A Note on the MIR closure

Transcription

1 A Note on the MIR closure Pierre Bonami Tepper School of Business, Carnegie Mellon University, Pittsburgh PA 53, USA. Gérard Cornuéjols Tepper School of Business, Carnegie Mellon University, Pittsburgh PA 53, USA; and LIF, Faculté des Sciences de Luminy, 388 Marseille, France. Abstract In 988, Nemhauser and Wolsey introduced the concept of MIR inequality for mixed integer linear programs. In 998, Wolsey defined MIR inequalities differently. In some sense these definitions are equivalent. However, this note points out that the natural concepts of MIR closures derived from these two definitions are distinct. Dash, Günlük and Lodi made the same observation independently. Let S := {(x, y) Z n + R p + : Ax + Gy b} be a mixed integer set. Here A R m n and G R m p are matrices and b R m is a vector. Let P := {(x, y) R n + R p + : Ax + Gy b} be the polyhedron that arises as the natural linear relaxation of S. We assume P. Nemhauser and Wolsey [6,7] define MIR NW inequalities by the following procedure. If and c x + hy c 0 c x + hy c 0 addresses: pbonami@andrew.cmu.edu (Pierre Bonami ), gc0v@andrew.cmu.edu (Gérard Cornuéjols ). Supported in part by a grant from IBM. Supported in part by NSF grant DMI and ONR grant N Preprint submitted to Elsevier 3 August 006

2 are valid inequalities for P, and π = c c Z n, π 0 = c 0 c 0 and γ = c 0 c 0 π 0, then is valid for S. πx + ( c x + hy c 0) /( γ) π0 Define the MIR NW closure as the intersection of all MIR NW inequalities. Nemhauser and Wolsey [7] proved that the MIR NW closure is identical to the split closure [] and the Gomory mixed integer closure [4] (see [] for another proof of the last identity). Later, Wolsey [8] (see also Marchand and Wolsey [5]) defined the MIR W inequality as being generated from a single constraint ax + gy b where (x, y) Z n + R p +. Specifically, let f 0 := b b and f j := a j a j. The MIR W inequality is n ( a j + (f j f 0 ) + ) j= x j + g j y j b. () For the mixed integer set S, let us define the MIR W closure as the set of all MIR W inequalities that can be generated from any valid inequality for the polyhedron P. (By Farkas Lemma, every valid inequality for P is of the form uax + ugy vx wy ub + t where u R m +, v R n +, w R p + and t R +.) In this note, we point out that MIR NW MIR W and that the inclusion is strict in general. This was also observed independently by Dash, Günlük and Lodi [3]. First, we give an example showing that MIR NW triangle in R defined as follows MIR W. Let P be the x + x 0 x + x x 0. Let S := P Z. The inequality x 0 is a MIR NW inequality. Indeed x + 4 x 0 x + 4 x are valid for P. Applying the MIR NW procedure we get π =, π 0 = = 0

3 x P 0 x Fig.. Examle showing that MIR NW MIR W. 0, γ =. Thus the following inequality is valid for S. x + x + 4 x 0, i.e. x 0. Therefore x 0 is valid for the MIR NW closure. However x 0 is not valid for the MIR W closure. We show this by contradiction. Let us assume that there exists a valid inequality αx + βx δ for P such that x 0 is a MIR W inequality. By Farkas Lemma, there exist multipliers u, u, v, t 0 satisfying α = u + u β = u + u v δ = u + t. Let f(η) = η η. Can we generate x 0 as the MIR W inequality ( α + ) (f(α) f(δ))+ x + ( β + f(δ) For this to be the case, we must have δ < since δ = 0, α 0 since ( ) α + (f(α) f(δ))+ f(δ) = 0, δ < β since ( ) β + (f(β) f(δ))+ f(δ) > 0. ) (f(β) f(δ))+ x δ? f(δ) α 0 is equivalent to u u. Furthermore δ < β and v, t 0 imply u < u. This is a contradiction, therefore there exists no valid inequality for P such that x 0 is a MIR W inequality. 3

4 To see that MIR NW MIR W, we express Gomory Mixed Integer (GMI) inequalities in a form similar to (). Recall that given an equality ax + gy = b where (x, y) Z n + R p +, the GMI inequality is f j x j + f j x j + g j y j j:f j f 0 f 0 j:f j >f 0 j:g j >0 f 0 where f j and f 0 are defined as above. g j y j () Lemma Consider a mixed integer set with m constraints S := {(x, y) Z n + R p + : Ax + Gy b}. We assume that the constraints Ax + Gy b contain the nonnegativity constraints on x and y. Let s := b Ax Gy be a nonnegative vector of slack variables. For any λ R m, let a := λa, g := λg, δ := λb, f j := a j a j and f 0 := δ δ. The Gomory mixed integer inequality generated from λax + λgy + λs = λb is n ( a j + (f j f 0 ) + ) x j + j= g j y j + i:λ i <0 λ i s i δ. (3) Proof: Applying the definition () to λax + λgy + λs = λb we get f j x j + f j x j + j:f j f 0 f 0 j:f j >f 0 j:g j >0 g j y j g j y j + f 0 i:λ i >0 λ i f 0 s i i:λ i <0 Substituting s = b Ax Gy in this inequality, it is straightforward to check that the result is inequality (3). Recall that MIR W inequalities are obtained from valid inequalities for P. This corresponds to λ 0 in Lemma. In this case (3) is identical to (). Therefore MIR W inequalities are GMI inequalities. Other authors have defined the MIR W closure when S is in equality form [5,3], in which case it is trivially identical to the Gomory mixed integer closure. λ i s i. References [] W. Cook, R. Kannan and A. Schrijver, Chvátal closures for mixed integer programming problems, Mathematical Programming 47 (990) [] G. Cornuéjols and Y. Li, On the rank of mixed 0, polyhedra, Mathematical Programming 9 (00)

5 [3] S. Dash, O. Günlük and A. Lodi, Separating from the MIR closure of polyhedra, presentation at the International Symposium on Mathematical Programming, Rio de Janeiro, Brazil, 006, paper in preparation. [4] R.E. Gomory, An algorithm for integer solution solutions to linear programs, in Recent Advances in Mathematical Programming R.L. Graves and P. Wolfe, editors, McGraw-Hill, New York (963) [5] H. Marchand and L. A. Wolsey, Aggregation and mixed integer rounding to solve MIPs. Operations Research 49 (00) [6] G.L. Nemhauser and L.A. Wolsey, Integer and Combinatorial Optimization, John Wiley & Sons, New York, 988. [7] G.L. Nemhauser and L.A. Wolsey, A recursive procedure to generate all cuts for 0- mixed integer programs, Mathematical Programming 46 (990) [8] L.A. Wolsey, Integer Programming, John Wiley & Sons, New York,