4 research outputs found
Automatic identification of embedded network rows in large-scale optimization models
The solution of a contemporary large-scale linear, integer, or mixed-integer programming problem is often facilitated by the exploitation of intrinsic special structure in the model. This paper deals with the problem of identifying embedded pure network rows within the coefficient matrix of such models and presents two heuristic algorithms for identifying such structure. The problem of identifying the maximum-size embedded pure network is shown to be among the class of NP-hard problems; therefore, the polynomially bounded, efficient algorithms presented here do not guarantee network sets of maximum size. However, upper bounds on the size of the maximum network set are developed and used to evaluate the algorithms. Computational tests with large-scale, real-world models are presented.Office of Naval Research, Code 434, Arlington, VAApproved for public release; distribution is unlimited
Fixed-Parameter Algorithms in Analysis of Heuristics for Extracting Networks in Linear Programs
We consider the problem of extracting a maximum-size reflected network in a
linear program. This problem has been studied before and a state-of-the-art SGA
heuristic with two variations have been proposed.
In this paper we apply a new approach to evaluate the quality of SGA\@. In
particular, we solve majority of the instances in the testbed to optimality
using a new fixed-parameter algorithm, i.e., an algorithm whose runtime is
polynomial in the input size but exponential in terms of an additional
parameter associated with the given problem.
This analysis allows us to conclude that the the existing SGA heuristic, in
fact, produces solutions of a very high quality and often reaches the optimal
objective values. However, SGA contain two components which leave some space
for improvement: building of a spanning tree and searching for an independent
set in a graph. In the hope of obtaining even better heuristic, we tried to
replace both of these components with some equivalent algorithms.
We tried to use a fixed-parameter algorithm instead of a greedy one for
searching of an independent set. But even the exact solution of this subproblem
improved the whole heuristic insignificantly. Hence, the crucial part of SGA is
building of a spanning tree. We tried three different algorithms, and it
appears that the Depth-First search is clearly superior to the other ones in
building of the spanning tree for SGA.
Thereby, by application of fixed-parameter algorithms, we managed to check
that the existing SGA heuristic is of a high quality and selected the component
which required an improvement. This allowed us to intensify the research in a
proper direction which yielded a superior variation of SGA
Automatic identification of embedded network rows in large-scale optimization models
The solution of a large-scale linear, integer, or mixed integer programming problem is often facilitated by the exploitation of special structure in the model. This paper presents heuristic algorithms for identifying embedded network rows within the coefficient matrix of such models. The problem of identifying a maximum-size embedded pure network is shown to be among the class of NP-hard problems. The polynomially-bounded, efficient algorithms presented here do not guarantee network sets of maximum size. However, upper bounds on the size of the maximum network set are developed and used to show that our algorithms identify embedded networks of close to maximum size. Computational tests with large-scale, real-world models are presented
Analysis of large scale linear programming problems with embedded network structures: Detection and solution algorithms
This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Linear programming (LP) models that contain a (substantial) network structure frequently
arise in many real life applications. In this thesis, we investigate two main questions; i) how an embedded network structure can be detected, ii) how the network structure can be exploited to create improved sparse simplex solution algorithms. In order to extract an embedded pure network structure from a general LP problem we
develop two new heuristics. The first heuristic is an alternative multi-stage generalised upper bounds (GUB) based approach which finds as many GUB subsets as possible. In order to identify a GUB subset two different approaches are introduced; the first is based on the notion of Markowitz merit count and the second exploits an independent set in the corresponding graph. The second heuristic is based on the generalised signed graph of the coefficient matrix. This heuristic determines whether the given LP problem is an entirely pure network; this is in contrast to all previously known heuristics. Using generalised signed graphs, we prove that the problem of detecting the maximum size embedded network structure within an LP problem is NP-hard. The two detection
algorithms perform very well computationally and make positive contributions to the
known body of results for the embedded network detection. For computational solution
a decomposition based approach is presented which solves a network problem with side constraints. In this approach, the original coefficient matrix is partitioned into the network and the non-network parts. For the partitioned problem, we investigate two alternative decomposition techniques namely, Lagrangean relaxation and Benders decomposition. Active variables identified by these procedures are then used to create
an advanced basis for the original problem. The computational results of applying these techniques to a selection of Netlib models are encouraging. The development and computational investigation of this solution algorithm constitute further contribution
made by the research reported in this thesis.This study is funded by the Turkish Educational Council and Mugla University