5,797 research outputs found
Exact Solution Methods for the -item Quadratic Knapsack Problem
The purpose of this paper is to solve the 0-1 -item quadratic knapsack
problem , a problem of maximizing a quadratic function subject to two
linear constraints. We propose an exact method based on semidefinite
optimization. The semidefinite relaxation used in our approach includes simple
rank one constraints, which can be handled efficiently by interior point
methods. Furthermore, we strengthen the relaxation by polyhedral constraints
and obtain approximate solutions to this semidefinite problem by applying a
bundle method. We review other exact solution methods and compare all these
approaches by experimenting with instances of various sizes and densities.Comment: 12 page
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Zero-one IP problems: Polyhedral descriptions & cutting plane procedures
A systematic way for tightening an IP formulation is by employing classes of linear inequalities that define facets of the convex hull of the feasible integer points of the respective problems. Describing as well as identifying these inequalities will help in the efficiency of the LP-based cutting plane methods. In this report, we review classes of inequalities that partially described zero-one poly topes such as the 0-1 knapsack polytope, the set packing polytope and the travelling salesman polytope. Facets or valid inequalities derived from the 0-1 knapsack and the set packing polytopes are algorithmically identifie
Algorithms for the continuous nonlinear resource allocation problem---new implementations and numerical studies
Patriksson (2008) provided a then up-to-date survey on the
continuous,separable, differentiable and convex resource allocation problem
with a single resource constraint. Since the publication of that paper the
interest in the problem has grown: several new applications have arisen where
the problem at hand constitutes a subproblem, and several new algorithms have
been developed for its efficient solution. This paper therefore serves three
purposes. First, it provides an up-to-date extension of the survey of the
literature of the field, complementing the survey in Patriksson (2008) with
more then 20 books and articles. Second, it contributes improvements of some of
these algorithms, in particular with an improvement of the pegging (that is,
variable fixing) process in the relaxation algorithm, and an improved means to
evaluate subsolutions. Third, it numerically evaluates several relaxation
(primal) and breakpoint (dual) algorithms, incorporating a variety of pegging
strategies, as well as a quasi-Newton method. Our conclusion is that our
modification of the relaxation algorithm performs the best. At least for
problem sizes up to 30 million variables the practical time complexity for the
breakpoint and relaxation algorithms is linear
A Weight-coded Evolutionary Algorithm for the Multidimensional Knapsack Problem
A revised weight-coded evolutionary algorithm (RWCEA) is proposed for solving
multidimensional knapsack problems. This RWCEA uses a new decoding method and
incorporates a heuristic method in initialization. Computational results show
that the RWCEA performs better than a weight-coded evolutionary algorithm
proposed by Raidl (1999) and to some existing benchmarks, it can yield better
results than the ones reported in the OR-library.Comment: Submitted to Applied Mathematics and Computation on April 8, 201
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