# Quadratic based primal-dual algorithms for multicommodity 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 multicommodity transportation problems with linear and convex costs. First, we consider a capacitated multicommodity transportation problem with an orthogonal quadratic objective function. We develop two new solution methods. Both exploit the fact that a projection on the conservation of flow constraints has an explicit form which was proved in an early paper by I. Chabini and M. Florian. The two algorithms deal differently with the remaining constraints namely the non negativity and capacity constraints. We prove the convergence of both algorithms using a basic general theory (developed by the author) which generalizes Bregman`s theory. The above algorithms can be extended for differentiable convex cost multicommodity transportation problems as follows. For strictly convex costs we use a variant of the projected gradient method. The quadratic proximal minimization algorithm is applied to the linear cost multicommodity transportation problems. For both cases we solve an orthogonal projection multicommodity transportation problem at each iteration. The algorithms developed are well-suited for a coarse grained parallelization. The different steps may be decomposed by nodes, by arcs and/or by commodities. We investigate different strategies depending on the structure of the problem, the numbermore »

- Authors:

- Publication Date:

- OSTI Identifier:
- 35886

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

TRN: 94:009753-0148

- 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.
```*Quadratic based primal-dual algorithms for multicommodity convex and linear cost transportation problems with serial and parallel implementations*. United States: N. p., 1994.
Web.

```
Chabini, I.
```*Quadratic based primal-dual algorithms for multicommodity convex and linear cost transportation problems with serial and parallel implementations*. United States.

```
Chabini, I. 1994.
"Quadratic based primal-dual algorithms for multicommodity convex and linear cost transportation problems with serial and parallel implementations". United States.
```

```
@article{osti_35886,
```

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

author = {Chabini, I},

abstractNote = {In this paper we present a new class of sequential and parallel algorithms for multicommodity transportation problems with linear and convex costs. First, we consider a capacitated multicommodity transportation problem with an orthogonal quadratic objective function. We develop two new solution methods. Both exploit the fact that a projection on the conservation of flow constraints has an explicit form which was proved in an early paper by I. Chabini and M. Florian. The two algorithms deal differently with the remaining constraints namely the non negativity and capacity constraints. We prove the convergence of both algorithms using a basic general theory (developed by the author) which generalizes Bregman`s theory. The above algorithms can be extended for differentiable convex cost multicommodity transportation problems as follows. For strictly convex costs we use a variant of the projected gradient method. The quadratic proximal minimization algorithm is applied to the linear cost multicommodity transportation problems. For both cases we solve an orthogonal projection multicommodity transportation problem at each iteration. The algorithms developed are well-suited for a coarse grained parallelization. The different steps may be decomposed by nodes, by arcs and/or by commodities. We investigate different strategies depending on the structure of the problem, the number of commodities and the architecture of the parallel machine. We present computational results for these different approaches on parallel and serial platforms such as a network of Transputers or Sun workstations. Very large problems are solved. The parallel implementations are analyzed using especially a new measure of performance developed previously by I. Chabini and M. Florian.},

doi = {},

url = {https://www.osti.gov/biblio/35886},
journal = {},

number = ,

volume = ,

place = {United States},

year = {1994},

month = {12}

}