Knapsack Constraints Research Articles

On tightening cover induced inequalities

Here we describe computationally efficient procedures for tightening cover induced inequalities by using 0–1 knapsack constraints and, if available, cliques whose variables are included in the cover. An interesting application is the case where the cover is implied by the knapsack constraint. The tightening is achieved by increasing the coefficients of the cover inequality. The new constraint is 0–1 equivalent to and LP tighter than the original one. The computational complexity of the procedures is O( n log n), where n is the number of variables in the cover.

Algorithms for the solution of quadratic knapsack problems

We consider the problem of minimizing a quadratic function with a knapsack constraint. Quadratic knapsack problems have numerous applications, including the least distance problem, quadratic programming defined on the convex hull of a set of points, and the maximum clique problem. We propose and analyze three algorithms for solving quadratic knapsack problems. Two algorithms are based on recently developed interior point methods. The first solves convex quadratic problems, and the second computes a stationary point for the indefinite case. For both, computational results on a variety of test problems are presented. The third algorithm, based on simplicial partitioning and convex underestimating functions, can be used to compute the global optimum of indefinite quadratic knapsack problems.

A nonlinear minimax allocation problem with multiple knapsack constraints

We consider a nonlinear minimax allocation problem with multiple knapsack-type resource constraints. Each term in the objective function is a nonlinear, strictly decreasing and continuous function of a single variable. All variables are continuous and nonnegative. A previous algorithm for such problems repeatedly solves relaxed problems without the nonnegativity constraints. That algorithm is particularly efficient for certain nonlinear functions for which there are closed-form solutions for the relaxed problems; for other functions, however, the algorithm must employ search methods. We present a new algorithm that uses at each iteration simple-to-compute algebraic expressions to check optimality conditions, instead of solving the relaxed minimax problems. The new algorithm is therefore significantly more efficient for more general nonlinear functions.

A hybrid algorithm for the generalized assignment problem

A hybrid algorithm for the generalized assignment problem is presented. The algorithm is based on the generation of valid inequalities from single knapsack constraints and a generalized branch and bound scheme. The pivot and complement heuristic, together with fathoming tests, are also carried out during the solution procedure. Computational results are presented from 5 test problems of size 100 0/1-variables.

The generalized assignment problem: Valid inequalities and facets

Three classes of valid inequalities based upon multiple knapsack constraints are derived for the generalized assignment problem. General properties of the facet defining inequalities are discussed and, for a special case, the convex hull is completely characterized. In addition, we prove that a basic fractional solution to the linear programming relaxation can be eliminated by a facet defining inequality associated with an individual knapsack constraint.

(1,k)-configuration facets for the generalized assignment problem

class of facet defining inequalities for the generalized assignment problem is derived. These inequalities are based upon multiple knapsack constraints and are derived from (1,k)-configuration inequalities.

On the supermodular knapsack problem

In this paper we introduce binary knapsack problems where the objective function is nonlinear, and investigate their Lagrangean and continuous relaxations. Some of our results generalize previously known theorems concerning linear and quadratic knapsack problems. We investigate in particular the case in which the objective function is supermodular. Under this hypothesis, although the problem remains NP-hard, we show that its Lagrangean dual and its continuous relaxation can be solved in polynomial time. We also comment on the complexity of recognizing supermodular functions. The particular case in which the knapsack constraint is of the cardinality type is also addressed and some properties of its optimal value as a function of the right hand side are derived.

The linear multiple choice knapsack problem

The multiple choice knapsack problem is defined as a knapsack problem with additional mutually exclusive multiple choice constraints. We present an algorithm for solving the linear programming relaxation of the multiple choice knapsack problem. The method is based on systematically perturbing the right-hand side of the knapsack constraint and the application of a dual-simplex based strategy. Computational experience with the method is also presented.

A Lagrangean dual ascent algorithm for simple plant location problems

We propose to strengthen the separable Lagrangean relaxation of the Simple Plant Location Problem (SPLP) by using Benders inequalities generated during a Lagrangean dual ascent procedure. These inequalities are expressed in terms of the 0–1 variables only, and they can be used as knapsack constraints in the pure integer part of the Lagrangean relaxation. We show how coupling this technique with a good primal heuristic can substantially reduce integrality gaps.

Maximising Real-Valued Submodular Functions: Primal and Dual Heuristics for Location Problems

Recently it has been shown how various results on the behaviour of exact and approximate primal algorithms for certain discrete location problems can be obtained by studying the more general problem of maximising a submodular set function. Unfortunately, this more general model tells us little about an important aspect of the discrete location problems, their linear programming relaxations and the possible use of dual solutions in obtaining bounds. Here we examine a slightly more specialised model that again generalises the location problems of interest, but now also includes the continuous aspects of the problems missing from the earlier model, namely the problem: max{w(y): ∑j=1najyj ≤ b, 0 ≤ yj < 1} of maximising a real-valued nondecreasing piecewise linear, concave submodular function subject to a knapsack constraint. We show that a continuous greedy heuristic always attains at least (1 − e−1) × 100% of the optimal value, and that for the discrete problem an adapted greedy heuristic always attains 35% of the optimal value. Specialised to the capacitated and uncapacitated location problems these results permit us to strengthen earlier results, and to obtain new results on dual greedy heuristics.

双線形ナップサック問題の解法

This paper proposes a cutting plane algorithm for solving 0-1 bilinear knapsack problem (BK), a special kind of integer quadratic programming problem in which a bilinear function c^tx + d^ty + x^tCy is to be maximized subject to disjoint knapsack constraints on x and y. This problem has applications in bipartite matching, multi-attribute utility analysis, cutting stock problems, and so on. The main purpose of this paper is to demonstrate the practicality of cutting plane algorithm which is an elaboration of similar algorithms proposed for bilinear linear programming problems and convex maximization problems. It will be shown that a very effective cut can be generated without excessive amount of computation and that our approach is potentially advantageous over the standard linearization approach.