145-153. On-Using an Automatic Scheme for Obtaining the. Convex Hull Defining Inequalities of a Weismantel. 0-1 Knapsack Constraint. Laureano F. Escudero.
Sociedad de Estadlstica e Investigacidn Operativa Top (1999) VoL 7, No. 1, pp. 145-153
On-Using an Automatic Scheme for Obtaining the Convex Hull Defining Inequalities of a Weismantel 0-1 Knapsack Constraint Laureano
F. Escudero
Dpto. de Estadlstica e LO., U.C.M., and Iberdrola lngenieria y Consultoria, Spain Garin DEAIII, Universidad del Pa(s Vaseo, Bilbao, Spain Gloria P~rez DMAEIO, Universidad del Pals Vasco, Bilbao, Spain
Araceli
Abstract 9 In this short note we obtain the full set of inequalities that define the convex hull of
a 0-1 knapsack constraint presented in Weismantel (1997). For that purpose we use our O(n) procedures for identifying maximal cliques and non-dominated extensions of consecutive minimal covers and alternates, as well as our schemes for coefficient increase based tightening cover induced inequalities and coefficient reduction based tightening general 0-1 knapsack constraints. K e y W o r d s : 0-1 program, cover, clique, knapsack constraint, coefficient increase, coefficient reduction. A M S s u b j e c t classification: 90C09, 90C10.
1
Introdution
Consider the 0-1 program
m a x { d x : A x _< b,x E {O, 1}m},
(1.1)
where x is the m-column vector of the 0-1 variables, d is the related row vector of the objective function coefficients, A is the coefficient m a t r i x of the constraints and b is the column vector of the right-hand-side (hereafter, rhs). All vectors and the m a t r i x are assumed to have the appropriate dimensions. Let I and J be the set of indices of the constraints and variables, respectively, a n d J / b e the set of indices of the variables with nonzero coefficient Received: December 1997; Accepted: March 1998
146
L.F. Escudero, A. Garin and G. Pdrez
in the i-th constraint for i E I. (The LP relaxation is the same system (1.1), where each xj is allowed to take any value in the range [0, 1]). We consider knapsack constraints of the form
arX ~_ br,
(1.2)
where ar is the r-th row vector of matrix A and br is the r-th element of rhs b. Let arj be the j - t h element of vector at; without loss of generality, we assume that arj and br are integer and strictly positive, arj _< brVj E Jr and ~-~deJ~arj > br for any r E I. Let n = ]Ji]. The knapsack polytope P is the convex hull of the 0-1 points satisfying (1.2). For any subset V C Ji, let Pv = con{x E {0, 1} Y / ~ j e y aijxj kc tec
Proof: See Escudero and Mufioz (1998).
jEC
(2.8)
L.F. Escudero, A. Garin and G. Pdrez
150
3
Weismantel
0-1 Knapsack
Constraint
T h e 0-1 knapsack constraint subject of our analysis is as follows, see Weismantel (1997). xl + x2 + x3 + x4 + 3x5 + 4x6 < 4
(3.1)
Table 1 shows the set of the inequalities t h a t define the convex hull of the 0-1 points satisfying (3.1). Xl
x2
x3
x4
55
56
1
1
