An ϵ\epsilonϵ-Relaxation Method for Separable Convex Cost Network Flow Problems (original) (raw)

AN ε-RELAXATION METHOD FOR SEPARABLE CONVEX COST NETWORK FLOW PROBLEMS1

We propose a new method for the solution of the single commodity, separable convex cost network flow problem. The method generalizes the � -relaxation method developed for linear cost problems, and reduces to that method when applied to linear cost problems. We show that the method terminates with a near optimal solution, and we provide an associated complexity analysis. We also present computational results showing that the method is much faster than earlier relaxation methods, particularly for ill-conditioned problems.

An C-Relaxation Method for Separable Convex Cost Network Flow PROBLEMS1

We propose a new method for the solution of the single commodity, separable convex cost network flow problem. The method generalizes the e-relaxation method developed for linear cost problems, and reduces to that method when applied to linear cost problems. We show that the method terminates with a near optimal solution, and we provide an associated complexity analysis. We also present computational results showing that the method is much faster than earlier relaxation methods, particularly for ill-conditioned problems. 1 Research supported by NSF under Grant CCR-9103804 and Grant 9300494-DMI. 2 Dept. of Electrical Engineering and Computer Science, M.I.T., Rm. 35-210, Cambridge, Mass., 02139. Email: dimitrib@mit.edu and lcpolyme@lids.mit.edu 3 Dept. of Math., Univ. of Washington, Seattle, Wash., 98195. Email: tseng@math.washington.edu

ɛ-Relaxation and Auction Methods for Separable Convex Cost Network Flow Problems

Lecture Notes in Economics and Mathematical Systems, 1997

We propose two new methods for the solution of the single commodity, separable convex cost network flow problem: the-relaxation method and the auction/sequential shortest path method. Both methods were originally developed for linear cost problems and reduce to their linear conterparts when applied to such problems. We show that both methods stem from a common algorithmic framework, that they terminate with a near optimal solution, and we provide an associated complexity analysis. We also present computational results showing that these methods are much faster than earlier relaxation methods, particularly for ill-conditioned problems.

Relaxation Methods for Network Flow Problems with Convex Arc Costs

SIAM Journal on Control and Optimization, 1987

We consider the standard single commodity network flow problem with both linear and strictly convex possibly nondifferentiable arc costs. For the case where all arc costs are strictly convex we study the convergence of a dual Gauss-Seide! type relaxation method that is well suited for parallel computation. We then extend this method to the case where some of the arc costs are linear. As a special case we recover a relaxation method for the linear minimum cost network flow problem proposed in Bertsekas [1] and Bertsekas and Tseng .

Parallel Algorithms for Solving the Convex Minimum Cost Flow Problem

Computational Optimization and Applications, 2001

In this paper we deal with the solution of the separable convex cost network flow problem. In particular, we propose a parallel asynchronous version of the ∈-relaxation method and we prove theoretically its correctness. We present two implementations of the parallel method for a shared memory multiprocessor system, and we empirically analyze their numerical performance on different test problems. The preliminary numerical results show a good reduction of the execution time of the parallel algorithm with the respect to the sequential counterpart.

Solving Concave Network Flow Problems

2012

The Minimum Cost Network Flow Problem (MCNFP) includes a wide range of combinatorial optimization problems. Many applications exist for MCNFPs for instance supply chains, logistics, production planning, communications and transportations. Concave costs are, in many applications, more realistic than linear ones because of the association of prices with economies of scale. When concave costs are introduced in MCNFPs, then the difficulty to solve them increases and they become NP-Hard. Solution methods developed for these problems comprise both exact and approximate algorithms, the latter ones usually of a heuristic type. What we propose to do in this work is to present an overview of the past and most recent literature published on the subject.

Minimum concave-cost network flow problems: Applications, complexity, and algorithms

Annals of Operations Research, 1990

We discuss a wide range of results for minimum concave-cost network flow problems, including related applications, complexity issues, and solution techniques. Applications from production and inventory planning, and transportation and communication network design are discussed. New complexity results are proved which show that this problem is NP-hard for cases with cost functions other than fixed charge. An overview of solution techniques for this problem is presented, with some new results given regarding the implementation of a particular branch-and-bound approach.

Minimum cost network flows: Problems, algorithms, and software

Yugoslav Journal of Operations Research, 2013

We present a wide range of problems concerning minimum cost network flows, and give an overview of the classic linear single-commodity Minimum Cost Network Flow Problem (MCNFP) and some other closely related problems, either tractable or intractable. We also discuss state-of-the-art algorithmic approaches and recent advances in the solution methods for the MCNFP. Finally, optimization software packages for the MCNFP are presented.

Algorithms for the single-source uncapacitated minimum concave-cost network flow problem

Journal of Global Optimization, 1991

We investigate algorithms, applications, and complexity issues for the single-source uncapacitated (SSU) version of the minimum concave-cost network flow problem (MCNFP). We present applications arising from production planning, and prove complexity results for both global and local search. We formally state the local search algorithm of Gallo and Sodini [5], and present alternative local search algorithms. Computational results are provided to compare the various local search algorithms proposed and the effects of initial solution techniques.

On Minimum Concave Cost Network Flow Problems

2008

Minimum concave Cost Network Flow Problems (MCNFPs) arise naturally in many practical applications such as communication, transporta- tion, distribution, and manufacturing, due to economic considerations. In addition, it has been shown that every MCNFP with general nonlinear cost functions can be transformed into a concave MCNFP on an expanded network. It must also be noted, that multiple source and capacitated networks can be transformed into single source and uncapacitated networks. The main feature defining the complexity of MCNFPs is the type of cost function for each arc. Concave MCNFPs are known to be NP-hard even for the simplest version (i.e. fixed-charge single source and uncapacitated). The review presented in this work describes several approaches to the design of Single Source Uncapacitated (SSU) flow networks involving concave costs.