Cutting planes for integer programming based on lattice-free sets - PowerPoint PPT Presentation

Cutting planes for integer programming based on lattice-free sets Ricardo Fukasawa Department of Combinatorics & Optimization University of Waterloo November 28th, 2013 Retrospective Workshop on Discrete Geometry, Optimization, and Symmetry Lattice-free cuts Nov 28, 2013 1 / 37

Mixed Integer Programming (MIP): c T x min s.t. Ax ≤ b x ∈ Z p × R n − p Lattice-free cuts Nov 28, 2013 2 / 37

Cutting plane approach Mixed Integer Programming (MIP): c T x min s.t. Ax ≤ b − c T x x ∈ Z p × R n − p Lattice-free cuts Nov 28, 2013 3 / 37

Cutting plane approach Mixed Integer Programming (MIP): c T x min s.t. Ax ≤ b − c T x x ∈ Z p × R n − p Lattice-free cuts Nov 28, 2013 3 / 37

Cutting plane approach Mixed Integer Programming (MIP): c T x min s.t. Ax ≤ b − c T x x ∈ Z p × R n − p Lattice-free cuts Nov 28, 2013 3 / 37

Cutting plane approach Mixed Integer Programming (MIP): c T x min s.t. Ax ≤ b − c T x x ∗ Lattice-free cuts Nov 28, 2013 3 / 37

Cutting plane approach Mixed Integer Programming (MIP): c T x min s.t. Ax ≤ b − c T x π 1 x ≤ π 1 o x ∗ Lattice-free cuts Nov 28, 2013 3 / 37

Cutting plane approach Mixed Integer Programming (MIP): c T x min s.t. Ax ≤ b − c T x π 1 x ≤ π 1 o x ∗ Lattice-free cuts Nov 28, 2013 3 / 37

Cutting plane approach Mixed Integer Programming (MIP): c T x min s.t. Ax ≤ b − c T x π 1 x ≤ π 1 o π 2 x ≤ π 2 o x ∗ Lattice-free cuts Nov 28, 2013 3 / 37

Cutting plane approach Mixed Integer Programming (MIP): c T x min s.t. Ax ≤ b − c T x π 1 x ≤ π 1 o π 2 x ≤ π 2 o x ∗ Lattice-free cuts Nov 28, 2013 3 / 37

Cutting plane approach Mixed Integer Programming (MIP): c T x min s.t. Ax ≤ b − c T x π 1 x ≤ π 1 o π 2 x ≤ π 2 o π 3 x ≤ π 3 o x ∗ Lattice-free cuts Nov 28, 2013 3 / 37

Cutting plane approach Mixed Integer Programming (MIP): c T x min s.t. Ax ≤ b − c T x π 1 x ≤ π 1 o π 2 x ≤ π 2 o π 3 x ≤ π 3 o x ∗ Lattice-free cuts Nov 28, 2013 3 / 37

Cutting plane approach Mixed Integer Programming (MIP): c T x min s.t. Ax ≤ b − c T x π 1 x ≤ π 1 o π 2 x ≤ π 2 o π 3 x ≤ π 3 o Valid inequalities/ Cutting planes/ Cuts Lattice-free cuts Nov 28, 2013 3 / 37

Cutting plane approach Mixed Integer Programming (MIP): c T x min s.t. Ax ≤ b − c T x π 1 x ≤ π 1 o π 2 x ≤ π 2 o π 3 x ≤ π 3 o Valid inequalities/ Cutting planes/ Cuts Want “strongest possible” valid inequalities (facet-defining): Get the convex hull of feasible solutions Lattice-free cuts Nov 28, 2013 3 / 37

Cutting plane approach Mixed Integer Programming (MIP): c T x min s.t. Ax ≤ b − c T x π 1 x ≤ π 1 o π 2 x ≤ π 2 o π 3 x ≤ π 3 o Valid inequalities/ Cutting planes/ Cuts Want “strongest possible” valid inequalities (facet-defining): Get the convex hull of feasible solutions Typically hard: Relax the problem and get facet-defining inequalities for the relaxation Lattice-free cuts Nov 28, 2013 3 / 37

Most important cuts Most important cutting planes used by commercial solvers: The Gomory mixed-integer cut (GMI). The Mixed Integer Rounding cut (MIR). Knapsack Cover and Lifted Knapsack Cover cuts. Bixby et. al (1999), “Closing the GAP”: three most important cuts Solution time increases by a factor of 2 . 52 without GMI cuts. Solution time increases by a factor of 1 . 83 without MIR cuts. Solution time increases by a factor of 1 . 4 without knapsack covers. (geometric averages after comparing the relative performance of 9 different cutting planes on 106 problems with CPLEX 8.0) Lattice-free cuts Nov 28, 2013 4 / 37

Multiple-row cutting planes Assume that we have the optimal tableau of an LP relaxation of a MIP c T min ¯ N x N x B − ¯ A N x N = ¯ s.t. b (1) x ≥ 0 x ∈ Z p × R n − p Now one can, in addition do the following relaxations: Pick a subset of rows associated with basic integer variables 1 Relax the nonnegativity of the basic variables 2 c T min ¯ N x N b i , ∀ i ∈ B ′ ⊆ B a ij x j = ¯ x i − � s.t. j ∈ N ¯ (2) x N ≥ 0 x ∈ Z p × R n − p (Gomory ’69) Lattice-free cuts Nov 28, 2013 5 / 37

Corner polyhedron Intuitively, what we are doing is relaxing all constraints that are not tight at the current optimal LP solution. − c T x x ∗ Lattice-free cuts Nov 28, 2013 6 / 37

Corner polyhedron Intuitively, what we are doing is relaxing all constraints that are not tight at the current optimal LP solution. − c T x x ∗ Lattice-free cuts Nov 28, 2013 6 / 37

Corner polyhedron Intuitively, what we are doing is relaxing all constraints that are not tight at the current optimal LP solution. − c T x x ∗ Lattice-free cuts Nov 28, 2013 6 / 37

Corner polyhedron Intuitively, what we are doing is relaxing all constraints that are not tight at the current optimal LP solution. − c T x x ∗ Still allows us to derive cutting planes for x ∗ , but much simpler to analyze. Lattice-free cuts Nov 28, 2013 6 / 37

Multiple-row cutting planes Assume that we have the optimal tableau of an LP relaxation of a MIP c T min ¯ N x N x B − ¯ A N x N = ¯ s.t. b (3) x ≥ 0 x ∈ Z p × R n − p Now one can, in addition do the following relaxations: Pick a subset of rows associated with basic integer variables 1 Relax the nonnegativity of the basic variables 2 c T min ¯ N x N b i , ∀ i ∈ B ′ ⊆ B a ij x j = ¯ s.t. x i − � j ∈ N ¯ (4) x N ≥ 0 x i ∈ Z , ∀ i ∈ B ′ ⊆ B Lattice-free cuts Nov 28, 2013 7 / 37

Multiple-row cutting planes Assume that we have the optimal tableau of an LP relaxation of a MIP c T min ¯ N x N x B − ¯ A N x N = ¯ s.t. b (3) x ≥ 0 x ∈ Z p × R n − p Now one can, in addition do the following relaxations: Pick a subset of rows associated with basic integer variables 1 Relax the nonnegativity of the basic variables 2 Relax the integrality of the non-basic variables. 3 (Andersen, Louveaux, Weismantel, Wolsey ’07) c T min ¯ N x N b i , ∀ i ∈ B ′ ⊆ B a ij x j = ¯ s.t. x i − � j ∈ N ¯ (4) x N ≥ 0 x i ∈ Z , ∀ i ∈ B ′ ⊆ B Lattice-free cuts Nov 28, 2013 7 / 37

Multiple-row cutting planes Assume that we have the optimal tableau of an LP relaxation of a MIP c T min ¯ N x N x B − ¯ A N x N = ¯ s.t. b (3) x ≥ 0 x ∈ Z p × R n − p Now one can, in addition do the following relaxations: Pick a subset of rows associated with basic integer variables 1 Relax the nonnegativity of the basic variables 2 Relax the integrality of the non-basic variables. 3 (Andersen, Louveaux, Weismantel, Wolsey ’07) c T min ¯ N x N b i , ∀ i ∈ B ′ ⊆ B a ij x j = ¯ s.t. x i − � j ∈ N ¯ (4) x N ≥ 0 x i ∈ Z , ∀ i ∈ B ′ ⊆ B This motivates the study of the following relaxation: � k � ( x , s ) ∈ Z q × R k � R q f ( r 1 , . . . , r k ) = conv r j s j + : x = f + j =1 Lattice-free cuts Nov 28, 2013 7 / 37

Multiple-row cutting planes � k � ( x , s ) ∈ Z q × R k f ( r 1 , . . . , r k ) = conv � R q r j s j + : x = f + j =1 Lattice-free cuts Nov 28, 2013 8 / 37

Multiple-row cutting planes � k � ( x , s ) ∈ Z q × R k f ( r 1 , . . . , r k ) = conv � R q r j s j + : x = f + j =1 Remark: If we have a basic feasible solution, we are at the point ( x , s ) = ( f , 0). Lattice-free cuts Nov 28, 2013 8 / 37

Multiple-row cutting planes � k � ( x , s ) ∈ Z q × R k f ( r 1 , . . . , r k ) = conv � R q r j s j + : x = f + j =1 Remark: If we have a basic feasible solution, we are at the point ( x , s ) = ( f , 0). If f ∈ Z q , then we are done, since we are at an integer feasible solution (and hence there is no cut to generate). ∈ Z q . So we may assume f / Lattice-free cuts Nov 28, 2013 8 / 37

Intersection Cut A Z m -free convex set B is a convex set with x 2 f ∈ int ( B ) and int ( B ) ∩ Z m = ∅ . Call it lattice-free f x 1 Figure: Picture of the x -space ( m = 2) Lattice-free cuts Nov 28, 2013 9 / 37

Intersection Cut A Z m -free convex set B is a convex set with x 2 f ∈ int ( B ) and int ( B ) ∩ Z m = ∅ . Call it lattice-free B lattice-free convex set with f in its interior. B f x 1 Figure: Picture of the x -space ( m = 2) Lattice-free cuts Nov 28, 2013 9 / 37

Intersection Cut A Z m -free convex set B is a convex set with x 2 f ∈ int ( B ) and int ( B ) ∩ Z m = ∅ . Call it lattice-free B lattice-free convex set with f in its interior. B f For any r , let α r ∈ R such that f + α r r is on the boundary of B . x 1 r Figure: Picture of the x -space ( m = 2) Lattice-free cuts Nov 28, 2013 9 / 37

Intersection Cut A Z m -free convex set B is a convex set with x 2 f ∈ int ( B ) and int ( B ) ∩ Z m = ∅ . Call it lattice-free B lattice-free convex set with f in its interior. B f For any r , let α r ∈ R such that f + α r r is on the boundary of B . x 1 r 2 r Figure: Picture of the x -space ( m = 2) Lattice-free cuts Nov 28, 2013 9 / 37