# Entropy based primal-dual algorithm for convex and linear cost transportation problems with serial and parallel implementations

## Abstract

In this paper we present a new class of sequential and parallel algorithms for transportation problems with linear and convex costs. First, we consider a capacitated transportation problem with an entropy type objective function. We show that this problem has some interesting properties, namely that its optimal solution verifies both the non negativity and capacity constraints. Then, we give a new solution method for this problem. The algorithm consists of a sequence of {open_quotes}balancing{close_quotes} iterations on the conservation of flow constraints which may be viewed as a generalization of the well known RAS algorithm for matrix balancing. Then we prove the convergence of this method and extend it to strictly convex and linear cost transportation problems. For differentiable convex costs we develop an adaptation where each projection is an entropy type capacitated transportation problem. For linear costs, we prove a triple equivalence between the entropy projection method, the proximal minimization approach (with our entropy type function) and an entropy barrier method. We give a convergence rate analysis for strongly convex costs and linear objectif functions. We show efficient implementations on both serial and parallel environments. Computational results indicate that this methods yields very encouraging results. We solve large problems withmore »

- Authors:

- Publication Date:

- OSTI Identifier:
- 35884

- Report Number(s):
- CONF-9408161-

TRN: 94:009753-0146

- Resource Type:
- Conference

- Resource Relation:
- Conference: 15. international symposium on mathematical programming, Ann Arbor, MI (United States), 15-19 Aug 1994; Other Information: PBD: 1994; Related Information: Is Part Of Mathematical programming: State of the art 1994; Birge, J.R.; Murty, K.G. [eds.]; PB: 312 p.

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATION; 99 MATHEMATICS, COMPUTERS, INFORMATION SCIENCE, MANAGEMENT, LAW, MISCELLANEOUS; TRAFFIC CONTROL; OPTIMIZATION; COMPUTERIZED SIMULATION; BUSES; PARALLEL PROCESSING; ALGORITHMS

### Citation Formats

```
Chabini, I, and Florian, M.
```*Entropy based primal-dual algorithm for convex and linear cost transportation problems with serial and parallel implementations*. United States: N. p., 1994.
Web.

```
Chabini, I, & Florian, M.
```*Entropy based primal-dual algorithm for convex and linear cost transportation problems with serial and parallel implementations*. United States.

```
Chabini, I, and Florian, M. 1994.
"Entropy based primal-dual algorithm for convex and linear cost transportation problems with serial and parallel implementations". United States.
```

```
@article{osti_35884,
```

title = {Entropy based primal-dual algorithm for convex and linear cost transportation problems with serial and parallel implementations},

author = {Chabini, I and Florian, M},

abstractNote = {In this paper we present a new class of sequential and parallel algorithms for transportation problems with linear and convex costs. First, we consider a capacitated transportation problem with an entropy type objective function. We show that this problem has some interesting properties, namely that its optimal solution verifies both the non negativity and capacity constraints. Then, we give a new solution method for this problem. The algorithm consists of a sequence of {open_quotes}balancing{close_quotes} iterations on the conservation of flow constraints which may be viewed as a generalization of the well known RAS algorithm for matrix balancing. Then we prove the convergence of this method and extend it to strictly convex and linear cost transportation problems. For differentiable convex costs we develop an adaptation where each projection is an entropy type capacitated transportation problem. For linear costs, we prove a triple equivalence between the entropy projection method, the proximal minimization approach (with our entropy type function) and an entropy barrier method. We give a convergence rate analysis for strongly convex costs and linear objectif functions. We show efficient implementations on both serial and parallel environments. Computational results indicate that this methods yields very encouraging results. We solve large problems with several million variables on a network of transputers and Sun workstations. For the linear case, the serial implementation is compared to some network simplex codes like RELAX and RNET. Computational experiments indicate that this algorithm can outperform both RELAX and RNET. The parallel implementations are analysed using especially a new measure of performance developed by the authors. The results demonstrate that this measure can give more information than the classical measure of speedup. Some unexpected behaviors are reported.},

doi = {},

url = {https://www.osti.gov/biblio/35884},
journal = {},

number = ,

volume = ,

place = {United States},

year = {1994},

month = {12}

}