Convergent Lagrangian heuristics for nonlinear minimum cost network flows (original) (raw)
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A Lagrangean heuristic for the capacitated concave minimum cost network flow problem
European Journal of Operational Research, 1994
We propose a heuristic solution technique for the capacitated concave minimum cost network flow problem based on a Lagrangean dualization of the problem. Despite its dual character the algorithm guarantees the generation of primal feasible solutions which are local optima and therefore candidates of being the global optimum. The Lagrangean dual is solved by a subgradient search procedure and provides a lower bound to the optimal value. The lower bound is, in general, stronger than the one obtained by a linear approximation of the original problem. It can be used as a judgement of the quality of the solution or in a branch and bound procedure. Computational results from randomly generated problems are presented.
Heuristic solutions for general concave minimum cost network flow problems
Networks, 2007
We address the single-source uncapacitated minimum cost network flow problem with general concave cost functions. Exact methods to solve this class of problems in their full generality are only able to address small to medium size instances, since this class of problems is known to be NP-Hard. Therefore, approximate methods are more suitable. In this work, we present a hybrid approach combining a genetic algorithm with a local search. Randomly generated test problems have been used to test the computational performance of the algorithm. The results obtained for these test problems are compared to optimal solutions obtained by a dynamic programming method for the smaller problem instances and to upper bounds obtained by a local search method for the larger problem instances. From the results reported it can be shown that the hybrid methodology improves upon previous approaches in terms of efficiency and also on the pure genetic algorithm, i.e., without using the local search procedure.
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.
A Primal Algorithm for Finding Minimum-Cost Flows in Capacitated Networks With Applications
Bell System Technical Journal, 1982
Algorithms for finding a minimum-cost, single-commodity flow in a capacitated network are based on variants of the simplex method of linear programming. We describe an implementation of a primal algorithm which is fast and can solve large problems. The major ideas incorporated are (i) the sparsity ofthe network is used to reduce the time and computer storage space requirements; (ii) basic solutions are stored compactly as spanning trees of the network; (iii) a candidate stack is used to allow flexible strategies in choosing an arc to enter the basis tree; (iv) the predecessor and thread data structures are used to efficiently traverse the tree and to update the solution at each iteration; (v) rules are implemented to avoid cycling or stalling caused by degeneracy; and (vi)piecewise-linear, convex arc costs are handled implicitly. The Primal Network Flow Convex (PNFC) code implements this algorithm and three examples, from communication networks, that can be solved with PNFC are discussed: (i) solving the area transfer problem; (ii) scheduling the collection of traffic data records; and (iii) planning the placement ofpair-gain systems.
Optimization, 1997
It is well known that linear programs can generally not be solved by straightforward Lagrangean relaxation and dual subgradient optimization, the reason being that the solutions to the Lagrangean relaxed problems are, normally, infeasible in the original linear program. This property is a consequence of the linearity of the problem and it holds even in the unlikely case that an exact optimal dual solution is found. We show that an optimal solution to the linear program can be obtained by calculating the simple average of the solutions to the relaxed problems which are solved during the subgradient search scheme, provided that the steplengths in this scheme are chosen according to a modified harmonic series. This method is similar to a procedure given earlier by Shor, which utilizes a particular weighted average and holds for any divergent series steplength rule. As an application of these two averaging schemes, we construct an approximate algorithm for the minimum cost multicommodity network flow problem. The algorithm has been implemented for solving multicommodity transportation problems and shows a good performance. The computational results indicate that it may be possible to use Lagrangean relaxation and dual subgradient optimization for constructing efficient special-purpose solution methods for structured large-scale linear programs.
An epsilon\epsilonepsilon-Relaxation Method for Separable Convex Cost Network Flow Problems
SIAM Journal on Optimization, 1997
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.
Parallel primal-dual methods for the minimum cost flow problem
Computational Optimization and Applications, 1993
In this paper we discuss the parallel asynchronous implementation of the classical primaldual method for solving the linear minimum cost network flow problem. Multiple augmentations and price rises are simultaneously attempted starting from several nodes with possibly outdated price and flow information. The results are then merged asynchronously subject to rather weak compatibility conditions. We show that this algorithm is valid, terminating finitely to an optimal solution. We also present computational results using an Encore Multimax that illustrate the speedup that can be obtained by parallel implementation.
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