Polyhedral Approach to Integer Linear Programming. Tepper School of Business Carnegie Mellon University, Pittsburgh
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1 Polyhedral Approach to Integer Linear Programming Gérard Cornuéjols Tepper School of Business Carnegie Mellon University, Pittsburgh 1 / 30
2 Brief history First Algorithms Polynomial Algorithms Solving systems of linear equations Babylonians 1700BC Gauss 1801 Solving systems of linear inequalities Fourier 1822 Dantzig 1951 Edmonds 1967 Khachyan 1979 Karmarkar 1984 Solving systems of linear inequalities in integers Gomory 1958 Lenstra / 30
3 Mixed Integer Linear Programming min cx x S where S := {x Z p + Rn p + : Ax b} x2 objective Linear Relaxation Ax b min cx x P where P := {x R n + : Ax b} x1 x2 objective x2 objective cut Ax b Ax b Branch-and-bound Land and Doig 1960 x Cutting Planes Dantzig, Fulkerson and Johnson 1954 Gomory 1958 x1 3 / 30
4 Polyhedral Theory P := {x R n + : Ax b} Polyhedron S := P (Z p + R n p + ) Mixed Integer Linear Set Conv S := {x R n : x 1,..., x k S, λ 0, λ i = 1 such that x = λ 1 x λ k x k } THEOREM Meyer 1974 If A, b have rational entries, then Conv S is a polyhedron. Proof Using a theorem of Minkowski 1896 and Weyl 1935 : P is a polyhedron if and only if P = Q + C where Q is a polytope and C is a polyhedral cone. P ConvS P S ConvS S 4 / 30
5 Thus min cx x S x2 objective Ax b x1 can be rewritten as the LP x2 objective min cx x Conv S Ax b x1 We are interested in the constructive aspects of Conv S. REMARK The number of constraints of Conv S can be exponential in the size of Ax b, BUT 1) sometimes a partial representation of Conv S suffices (Example : Dantzig, Fulkerson, Johnson 1954) ; 2) Conv S can sometimes be obtained as the projection of a polyhedron with a polynomial number of variables and constraints. 5 / 30
6 Projections Let P := {(x, y) R n R k : Ax + Gy b} DEFINITION Proj x (P) := {x R n : y R k such that Ax + Gy b} y P Proj x (P) x THEOREM Proj x (P) = {x R n : vax vb for all v Q} where Q := {v R m : vg = 0, v 0}. PROOF Let x R n. Farkas s lemma (Farkas 1894) implies that Gy b Ax has a solution y if and only if v(b Ax) 0 for all v 0 such that vg = 0. 6 / 30
7 Fractional Cuts Gomory 1958 Consider a single constraint : S := {x Z n + : n j=1 a jx j = b}. Let b = b + f 0 where 0 < f 0 < 1, and a j = a j + f j where 0 f j < 1. THEOREM j f jx j f 0 is a valid inequality for S. EQUIVALENT FORM j a j x j b. APPLICATION 3 x objective x 1 max z = x 1 + 2x 2 x 1 + x 2 2 x 1 + x 2 5 x 1 Z + x 2 R + z + 0.5x x 4 = 8.5 x 1 0.5x x 4 = 1.5 x x x 4 = 3.5 Cut or 0.5x x x / 30
8 Mixed Integer Cuts Gomory 1963 Consider a single constraint : S := {x Z p + Rn p + : n j=1 a jx j = b}. Let b = b + f 0 where 0 < f 0 < 1, and a j = a j + f j where 0 f j < 1. THEOREM j p: j p: f j 1 f j x j + x j + f 0 1 f 0 f j f 0 f j >f 0 is a valid inequality for S. j p+1: a j >0 j p+1: a j x j f 0 a j <0 a j 1 f 0 x j 1 NOTE The mixed integer cuts dominate the fractional cuts. Experiments of Bonami and Minoux 2005 on MIPLIB 3 instances give the amount of duality gap = min x S cx min x P cx closed by strengthening P with mixed integer cuts from the optimal basis : gap closed : 24 % 8 / 30
9 Yet, for thirty years, fractional cuts and mixed integer cuts were not used in MILP solvers. In 1991, Gomory remembered his experience with fractional cuts as follows : In the summer of 1959, I joined IBM research and was able to compute in earnest... We started to experience the unpredictability of the computational results rather steadily. In 1991, Padberg and Rinaldi made the following comments : These cutting planes have poor convergence properties... classical cutting planes furnish weak cuts... A marriage of classical cutting planes and tree search is out of the question as far as the solution of large-scale combinatorial optimization problems is concerned. 9 / 30
10 In 1989, Nemhauser and Wolsey had this to say : They do not work well in practice. They fail because an extremely large number of these cuts frequently are required for convergence. In 1985, Williams says : Although cutting plane methods may appear mathematically elegant, they have not proved very successful on large problems. In 1988, Parker and Rardin give the following explanation for this lack of success : The main difficulty has come, not from the number of iterations, but from numerical errors in computer arithmetic. 10 / 30
11 GOMORY CLOSURE Ax b x Z p + Rn p + Every valid inequality for P := {x 0 : Ax b} ( ) is of the form uax + vx ub t, where u, v, t 0. Subtract a nonnegative surplus variable αx s = β. Generate a Gomory inequality. Eliminate s = αx β to get the inequality in the x-space. The convex set obtained by intersecting all these inequalities with P is called the Gomory closure. THEOREM Cook, Kannan, Schrijver 1990 The Gomory closure is a polyhedron. THEOREM Caprara, Letchford 2002 et Cornuéjols, Li 2002 It is NP-hard to optimize a linear function over the Gomory closure. 11 / 30
12 Nevertheless, Balas and Saxena 2006 and Dash, Günlück and Lodi 2007 were able to optimize over the Gomory closure by solving a sequence of parametric MILPs. DUALITY GAP CLOSED BY DIFFERENT CUTS MIPLIB 3 Gomory cuts (optimal basis) 24 % Gomory closure 80 % 12 / 30
13 Split Inequalities Cook-Kannan-Schrijver 1990 P := {x R n : Ax b} S := P (Z p R n p ). For π Z n such that π p+1 =... = π n = 0 and π 0 Z, define πx π 0 πx π split inequality Π 1 := P {x : πx π 0 } Π 2 := P {x : πx π 0 + 1} Π 1 P Π 2 We call cx c 0 a split inequality if there exists (π, π 0 ) Z p Z such that cx c 0 is valid for Π 1 Π 2. The split closure is the intersection of all split inequalities. THEOREM Nemhauser-Wolsey 1990, Cornuéjols-Li 2002 The split closure is identical to the Gomory closure. 13 / 30
14 Chvátal Inequalities Chvátal 1973 A Chvátal inequality is a split inequality where Π 2 =. πx π 0 πx π Π 1 P The Chvátal closure is the intersection of all these inequalities. REMARK Chvátal defined this concept in 1973 in the context of pure integer programs. The Chvátal closure reduces the duality gap by around 63 % on the pure integer MIPLIB 03 instances (Fischetti-Lodi 2006) and around 28 % on the mixed instances (Bonami-Cornuéjols- Dash-Fischetti-Lodi 2007). 14 / 30
15 Lift-and-Project Sherali-Adams 1990 Lovász-Schrijver 1991 Balas-Ceria-Cornuéjols 1993 Let S := {x {0, 1} p R n p + : Ax b} P := {x R n + : Ax b} LIFT-AND-PROJECT PROCEDURE STEP 0 Choose an index j {1,..., p}. STEP 1 Generate the nonlinear system x j (Ax b) 0 (1 x j )(Ax b) 0 STEP 2 Linearize the system by substituting x i x j by y i for i j, and x 2 j by x j. Denote this polyhedron by M j. STEP 3 Project M j on the x-space. Denote this polyhedron by P j. PROPOSITION Conv(S) P j P. 15 / 30
16 THEOREM ( Ax b P j = Conv{ x j = 0 ) ( Ax b x j = 1 ) }. Ax b Ax b xj = 0 Pj Ax b xj = 1 xj 0 1 THEOREM Balas 1979 Conv(S) = P p (... P 2 (P 1 )...). LIFT-AND-PROJECT CUT Given a fractional solution x of the linear relaxation Ax b, find a cutting plane αx β (namely α x < β) that is valid for P j (and therefore for S ). max β α x DEEPEST CUT αx β valid for P j 16 / 30
17 CUT GENERATION LINEAR PROGRAM M j := {x R n +, y R n + : Ay bx j 0, Ax + bx j Ay b, y j = x j } In fact, one does not use the variable y j To project onto the x-space, we use the cone Q := {u, v 0 : ua j va j = 0} The first two constraints come from the linearization in STEP 1. M j := {x R n +, y R n 1 + : B j x + A j y 0, D j x A j y b} P j = {x R n + : (ub j + vd j )x vb for all (u, v) Q} DEEPEST CUT max vb (ub j + vd j ) x ua j va j = 0 u 0, v 0 ui + v i = 1 17 / 30
18 SIZE OF THE CUT GENERATION LP max vb (ub j + vd j ) x ua j va j = 0 ui + v i = 1 u 0, v 0 Number of variables : 2m Number of constraints : n + nonnegativity Balas and Perregaard 2003 give a precise correspondance between the basic feasible solutions of the cut generation LP and the basic min cx solutions of the LP Ax b Ax b 0 1 x j 18 / 30
19 LIFT-AND-PROJECT CLOSURE OF P 1 x 2 F := p j=1 P j P 1 F P 2 P x REMARK Balas and Jeroslow 1980 show how to strengthen cutting planes by using the integrality of the other integer variables (lift-and-project only considers the integrality of one x j at a time). Experiments of Bonami and Minoux 2005 on MIPLIB03 instances : Gap closed Lift-and-project closure 37 % Lift-and-project + strengthening 45 % 19 / 30
20 Duality gap closed by different types of cutting planes MIPLIB 3 instances 28% 63% Chvatal MIR 23% MIR heuristic Reduce and split ~30 % Gomory Split Lift and project + strengthening Gomory from 24% 80 % the optimal basis 45 % Lift and project 37 % All these cuts are generated from integrality arguments applied to one linear equation. Can we generate deeper cuts by considering several equations? 20 / 30
21 Corner Polyhedron [Gomory 1969] Relax nonnegativity on basic variables x j. In our current work [Basu, Bonami, Borozan, Conforti, Cornuéjols, Margot, Zambelli 2009], we make a further relaxation : Relax integrality on nonbasic variables. f r 1 x = f + k j=1 r j s j x Z q s 0 Example ) r 2 x1 Feasible set {( x 2 ( x1 x 2 Z 2 : ) = f + r 1 s 1 + r 2 s 2 where s 1 0, s 2 0} 21 / 30
22 Intersection Cuts [Balas 1971] Assume f Z q. Want to cut off the basic solution s = 0, x = f. r 2 S r 1 f Any convex set S with f int(s) with no integer point in int(s). 22 / 30
23 Intersection Cuts [Balas 1971] Assume f Z q. Want to cut off the basic solution s = 0, x = f. r 2 S r 1 f intersection cut Any convex set S with f int(s) with no integer point in int(s). Compute intersection of the rays with the boundary of S. Cut defined by these points is valid : ψ(r 1 )s 1 + ψ(r 2 )s / 30
24 A Better Intersection Cut [Balas 1971] r 2 S r 1 f Bigger convex set : Octahedron f int(s) with no integral point in int(s). 24 / 30
25 A Better Intersection Cut [Balas 1971] r 2 S r 1 f intersection cut Bigger convex set : Octahedron f int(s) with no integral point in int(s). Better cut : ψ(r 1 )s 1 + ψ(r 2 )s / 30
26 Maximal Lattice-Free Convex Sets in the Plane Split, triangles and quadrilaterals f f f generate split, triangle and quadrilateral inequalities ψ(r)s r / 30
27 Split closure := the intersection of all split inequalities. Triangle closure := the intersection of all inequalities arising from maximal lattice-free triangles. Quadrilateral closure := the intersection of all inequalities arising from maximal lattice-free quadrilaterals. Since all the facets of Integer Hull are induced by these three families of maximal lattice-free convex sets (Andersen, Louveaux, Wolsey, Weismantel 2007), we have Integer Hull = Split closure Triangle closure Quad closure The split closure is a polyhedron (Cook, Kannan and Schrijver) but such a result is not known for the triangle closure and the quadrilateral closure. THEOREM Basu, Bonami, Cornuéjols, Margot 2008 Triangle closure Split closure and Quad closure Split closure. 27 / 30
28 A theorem of Goemans 1995 Let Q := {x : a i x b i for i = 1,..., m} R n + \ {0} where a i 0 and b i 0 for all i. For α > 0 let αq := {x : αa i x b i for i = 1,..., m}. THEOREM If convex set P R n + contains Q, then the smallest value of α 1 such that P αq is { } b i max i=1,...,m inf{a i x : x P} : b i > 0. In other words, the only directions that need to be considered to compute α are those defined by the nontrivial facets of Q. Q P αq 28 / 30
29 Relative strength of closures Basu, Bonami, Cornuéjols, Margot 2008 Both the triangle closure and the quadrilateral closure are good approximations of the integer hull : THEOREM Integer Hull Triangle closure 2 (Integer Hull) and Integer Hull Quad closure 2 (Integer Hull) We also show that the split closure may not be a good approximation of the integer hull : THEOREM For any α > 1, there is a choice of f, r 1,..., r k such that Split closure α (Integer hull). 29 / 30
30 Papers available on http ://integer.tepper.cmu.edu/ 30 / 30
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