Some Relationships between Disjunctive Cuts and Cuts based on S-free Convex Sets

Transcription

1 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 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 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 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.

5 1 MIR Procedure

6 6 Looking at MIR cuts Traditional MIR Set x + y + b, x Z, y + R +

7 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 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 f (b) y 1

9 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 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 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 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 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 f (b) y 1 b f(b) 1 f(b)

12 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 a b 0 a 12 a 4

13 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

14 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 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)??

16 2 Split Procedure

17 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

18 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 19 Maximal Lattice-free convex Sets [Lovász (1989)] [D., Wolsey (2007)]

20 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.

21 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 22 Unimodular vs Nonunimodular Crosses ( x1 x 2 ) = ( ) ( ) ( 1 y Now consider the non-unimodular cross set ) ( 11 y ) ( 1 y ) 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 y 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 23 Type 3 Triangles vs Cross Disjunctions

24 24 Type 3 Triangles vs Cross Disjunctions

25 25 Type 3 Triangles vs Cross Disjunctions

26 26 Type 3 Triangles vs Cross Disjunctions Not covered by two splits

27 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 28 Crooked Cross Cut Proposition All maximal lattice-free convex sets in R 2 are contained in crooked cross sets.

29 29 Consequences for canonical set Theorem All valid inequalities of the canonical set are obtained by disjunctions based on crooked cross set.

30 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 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 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 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 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 35 A partial answer 1 Crooked cross closure 2D lattice-free closure.

36 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 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 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 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.

40 2.5 Split Closure Again

41 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 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 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 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 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 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})

47 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.

48 3 Gomory Mixed Integer Cut

49 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 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 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 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 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 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 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 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 57 Thank You.