Projected Chv´ atal-Gomory cuts for Mixed Integer Linear Programs Pierre Bonami CMU, USA Gerard Cornu´ ejols CMU, USA and LIF Marseille, France Sanjeeb Dash IBM T.J. Watson, USA Matteo Fischetti University of Padova, Italy Andrea Lodi University of Bologna, Italy alodi@deis.unibo.it Aussois X, January 9, 2006 A. Lodi, Projected Chv´ atal-Gomory cuts for Mixed Integer Linear Programs
Notation and background • Consider an Integer Linear Program (ILP) of the form: min { c T x : Ax ≤ b, x ≥ 0 integer } A. Lodi, Projected Chv´ atal-Gomory cuts for Mixed Integer Linear Programs 1
Notation and background • Consider an Integer Linear Program (ILP) of the form: min { c T x : Ax ≤ b, x ≥ 0 integer } and two associated polyhedra: R n P := { x ∈ I + : Ax ≤ b } P I := conv { x ∈ Z n + : Ax ≤ b } = conv ( P ∩ Z n ) A. Lodi, Projected Chv´ atal-Gomory cuts for Mixed Integer Linear Programs 1
Notation and background • Consider an Integer Linear Program (ILP) of the form: min { c T x : Ax ≤ b, x ≥ 0 integer } and two associated polyhedra: R n P := { x ∈ I + : Ax ≤ b } P I := conv { x ∈ Z n + : Ax ≤ b } = conv ( P ∩ Z n ) • A Chv´ atal-Gomory ( CG ) cut is a valid inequality for P I of the form: A. Lodi, Projected Chv´ atal-Gomory cuts for Mixed Integer Linear Programs 1
Notation and background • Consider an Integer Linear Program (ILP) of the form: min { c T x : Ax ≤ b, x ≥ 0 integer } and two associated polyhedra: R n P := { x ∈ I + : Ax ≤ b } P I := conv { x ∈ Z n + : Ax ≤ b } = conv ( P ∩ Z n ) • A Chv´ atal-Gomory ( CG ) cut is a valid inequality for P I of the form: ⌊ u T A ⌋ x ≤ ⌊ u T b ⌋ where u ∈ R m + is called the CG multiplier vector, and ⌊·⌋ denotes lower integer part. A. Lodi, Projected Chv´ atal-Gomory cuts for Mixed Integer Linear Programs 1
Notation and background • Consider an Integer Linear Program (ILP) of the form: min { c T x : Ax ≤ b, x ≥ 0 integer } and two associated polyhedra: R n P := { x ∈ I + : Ax ≤ b } P I := conv { x ∈ Z n + : Ax ≤ b } = conv ( P ∩ Z n ) • A Chv´ atal-Gomory ( CG ) cut is a valid inequality for P I of the form: ⌊ u T A ⌋ x ≤ ⌊ u T b ⌋ where u ∈ R m + is called the CG multiplier vector, and ⌊·⌋ denotes lower integer part. • The first Chv´ atal closure of P is defined as: P 1 := { x ≥ 0 : Ax ≤ b, ⌊ u T A ⌋ x ≤ ⌊ u T b ⌋ for all u ∈ I R m + } A. Lodi, Projected Chv´ atal-Gomory cuts for Mixed Integer Linear Programs 1
Notation and background • Consider an Integer Linear Program (ILP) of the form: min { c T x : Ax ≤ b, x ≥ 0 integer } and two associated polyhedra: R n P := { x ∈ I + : Ax ≤ b } P I := conv { x ∈ Z n + : Ax ≤ b } = conv ( P ∩ Z n ) • A Chv´ atal-Gomory ( CG ) cut is a valid inequality for P I of the form: ⌊ u T A ⌋ x ≤ ⌊ u T b ⌋ where u ∈ R m + is called the CG multiplier vector, and ⌊·⌋ denotes lower integer part. • The first Chv´ atal closure of P is defined as: P 1 := { x ≥ 0 : Ax ≤ b, ⌊ u T A ⌋ x ≤ ⌊ u T b ⌋ for all u ∈ I R m + } [Chv´ atal 1973] • P 1 is indeed a polyhedron, i.e., a finite number of CG cuts suffice to define it. A. Lodi, Projected Chv´ atal-Gomory cuts for Mixed Integer Linear Programs 1
Notation and background (cont.d) • Clearly, P I ⊆ P 1 ⊆ P . A. Lodi, Projected Chv´ atal-Gomory cuts for Mixed Integer Linear Programs 2
Notation and background (cont.d) • Clearly, P I ⊆ P 1 ⊆ P . • Chv´ atal-Gomory separation problem (CG-SEP) is NP-hard. [Eisenbrand 1999] A. Lodi, Projected Chv´ atal-Gomory cuts for Mixed Integer Linear Programs 2
Notation and background (cont.d) • Clearly, P I ⊆ P 1 ⊆ P . • Chv´ atal-Gomory separation problem (CG-SEP) is NP-hard. [Eisenbrand 1999] • Recently Fischetti & Lodi have shown that: Optimizing over P 1 is possible in practice via an MIP model (MIPping). A. Lodi, Projected Chv´ atal-Gomory cuts for Mixed Integer Linear Programs 2
Notation and background (cont.d) • Clearly, P I ⊆ P 1 ⊆ P . • Chv´ atal-Gomory separation problem (CG-SEP) is NP-hard. [Eisenbrand 1999] • Recently Fischetti & Lodi have shown that: Optimizing over P 1 is possible in practice via an MIP model (MIPping). P 1 is a good (often excellent) approximation of P I in practice. A. Lodi, Projected Chv´ atal-Gomory cuts for Mixed Integer Linear Programs 2
Notation and background (cont.d) • Clearly, P I ⊆ P 1 ⊆ P . • Chv´ atal-Gomory separation problem (CG-SEP) is NP-hard. [Eisenbrand 1999] • Recently Fischetti & Lodi have shown that: Optimizing over P 1 is possible in practice via an MIP model (MIPping). P 1 is a good (often excellent) approximation of P I in practice. CG cuts in the first closure have a nice numerical behavior and stability. A. Lodi, Projected Chv´ atal-Gomory cuts for Mixed Integer Linear Programs 2
Notation and background (cont.d) • Clearly, P I ⊆ P 1 ⊆ P . • Chv´ atal-Gomory separation problem (CG-SEP) is NP-hard. [Eisenbrand 1999] • Recently Fischetti & Lodi have shown that: Optimizing over P 1 is possible in practice via an MIP model (MIPping). P 1 is a good (often excellent) approximation of P I in practice. CG cuts in the first closure have a nice numerical behavior and stability. • Thus, the natural question is: What does it happen in the Mixed IP case? A. Lodi, Projected Chv´ atal-Gomory cuts for Mixed Integer Linear Programs 2
Notation and background (cont.d) • Clearly, P I ⊆ P 1 ⊆ P . • Chv´ atal-Gomory separation problem (CG-SEP) is NP-hard. [Eisenbrand 1999] • Recently Fischetti & Lodi have shown that: Optimizing over P 1 is possible in practice via an MIP model (MIPping). P 1 is a good (often excellent) approximation of P I in practice. CG cuts in the first closure have a nice numerical behavior and stability. • Thus, the natural question is: What does it happen in the Mixed IP case? • Of course, the natural answer would be using Gomory Mixed Integer cuts (GMI) (also known as MIR cuts and split cuts) but their separation is much more involved than CG separation: nobody knows a MIP model for GMI yet! A. Lodi, Projected Chv´ atal-Gomory cuts for Mixed Integer Linear Programs 2
Projected Chv´ atal-Gomory cuts • Our first order of business is to extend the classical definition of Chv´ atal-Gomory cuts to the mixed integer case, in such a way that the resulting separation problem remains a clean MIP. A. Lodi, Projected Chv´ atal-Gomory cuts for Mixed Integer Linear Programs 3
Projected Chv´ atal-Gomory cuts • Our first order of business is to extend the classical definition of Chv´ atal-Gomory cuts to the mixed integer case, in such a way that the resulting separation problem remains a clean MIP. • We then consider the MIP: min { c T x + f T y : Ax + Cy ≤ b, x ≥ 0 , x integral , y ≥ 0 } • with the two associated polyhedra: P ( x, y ) := { ( x, y ) ∈ R n + × R r + : Ax + Cy ≤ b } P I ( x, y ) := conv ( { ( x, y ) ∈ P ( x, y ) : x integral } ) A. Lodi, Projected Chv´ atal-Gomory cuts for Mixed Integer Linear Programs 3
Projected Chv´ atal-Gomory cuts • Our first order of business is to extend the classical definition of Chv´ atal-Gomory cuts to the mixed integer case, in such a way that the resulting separation problem remains a clean MIP. • We then consider the MIP: min { c T x + f T y : Ax + Cy ≤ b, x ≥ 0 , x integral , y ≥ 0 } • with the two associated polyhedra: P ( x, y ) := { ( x, y ) ∈ R n + × R r + : Ax + Cy ≤ b } P I ( x, y ) := conv ( { ( x, y ) ∈ P ( x, y ) : x integral } ) • and we project P ( x, y ) onto the space of x variables as: := { x ∈ R n + : there exists y ∈ R r P ( x ) + s.t. Ax + Cy ≤ b } A. Lodi, Projected Chv´ atal-Gomory cuts for Mixed Integer Linear Programs 3
Projected Chv´ atal-Gomory cuts • Our first order of business is to extend the classical definition of Chv´ atal-Gomory cuts to the mixed integer case, in such a way that the resulting separation problem remains a clean MIP. • We then consider the MIP: min { c T x + f T y : Ax + Cy ≤ b, x ≥ 0 , x integral , y ≥ 0 } • with the two associated polyhedra: P ( x, y ) := { ( x, y ) ∈ R n + × R r + : Ax + Cy ≤ b } P I ( x, y ) := conv ( { ( x, y ) ∈ P ( x, y ) : x integral } ) • and we project P ( x, y ) onto the space of x variables as: := { x ∈ R n + : there exists y ∈ R r P ( x ) + s.t. Ax + Cy ≤ b } = { x ∈ R n + : u k A ≤ u k b, k = 1 , . . . , K } A. Lodi, Projected Chv´ atal-Gomory cuts for Mixed Integer Linear Programs 3
Projected Chv´ atal-Gomory cuts • Our first order of business is to extend the classical definition of Chv´ atal-Gomory cuts to the mixed integer case, in such a way that the resulting separation problem remains a clean MIP. • We then consider the MIP: min { c T x + f T y : Ax + Cy ≤ b, x ≥ 0 , x integral , y ≥ 0 } • with the two associated polyhedra: P ( x, y ) := { ( x, y ) ∈ R n + × R r + : Ax + Cy ≤ b } P I ( x, y ) := conv ( { ( x, y ) ∈ P ( x, y ) : x integral } ) • and we project P ( x, y ) onto the space of x variables as: := { x ∈ R n + : there exists y ∈ R r P ( x ) + s.t. Ax + Cy ≤ b } = { x ∈ R n + : u k A ≤ u k b, k = 1 , . . . , K } =: { x ∈ R n + : ¯ Ax ≤ ¯ b } where u 1 , . . . , u K are the (finitely many) extreme rays of the projection cone { u ∈ R m + : u T C ≥ 0 T } . A. Lodi, Projected Chv´ atal-Gomory cuts for Mixed Integer Linear Programs 3
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