Linear Programming Decomposition Research Articles

Filtering Algorithms for Biobjective Mixed Binary Linear Optimization Problems with a Multiple-Choice Constraint

This paper deals with a class of biobjective mixed binary linear programs having a multiple-choice constraint, which are found in applications such as Pareto set–reduction problems, single-supplier selection, and investment decisions, among others. Two objective space–search algorithms are presented. The first algorithm, termed line search and linear programming filtering, is a two-phase procedure. Phase 1 searches for supported Pareto outcomes using the parametric weighted sum method, and Phase 2 searches for unsupported Pareto outcomes by solving a sequence of auxiliary mixed binary linear programs. An effective linear programming filtering procedure excludes any previous outcomes found to be dominated. The second algorithm, termed linear programming decomposition and filtering, decomposes the mixed binary problem by iteratively fixing binary variables and uses the linear programming filtering procedure to prune out any dominated outcomes. Computational experiments show the effectiveness of the linear programming filtering and suggest that both algorithms run faster than existing general-purpose objective space–search procedures.

A linear programming decomposition focusing on the span of the nondegenerate columns

The improved primal simplex (IPS) was recently developed by Elhalaloui et al. to take advantage of degeneracy when solving linear programs with the primal simplex. It implements a dynamic constraint reduction based on the compatible columns, i.e., those that belong to the span of a given subset of basic columns including the nondegenerate ones. The identification of the compatible variables may however be computationally costly and a large number of linear programs are solved to enlarge the subset of basic variables. In this article, we first show how the positive edge criterion of Raymond et al. can be included in IPS for a fast identification of the compatible variables. Our algorithm then proceeds through a series of reduction and augmentation phases until optimality is reached. In a reduction phase, we identify compatible variables and focus on them to make quick progress toward optimality. During an augmentation phase, we compute one greatest normalized improving direction and select a subset of variables that should be considered in the reduced problem. Compared with IPS, the linear program that is solved to find this direction involves the data of the original constraint matrix. This new algorithm is tested over Mittelmann’s benchmark for linear programming and on instances arising from industrial applications. The results show that the new algorithm outperforms the primal simplex of CPLEX on most highly degenerate instances in which a sufficient number of nonbasic variables are compatible. In contrast, IPS has difficulties on the eleven largest Mittelmann instances.

A class of Dantzig–Wolfe type decomposition methods for variational inequality problems

We consider a class of decomposition methods for variational inequalities, which is related to the classical Dantzig–Wolfe decomposition of linear programs. Our approach is rather general, in that it can be used with certain types of set-valued or nonmonotone operators, as well as with various kinds of approximations in the subproblems of the functions and derivatives in the single-valued case. Also, subproblems may be solved approximately. Convergence is established under reasonable assumptions. We also report numerical experiments for computing variational equilibria of the game-theoretic models of electricity markets. Our numerical results illustrate that the decomposition approach allows to solve large-scale problem instances otherwise intractable if the widely used PATH solver is applied directly, without decomposition.

Assigning spare capacities in mesh survivable networks

This paper analyses one key issue of designing reliable networks: assignment of spare capacities in transmission networks. The spare capacities are optimized to facilitate the restoration of single failures. This problem can be formulated as an integer linear program and approximated by its continuous relaxation. This model is based on arc-path formulation especially efficient for dealing with end-to-end rerouting and providing appreciable economies in comparison with local rerouting. The main idea of our method resides in a linear programming decomposition, which permits us to solve problems for medium and large networks. Our approach could be applicable to both STM and ATM-based networks. This method was tested successfully on medium and large DCS-meshed networks and some numerical examples are given to illustrate its performances in terms of CPU time and ratio of optimality.

Annotated Bibliographies in Combinatorial Optimization.

Part I: General methodologies complexity and approximability polyhedral combinatorics branch-and-cut algorithms matroids and submodular functions advances in linear programming decomposition and column generation stochastic integer programming randomized algorithms local search graphs and matrices. Part II: Specific topics and applications sequencing and scheduling Travelling Salesman Problem max cut location problems network design flows and paths quadratic and 3-dimensional assignments linear assignment vehicle routing cutting and packing combinatorial topics in VLSI design applications in computational biology.

Using central prices in the decomposition of linear programs

This paper deals with a decomposition technique for linear programs which proposes a new treatment of the master program in the classical Dantzig-Wolfe algorithm. This approach exploits (a) the relationship between the master program and the minimization of a convex piecewise linear function and (b) a recently proposed cutting plane algorithm for convex nondifferentiable optimization. The new algorithm seeks to achieve ‘deep’ cuts by generating them at, or near to, the analytic centre of a set of localization containing the solution. Therefore each iteration entails a sizeable decrease of the set of localization and the overall algorithm shows fast convergence. This algorithm has been used to solve reputedly difficult convex nondifferentiable optimization problems. Its implementation, as a decomposition algorithm for large-scale structured linear program, seems to be a promising alternative to the standard Dantzig-Wolfe approach which suffers from very slow convergence in many practical problems. The new approach differs from the Dantzig-Wolfe algorithm through the fact that it does not use prices corresponding to an optimal combination of the past proposals of the subproblem, but rather ‘central’ prices (i.e. analytic centres) which balance them all. This property seems to explain the good behaviour of the new algorithm.

Decomposition of linear programs using parallel computation

This paper describes DECOMPAR: an implementation of the Dantzig-Wolfe decomposition algorithm for block-angular linear programs using parallel processing of the subproblems. The software is based on a robust experimental code for LP decomposition and runs on the CRYSTAL multicomputer at the University of Wisconsin-Madison. Initial computational experience is reported. Promising directions in future development of this approach are discussed.

A Decomposition Method Based On The Augmented Lagrangian

AbstractIn this paper, we explore the use of the augmented Lagrangian in a decomposition method which is a direct application of the “method of multipliers”. Contrary to popular opinion, the use of the augmented Lagrangian does not necessarily destroy separability inherent in a problem. This separability can be recovered by the use of sequential linearization algorithms, of which the Frank-Wolfe algorithm is an example. We outline this decomposition approach in a fairly general framework and then specialize it to the decomposition of linear programs. Linear programs are, in at least one sense, the worst case for a method of multipliers approach since little is known about the rate of convergence of the dual multipliers in tMs case. We report some computational evidence from two linear programming problems related to forest management which indicates that empirical convergence rates can be quite satisfactory.

A Note on Garfinkel, Neebe and Rao's LP Decomposition for the p-Median Problem

The convergence problems of Garfinkel, Neebe and Rao's linear programming decomposition algorithm for the relaxed p-median problem are analyzed and traced back to the very degenerate nature of the LP formulation. Extensive computational experience with this algorithm is described. One surprising outcome of the tests conducted with the decomposition algorithm was that convergence was obtained more consistently when a random (and usually worse) initial solution was used instead of a “good” initial solution provided by heuristic methods.

Simulation of Decentralized Planning in Two Danish Organizations Using Linear Programming Decomposition

This paper describes the results of two studies where decentralized planning has been simulated. The coordination mechanisms employed were based on the Dantzig-Wolfe decomposition algorithm, and the Ten Kate decomposition algorithm, which are price directive and budget directive planning procedures, respectively. The importance of the results is that the data and the structure of the two planning problems come from real organizations—a cooperative slaughterhouse and a forecast model for Danish agriculture. Thus conclusions based on using a price-directive and a budget-directive planning method to simulate decentralized planning in these organizations have relevance to the potential value of using such methods in real organizations. The main conclusions concern the importance of and the difficulties in finding good starting strategies for the two methods and the impact of the organizational structure on the rate of improvement for the methods.

Implications of Resource Directive Allocation Models for Organizational Design

This paper develops theoretical results, for a class of coordination mechanisms for resource allocation in decentralized firms, using as a model the decomposition of linear programs. Of the two pure decentralized allocation mechanisms (price-directive and resource-directive), resource-directive mechanisms, based in concept upon rationing or budgeting procedures, have received less attention in the literature. This paper focuses on these mechanisms and reports that under rather mild conditions the use of the plausible and economically appealing rule of allocating a global resource so as to equalize its marginal value over alternate uses is almost surely doomed to failure. In particular it is found that (1) resource-directive decompositions induce a kind of structural degeneracy in the decentralized divisions; (2) the amount of degeneracy increases with the number of divisions and the number of scarce global resources; and (3) degeneracy “seeks its own level” in that degeneracies spread over the divisions. The implication of these results for those interested in organizational design is that caution should be observed in utilizing such a rule as equalizing marginal values, and that more complicated and costly procedures for effecting resource allocation may be required. Finally, some symmetries between price-directive and resource-directive mechanisms are summarized.

A PRACTICAL APPROACH TO THE LARGE‐SCALE FOREST SCHEDULING PROBLEM*

ABSTRACTA large‐scale forest cutting schedule problem involving 1166 forest units to be cut over a 24–year planning period is discussed. The problem is formulated as a generalized version of the basic transportation problem. The conversion procedure for such a problem to the standard transportation format is outlined. The proposed approach is then compared with a linear programming decomposition approach on the basis of operating results obtained and computer time required by each approach. It is shown that this new approach will solve a real world problem about 40 times faster than the usual linear programming decomposition approach.

Decomposition of Linear Programs by Direct Distribution

Decomposition of Linear Programs by Direct Distribution

Optimum Design of Complex Water Resource Projects

Permissible draft from reservoirs is related to reservoir capacity by draft-storage-frequency analysis and the resulting draft-storage curves are expressed in algebraic form suitable for the optimum design of the reservoir system by linear programming. The curves are general and can apply to a number of reservoirs together. The principle of decomposition of linear programs is used to divide complex systems into projects, at the same time considering interbasin links. The river basins are then optimized using operations research techniques on individual basins or projects. A mathematical model of the Vaal and Tugela river basins in South Africa is used as an example.

Decomposition of linear programs by dynamic programming

AbstractThe decomposition principle of Dantzig and Wolfe is a method for breaking large linear programs with a block diagonal structure into a set of smaller subprograms. An alternative decomposition scheme derived from a dynamic programming approach is proposed here. This results in a series of parametric linear subprograms whose recursive solution yields the solution to the original linear program.