Some Relationships between Disjunctive Cuts and Cuts based on S-free Convex Sets Sanjeeb Dash a Santanu S. Dey b Oktay Günlük a a Business Analytics and Mathematical Sciences Department, IBM T. J. Watson Research Center, Yorktown Heights, USA. b H. Milton Stewart Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, USA. July 2010.
2 A Fundamental Relationship in Mixed Integer Programming Theorem ([Nemhauser, Wolsey (1990)], [Cournuéjols, Li (2002)]) Mixed Integer Rounding (MIR) Closure = Gomory Mixed Integer Cuts (GMIC) Closure = Split Closure (1) Each of the cutting planes above is based on information from Z 1 ". Observation The convex hull of a MIP with one integer variable is given by the split closure.
3 A Fundamental Relationship in Mixed Integer Programming Theorem ([Nemhauser, Wolsey (1990)], [Cournuéjols, Li (2002)]) Mixed Integer Rounding (MIR) Closure = Gomory Mixed Integer Cuts (GMIC) Closure = Split Closure (1) Each of the cutting planes above is based on information from Z 1 ". Observation The convex hull of a MIP with one integer variable is given by the split closure. The goal here: To understand and establish generalization of (1) with respect to cuts based on information from Z 2 ".
4 Definitions: S-free and Lattice-free convex set Definition (Maximal S-free Convex Sets) Let S = P Z m where P is a rational polyhedron. A convex set K is called S-free (resp. lattice-free) convex set, if int(k ) S = (resp. int(k ) Z m = ). An S-free (resp. lattice-free) convex set K is called a maximal S-free (resp. lattice-free) convex set if K is not contained properly in another S-free (resp. lattice-free) convex set. Theorem (Lovász (1989), D., Wolsey (2009), Basu et al. (2009), Fukasawa, Günlük (2009), Moran, D. (2010)) Maximal S-free convex sets are polyhedron.
1 MIR Procedure
6 Looking at MIR cuts Traditional MIR Set x + y + b, x Z, y + R +
7 Looking at MIR cuts Traditional MIR Set x + y + b, x Z, y + R + x = b y + + y x Z, y +, y R +
8 Looking at MIR cuts Traditional MIR Set x + y + b, x Z, y + R + x = b y + + y x Z, y +, y R + The MIR cut (written differently) is: 1 f (b) y + 1 + 1 f (b) y 1
9 Looking at MIR cuts Set introduced by [Andersen et al. (2007)] Traditional MIR Set x + y + b, x Z, y + R + x = b y + + y x Z, y +, y R + The MIR cut (written differently) is: 1 f (b) y + 1 + 1 f (b) y 1 Two rows Canonical Set : n x 1 = b 1 + a 1i y i n x 2 = b 2 + a 2i y i x Z 2, y i R + i {1,...n} The cuts are based on lattice-free convex sets... [Cornuéjols, Margot (2008)] [Borozan, Cornuéjols (2008)]
10 Looking at MIR cuts Set introduced by [Andersen et al. (2007)] Traditional MIR Set x + y + b, x Z, y + R + Two rows Canonical Set : x = b y + + y x Z, y +, y R + The MIR cut (written differently) is: 1 f (b) y + + 1 1 f (b) y 1 n x 1 = b 1 + a 1i y i n x 2 = b 2 + a 2i y i x Z 2, y i R + i {1,...n} The cuts are based on lattice-free convex sets... [Cornuéjols, Margot (2008)] [Borozan, Cornuéjols (2008)] Both sets obtained from simplex tableau by relaxing: (1) Bounds on basic integer variables (2) Integrality of non-basic variables.
11 Intersection Cuts Intersection Cut [Balas (1971)] x = b y + + y x Z, y +, y R + The MIR cut (written differently) is: 1 f (b) y + 1 + 1 f (b) y 1 b f(b) 1 f(b)
Intersection Cuts II Intersection Cut [Balas (1971)] n x 1 = b 1 + a 1i y i n x 2 = b 2 + a 2i y i x Z 2, y i R + i {1,...n} Let K be a lattice-free convex set containing b in its interior. The cut n α i y i 1 obtained using K is of the form: { 0 if ai recc.cone(k ) α i = λ λ > 0, s.t. b + 1 λ a i bnd(k ) 2 a 2 1.5 a 3 1 0.5 b 0 a 12 a 4
13 2D lattice-free closure Definition (2D Lattice-free Cut Closure) Rewrite the MIP set as P := {(x, y) Z n 1 R n 2 + Ax + Gy = b} (by possibly adding slacks). Let λ 1, λ 2 R m such that λ 1 A, λ 2 A Z n 1 λ 1 Ax + λ 1 G y = λ 1 b }{{}}{{}}{{} z 1 g 1 b 1
2D lattice-free closure Definition (2D Lattice-free Cut Closure) Rewrite the MIP set as P := {(x, y) Z n 1 R n 2 + Ax + Gy = b} (by possibly adding slacks). Let λ 1, λ 2 R m such that λ 1 A, λ 2 A Z n 1 λ 1 Ax + λ 1 G y = λ 1 b }{{}}{{}}{{} z 1 g 1 b 1 P 2 (λ 1, λ 2 ) = {(z, y) Z 2 R n 2 + z 1 + g 1 y = b 1, z 2 + g 2 y = b 2, y 0} A general 2D lattice-free cut for the mixed-integer set P is an inequality αy 1 which can be obtained as a intersection cut from a 2D maximal lattice-free convex set applied to above set. The set of points of P LP that satisfy all general 2D lattice-free cut is called the 2D lattice-free closure.
15 Taking Stock... One row/one integer variable based MIR Set (MIR closure) GMIC Closure Split Cut Closure Two row/two integer variables based Canonical Set (2D Lattice-free cut closure)??
2 Split Procedure
17 Split cuts [Balas(1979)], [Cook et al.(1990)] Let P = {(x, y) Z n 1 R n 2 + Ax + Gy = b} be a mixed integer linear set and let P LP be its linear relaxation. Let π Z n 1 and γ Z. A split cut is an inequality valid for P LP {(x, y) πx γ} P LP {(x, y) πx γ + 1} split cut
Towards a generalization of split cuts Let π 1, π 2 Z n 1 and γ 1, γ 2 Z. A cross cut is an inequality valid for P LP {(x, y) π 1 x γ 1, π 2 x γ 2 } P LP {(x, y) π 1 x γ 1, π 2 x γ 2 + 1} P LP {(x, y) π 1 x γ 1 + 1, π 2 x γ 2 } P LP {(x, y) π 1 x γ 1 + 1, π 2 x γ 2 + 1} [Balas (ISMP 2009)] [Balas and Qualizza (2009)] [Li and Richard (2008)] 18
19 Maximal Lattice-free convex Sets [Lovász (1989)] [D., Wolsey (2007)]
20 2D Lattice-free convex cuts vs Unimodular Cross Cut Definition A unimodular cross cut is one where π 1, π 2 Z 2 and form a unimodular matrix.
2D Lattice-free convex cuts vs Unimodular Cross Cut Definition A unimodular cross cut is one where π 1, π 2 Z 2 and form a unimodular matrix. Related result first shown by [Balas (ISMP 2009)], [Balas and Qualizza (2009)] Proposition For the canonical set, all unimodular cross cuts are either split cuts, quadrilateral cuts or triangle cuts of type 1 or 2. Proposition For the canonical set, split cuts, quadrilateral cuts and triangle cuts of type 1 or 2 are dominated by unimodular cross cuts.
22 Unimodular vs Nonunimodular Crosses ( x1 x 2 ) = ( 1 4 1 2 ) ( 0 + 1 ) ( 1 y 1 + 1 Now consider the non-unimodular cross set ) ( 11 y 2 + 6 ) ( 1 y 3 + 2 ) y 4. {(x 1, x 2 ) R 2 : 0 x 1 + x 2 1} {(x 1, x 2 ) R 2 : 0 x 1 x 2 1}. (2) The inequality y 1 + y 2 + 14y 3 + 2y 4 1 is obtained by the above disjunction and cannot be obtained by any single unimodular cross cut. Observation Some cross cuts are not unimodular cross cuts.
23 Type 3 Triangles vs Cross Disjunctions 2 1.5 1 0.5 0 0.5 1 1 0.5 0 0.5 1 1.5 2
24 Type 3 Triangles vs Cross Disjunctions 2 1.5 1 0.5 0 0.5 1 1 0.5 0 0.5 1 1.5 2
25 Type 3 Triangles vs Cross Disjunctions 2 1.5 1 0.5 0 0.5 1 1 0.5 0 0.5 1 1.5 2
26 Type 3 Triangles vs Cross Disjunctions 2 1.5 1 Not covered by two splits 0.5 0 0.5 1 1 0.5 0 0.5 1 1.5 2
Crooked Cross Cut Let P = {(x, y) Z n 1 R n 2 + Ax + Gy = b} be a mixed integer linear set and let P LP be its linear relaxation. Let π 1, π 2 Z n 1 and γ 1, γ 2 Z. A crooked cross cut is an inequality valid for P LP {(x, y) π 1 x γ 1, (π 2 π 1 )x γ 2 γ 1 } P LP {(x, y) π 1 x γ 1, (π 2 π 1 )x γ 2 γ 1 + 1} P LP {(x, y) π 1 x γ 1 + 1, π 2 x γ 2 } P LP {(x, y) π 1 x γ 1 + 1, π 2 x γ 2 + 1}
28 Crooked Cross Cut Proposition All maximal lattice-free convex sets in R 2 are contained in crooked cross sets.
29 Consequences for canonical set Theorem All valid inequalities of the canonical set are obtained by disjunctions based on crooked cross set.
30 A generalization of Split Closure Definition (Crooked Cross Closure) For the mixed-integer set P := {(x, y) Z n 1 R n 2 + Ax + Gy = b} the crooked cross closure is defined as P LP {(x, y) π 1 x γ 1, (π 2 π 1 )x γ 2 γ 1 } conv P LP {(x, y) π 1 x γ 1, (π 2 π 1 )x γ 2 γ 1 + 1} P LP {(x, y) π π 1 Z n 1,π 2 Z n 1 x γ 1 + 1, π 2 x γ 2 } 1,γ 1 Z,γ 2 Z P LP {(x, y) π 1 x γ 1 + 1, π 2 x γ 2 + 1} Theorem Let P := {(x, y) Z n 1 R n 2 + Ax + Gy = b}. If rank(a) = 2, then the crooked cross closure is the convex hull of P.
31 A generalization of Split Closure Definition (Crooked Cross Closure) For the mixed-integer set P := {(x, y) Z n 1 R n 2 + Ax + Gy = b} the crooked cross closure is defined as P LP {(x, y) π 1 x γ 1, (π 2 π 1 )x γ 2 γ 1 } conv P LP {(x, y) π 1 x γ 1, (π 2 π 1 )x γ 2 γ 1 + 1} P LP {(x, y) π π 1 Z n 1,π 2 Z n 1 x γ 1 + 1, π 2 x γ 2 } 1,γ 1 Z,γ 2 Z P LP {(x, y) π 1 x γ 1 + 1, π 2 x γ 2 + 1} Theorem Let P := {(x, y) Z n 1 R n 2 + Ax + Gy = b}. If rank(a) = 2, then the crooked cross closure is the convex hull of P. Corollary The convex hull of a MIP with two integer variable is given by the crooked cross closure. Corollary Let P := {x Z n Ax b} where A and b are integral and on removing two columns from A the remaining matrix is totally unimodular, then the convex hull of P is obtained by the crooked cross closure.
32 Some Other Sets in the Literature D., Wolsey (2009), Basu et al. (2009), Fukasawa, Günlük (2009) Two row Canonical Set + Constraints: n x 1 = b 1 + a 1i y i n x 2 = b 2 + a 2i y i where P is a rational polyhedron. x P Z 2, y i R + i {1,...n}, Let S = P Z 2. Let K be a maximal S-free convex set containing b in its interior, then we can generate facet-defining inequalities as follows: Let K f be written as a set {x (g j ) T x 1, j {1,..., l}}. Let π K (u) = max 1 j l {(g j ) T u}. Then the inequality n π K (r i )y i 1 Corollary The inequalities obtained by S-free convex sets are dominated by crooked cross cuts.
33 Taking Stock... One row/one integer variable based MIR Set (MIR closure) GMIC Closure Split Cut Closure Two row/two integer variables based Canonical Set (2D Lattice-free cut closure)? Crooked Cross Closure Convex hull of canonical set = 2D lattice-free closure = Crooked Cross Closure.
34 Taking Stock... One row/one integer variable based MIR Set (MIR closure) GMIC Closure Split Cut Closure Two row/two integer variables based Canonical Set (2D Lattice-free cut closure)? Crooked Cross Closure Convex hull of canonical set = 2D lattice-free closure = Crooked Cross Closure. Question: Is 2D lattice-free closure = Crooked Cross Closure for general MILPs?
35 A partial answer 1 Crooked cross closure 2D lattice-free closure.
36 A partial answer 1 Crooked cross closure 2D lattice-free closure. 2 Let (π 1, γ 1 ), (π 2, γ 2 ) Z n1+1 and consider the crooked cross disjunction D 1 = {(x, y) R n 1+n 2 : π 1 x γ 1, (π 2 π 1 )x (γ 2 γ 1 )}, D 2 = {(x, y) R n 1+n 2 : π 1 x γ 1, (π 2 π 1 )x γ 2 γ 1 + 1} D 3 = {(x, y) R n 1+n 2 : π 1 x γ 1 + 1, π 2 x γ 2 }, and, D 4 = {(x, y) R n 1+n 2 : π 1 x γ 1 + 1, π 2 x γ 2 + 1}. 3 We say that an inequality cx + dy f is a translation of c x + d y f w.r.t. P := {(x, y) Z n 1 R n 2 + Ax + Gy = b}, if there exists a vector µ Rm and a positive scalar θ such that [c, d, f ] = µ[a, G, b] + θ[c, d, f ].
37 A partial answer 1 Crooked cross closure 2D lattice-free closure. 2 Let (π 1, γ 1 ), (π 2, γ 2 ) Z n1+1 and consider the crooked cross disjunction D 1 = {(x, y) R n 1+n 2 : π 1 x γ 1, (π 2 π 1 )x (γ 2 γ 1 )}, D 2 = {(x, y) R n 1+n 2 : π 1 x γ 1, (π 2 π 1 )x γ 2 γ 1 + 1} D 3 = {(x, y) R n 1+n 2 : π 1 x γ 1 + 1, π 2 x γ 2 }, and, D 4 = {(x, y) R n 1+n 2 : π 1 x γ 1 + 1, π 2 x γ 2 + 1}. 3 We say that an inequality cx + dy f is a translation of c x + d y f w.r.t. P := {(x, y) Z n 1 R n 2 + Ax + Gy = b}, if there exists a vector µ Rm and a positive scalar θ such that [c, d, f ] = µ[a, G, b] + θ[c, d, f ]. Theorem (Three Rows) Let cx + dy f be a non-trivial crooked cross cut for P derived from the disjunction 4 D i. Then a translation of cx + dy f can be obtained as a crossed cross cut using the same disjunction from a 3-row relaxation of P, namely P 3 (λ 1, λ 2, λ 3 ) = {(x, y) Z n 1 R n 2 : π 1 x+g 1 y = b 1, π 2 x+g 2 y = b 2, g 3 y = b 3, y 0}, where λ 1, λ 2, λ 3 R m, π 3 = 0 and π i = λ i A, for i = 1, 2, 3 and g i = λ i G, b i = λ i b for i = 1, 2, 3.
38 A Corollary Corollary Let P = {(x, y) Z n 1 R n 2 + Ax + Gy = b}. If A has full row rank, then 2D lattice-free closure = Crooked Cross Closure.
39 A Corollary Corollary Let P = {(x, y) Z n 1 R n 2 + Ax + Gy = b}. If A has full row rank, then 2D lattice-free closure = Crooked Cross Closure. If P is a Corner relaxation of an MILP, then 2D lattice-free closure = Crooked Cross Closure for P.
2.5 Split Closure Again
41 A slightly different view of split closure 1 Let P = {(x, y) Z n 1 R n 2 + Ax + Gy = b} and PLP the LP relaxation.
42 A slightly different view of split closure 1 Let P = {(x, y) Z n 1 R n 2 + Ax + Gy = b} and PLP the LP relaxation. 2 Split closure 1 [Balas (1979)] π Z n 1,γ Z conv(p LP {(x, y) πx γ} P LP {(x, y) πx γ + 1})
43 A slightly different view of split closure 1 Let P = {(x, y) Z n 1 R n 2 + Ax + Gy = b} and PLP the LP relaxation. 2 Split closure 1 [Balas (1979)] π Z n 1,γ Z conv(p LP {(x, y) πx γ} P LP {(x, y) πx γ + 1}) 3 Split closure 2 [Cook et al. (1990)] π Z n 1 conv(p LP {(x, y) πx Z})
44 A slightly different view of split closure 1 Let P = {(x, y) Z n 1 R n 2 + Ax + Gy = b} and PLP the LP relaxation. 2 Split closure 1 [Balas (1979)] π Z n 1,γ Z conv(p LP {(x, y) πx γ} P LP {(x, y) πx γ + 1}) 3 Split closure 2 [Cook et al. (1990)] π Z n 1 conv(p LP {(x, y) πx Z}) 4 Split closure 1 = Split closure 2.
45 Two row/two integer variable case Parametric Cross Closure (t Z +) P t := π 1 Z n 1,π 2 Z n 1,γ 1 Z,γ 2 Z P 1 is crooked cross closure. conv P LP {(x, y) π 1 x γ 1, (π 2 tπ 1 )x γ 2 tγ 1 } P LP {(x, y) π 1 x γ 1, (π 2 tπ 1 )x γ 2 tγ 1 + 1} P LP {(x, y) π 1 x γ 1 + 1, π 2 x γ 2 } P LP {(x, y) π 1 x γ 1 + 1, π 2 x γ 2 + 1}
46 Two row/two integer variable case Parametric Cross Closure (t Z +) P t := π 1 Z n 1,π 2 Z n 1,γ 1 Z,γ 2 Z P 1 is crooked cross closure. conv P LP {(x, y) π 1 x γ 1, (π 2 tπ 1 )x γ 2 tγ 1 } P LP {(x, y) π 1 x γ 1, (π 2 tπ 1 )x γ 2 tγ 1 + 1} P LP {(x, y) π 1 x γ 1 + 1, π 2 x γ 2 } P LP {(x, y) π 1 x γ 1 + 1, π 2 x γ 2 + 1} P Π := π1,π 2 Z n 1 conv(p LP {(x, y) π 1 x, π 2 x Z})
Two row/two integer variable case Parametric Cross Closure (t Z +) P t := π 1 Z n 1,π 2 Z n 1,γ 1 Z,γ 2 Z P 1 is crooked cross closure. conv P LP {(x, y) π 1 x γ 1, (π 2 tπ 1 )x γ 2 tγ 1 } P LP {(x, y) π 1 x γ 1, (π 2 tπ 1 )x γ 2 tγ 1 + 1} P LP {(x, y) π 1 x γ 1 + 1, π 2 x γ 2 } P LP {(x, y) π 1 x γ 1 + 1, π 2 x γ 2 + 1} P Π := π1,π 2 Z n 1 conv(p LP {(x, y) π 1 x, π 2 x Z}) Theorem t Z +, P t P 1 = P Π.
3 Gomory Mixed Integer Cut
49 Monoidal Strengthening/Trivial Fill-in: From MIR to GMIC [Balas and Jeroslow (1984)] Let t i = argmin{t Z max{ t a i, f (b) Then rewrite as n 1 n 2 x B = b + a i z i + c i y i x B Z, z i Z +, y i R + a i +t 1 f (b) }}. n 1 n 1 n 2 x B t i z i = b + (a i t i )z i + c i y i x B Z, z i Z +, y i R + Aggregate all the integer terms on the left-hand-side.
50 Monoidal Strengthening/Trivial Fill-in: From MIR to GMIC [Balas and Jeroslow (1984)] Let t i = argmin{t Z max{ t a i, f (b) Then rewrite as n 1 n 2 x B = b + a i z i + c i y i x B Z, z i Z +, y i R + a i +t 1 f (b) }}. n 1 n 1 n 2 x B t i z i = b + (a i t i )z i + c i y i x B Z, z i Z +, y i R + Aggregate all the integer terms on the left-hand-side. Aggregate all variables with positive coefficients on the right-hand-side. Aggregate all variables with negative coefficients on the right-hand-side.
51 Monoidal Strengthening/Trivial Fill-in: From MIR to GMIC [Balas and Jeroslow (1984)] Let t i = argmin{t Z max{ t a i, f (b) Then rewrite as n 1 n 2 x B = b + a i z i + c i y i x B Z, z i Z +, y i R + a i +t 1 f (b) }}. n 1 n 1 n 2 x B t i z i = b + (a i t i )z i + c i y i x B Z, z i Z +, y i R + Aggregate all the integer terms on the left-hand-side. Aggregate all variables with positive coefficients on the right-hand-side. Aggregate all variables with negative coefficients on the right-hand-side. Apply MIR.
52 Monoidal Strengthening/Trivial Fill-in [Balas and Jeroslow (1984)] n 1 n 2 x 1 = b 1 + a 1i z i + c 1i y i n 2 n 2 x 2 = b 2 + a 1i z i + c 2i y i x Z 2, z i Z + i {1,...n 1 }, y i R + i {1,...n 2 } Given a lattice-free convex set K containing b, let { 0 if u recc.cone(k ) π(u) = λ λ > 0, s.t.b + 1 λ u bnd(k ) Let t i = argmin{t Z 2 π(t + a i )}. Then rewrite: Apply 2D lattice-free cut. n 1 n 1 n 2 x 1 t 1i z i = b 1 + (a 1i t 1i )z i + c 1i y i n 1 n 1 n 2 x 2 t 2i z i = b 2 + (a 2i t 2i )z i + c 2i y i
53 2D lattice-free cuts + Monoidal Strengthening closure 1 Rewrite the MILP set as P := {(z, y) Z n 1 2 Construct two row relaxation as + Rn 2 + Az + Gy = b}. Ez + Fy + d = 0, z Z n 1 +, y Rn 1, where E = [λ 1 ; λ 2 ]A, F = [λ 1 ; λ 2 ]G, d = [λ 1 ; λ 2 ]b and λ 1, λ 2 R 1 m. 3 Relax the set to x = Ez + Fy + d, x Z 2, z Z n 1 +, y Rn 1 4 Apply all possible 2D lattice-free closure + Monoidal Strengthening cut.
54 2D lattice-free closure = 2D lattice-free cut + Monoidal Strengthening closure 1 Let P := {(x, y) Z n 1 R n 2 + Ax + Gy = b}. Then (2D lattice-free cut + Monoidal Strengthening closure) (2D lattice-free closure), by rewriting P as P := {(x +, x, y) Z n 1 Z n 1 R n 2 + Ax+ Ax + Gy = b} 2 Let P := {(z, y) Z n 1 + R n 2 + Az + Gy = b}. Now can be rewritten as n 1 n 2 x 1 = b 1 + a 1i z i + c 1i y i (3) n 1 n 2 x 2 = b 2 + a 1i z i + c 2i y i (4) x Z 2, z i Z + i {1,...n 1 }, y i R + i {1,...n 2 } (5) n 1 n 2 x 1 = b 1 + a 1i z i + c 1i y i n 1 n 2 x 2 = b 2 + a 1i z i + c 2i y i z i s i = 0 i {1,...n 1 } x 1, x 2 Z, z i Z, s i, y i 0 Now by taking suitable combination of above system, every cut from the monoidal strengthening can be obtained using 2D lattice-free closure, i.e. (2D lattice-free closure + Monoidal Strengthening closure) (2D lattice-free closure)
55 Taking Stock... One row/one integer variable based Two row/two integer variables based MIR Set (MIR closure) Canonical Set (2D Lattice-free cut closure) GMIC Closure 2D Lattice-free cut + Monoidal Strengthening Split Cut Closure Crooked Cross Closure Theorem Let P be a mixed integer linear set. 2D Lattice-free cut closure = 2D Lattice-free cut + Monoidal Strengthening closure Crooked Cross Closure (6) Moreover, for the mixed-integer set P := {(x, y) Z n 1 R n 2 + Ax + Gy = b}: 1 If A has full row rank, then (6) holds at equality. 2 If rank(a) = 2, then the crooked cross closure is the convex hull of P.
56 Questions 1 Is the crooked cross closure or cross closure a polyhedron? 2 Is the crooked cross closure strictly contained in the cross closure?
57 Thank You.