-bound alone. Our interest in return-ing to the ideas of disjunctive programming was prompted by the exciting workof Lovasz and Schrijver on matrix cones [LS91]. We discovered that a streamlinedversion of the Lovasz-Schrijver procedure was isomorphic to the disjunctive pro-gramming procedure for generating the integer hull of a 0-1 program by startingwith the higher dimensional representation and projecting it onto the original space.Thus the Lovasz-Schrijver Theorem according to which n applications of this pro-cedure (n being the number of 0-1 variables) yields the integer hull, follows directlyfrom the sequential convexification procedure for facial disjunctive programs, ofwhich 0-1 programs are a prime example (see Section 6 below). The reader will nodoubt recognize in the linear program P∗

1 (g,α0) preceding Theorem 4.4 of section4 below, the ancestor of the cut generating linear program of the lift-and-project(L&P) algorithm [BCC93], which is the specialization of P∗

1 (g,α0) to the case ofthe disjunction xk = 0 or xk = 1.

The computational success of L&P cuts triggered a strong revival of interest incutting planes. Gerard and Sebastian soon discovered [BCCN96] that mixed inte-ger Gomory (MIG) cuts, when used in the MIPO fashion, i.e., generated in roundsand embedded into a branch-and-bound framework, could also solve many of theproblem instances unsolved at the time. Since the MIG cuts were easier to imple-ment than the L&P ones, they were the first to find their way into the commercialcodes. The combination of cutting planes with branch and bound played a centralrole in the revolution in the state of the art in integer programming that started inthe mid-90s. The commercial implementation of lift-and-project cuts had to awaitthe discovery of a method [BP03], [P03] for generating them directly from the LPsimplex tableau, without explicit recourse to the higher dimensional cut generatinglinear program. Today, due to the efforts of Pierre Bonami [BB07], an open-sourceimplementation is also publicly available [COIN-OR].

References

[A06] A. Atamturk, Strong formulations of robust mixed 0-1 programming, MathematicalProgramming 108 (2006) 235–250.

[ALWW07] K. Andersen, Q. Louveaux, R. Weismantel and L.A. Wolsey, Inequalities from tworows of the simplex tableau, Integer Programming and Combinatorial OptimizationIPCO 12 (M. Fischetti and D.P. Williamson, eds.), Springer, 2007, pp. 1–16.

[B65] E. Balas, An additive algorithm for solving linear programs in 0-1 variables, Opera-tions Research 13 (1965) 517–546.

[B85] E. Balas, Disjunctive programming and a hierarchy of relaxations for discrete op-timization problems, SIAM Journal on Algebraic and Discrete Methods 6 (1985)466–486.

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[B98] E. Balas, Disjunctive programming: Properties of the convex hull of feasible points,Invited paper with a Foreword by G. Cornuejols and G. Pulleyblank, Discrete Ap-plied Mathematics 89 (1998) 1–44.

[BB07] E. Balas and P. Bonami, New variants of lift-and-project cut generation from the LPtableau: Open source implementation and testing, Integer Programming and Combi-natorial Oprtimization IPCO 12 (M. Fischetti and D.P. Williamson, eds.), Springer,2007, pp. 89–103.

[BC07] V. Borozan and G. Cornuejols, Minimal inequalities for integer constraints, Techni-cal Report, Tepper School, Carnegie Mellon University, 2007.

[BCC93] E. Balas, S. Ceria and G. Cornuejols, A lift-and-project cutting plane algorithm formixed 0-1 programs, Mathematical Programming 58 (1993) 295–324.

[BCCN96] E. Balas, S. Ceria, G. Cornuejols and N. Natraj, Gomory cuts revisited, OperationsResearch Letters 19 (1996) 1–10.

[BLP89] M. Ball, W. Liu and W.R. Pulleyblank, Two-terminal Steiner tree polyhedra, Contri-butions to Operations Research and Economics, MIT Press, 1989, pp. 251–284.

[BP83] E. Balas and W.R. Pulleyblank, The perfectly matchable subgraph polytope of a bi-partite graph, Networks 13 (1983) 495–516.

[BP89] E. Balas and W.R. Pulleyblank, The perfectly matchable subgraph polytope of anarbitrary graph, Combinatorica 9 (1989) 321–337.

[BP03] E. Balas and M. Perregaard, A precise correspondence between lift-and-project cuts,simple disjunctive cuts, and mixed integer Gomory cuts for 0-1 programming, Math-ematical Programming 94 (2003) 221–245.

[CM07] G. Cornuejols and F. Margot, On the facets of mixed integer programs with two inte-ger variables and two constraints, Technical Report, Tepper School, Carnegie Mel-lon University, 2007.

[COIN-OR] http://www.coin-or.org[CW08] M. Conforti and L.A. Wolsey, Compact formulations as a union of polyhedra, Math-

ematical Programming 114 (2008) 277–289.[DW07] S.S. Dey and L.A. Wolsey, Lifting integer variables in minimal inequalities cor-

responding to lattice-free triangles, Integer Programming and Combinatorial Opti-mization IPCO 13 (A. Lodi, A. Panconesi, and G. Rinaldi, eds.), Springer, 2008, pp.463–475.

[LS91] L. Lovasz and A. Schrijver, Cones of matrices and set functions and 0-1 optimization,SIAM Journal of Optimization 1 (1991) 166–190.

[P03] M. Perregaard, A Practical implementation of lift-and-project cuts: a computationalexploration of lift-and-project with XPRESS-MP, International Symposium on Math-ematical Programming, Copenhagen, August 2003.

[PW06] Y. Pochet and L.A. Wolsey, Production Planning by Mixed Integer Programming,Springer, 2006.

[SM99] R. Stubbs and S. Mehrotra, A branch and cut method for 0-1 mixed integer convexprogramming, Mathematical Programming 86 (1999) 515–532.

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The following article originally appeared as:

E. Balas, Disjunctive Programming, Discrete Optimization II (P.L. Hammer, E.L.Johnson, and B.H. Korte, eds.), Annals of Discrete Mathematics 5 (1979) 3–51.

Copyright c© 1979 North-Holland Publishing Company.

Reprinted by permission from Elsevier.

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The following article originally appeared as:

E. Balas, Erratum to: Disjunctive Programming, Discrete Applied Mathematics 5(1983) 247–248.

Copyright c© 1983 North-Holland Publishing Company.

Reprinted by permission from Elsevier.

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