# Decomposition and parallel solution of network-structured optimization problems

#### Abstract

Optimization problems characterized by an embedded network structure in the constraint matrix are frequently used as a modeling tool in many diverse application areas such as transportation, logistics, finance, telecommunications and so on. We develop techniques that exploit the special structure present in this class of problems. A solution methodology where we propose to place the complicating or side constraints into the objective function using the 1-norm penalty function is developed. Thus we obtain a nondifferentiable penalty problem with network constraints. We develop smoothing techniques for the 1-norm penalty function and establish their properties. By using a quadratic smoothing term we obtain a nonlinear nonseparable problem with network constraints. The penalty problem is solved iteratively using a decomposition technique based on a simplicial decomposition of the network constraint set. This decomposition scheme induces separability in the objective function through linearization in the subproblem phase and a nonlinear nonseparable master problem is solved based on the information obtained from the subproblem phase. We develop two specializations of the algorithm: (1) for the network flow problem with side constraints, (2) for the multicommodity flow problem. We present numerical results with Patient Distribution System (PDS) multicommodity flow problems and with network flow problems with side constraints derived from matrix estimation problems and the NETLIB Linear Programming Library problems. The decomposition is particularly suitable for vector multiprocessor systems. We develop a parallel implementation of the linear-quadratic penalty algorithm. Numerical results with a set of large linear multicommodity network flow problems drawn from a military planning application are presented. The impact of parallel decomposition is investigated using a CRAY Y-MP supercomputer system and a Connection Machine CM-2. The parallelism is exploited both at the tightly coupled linear algebra level in the master problem and at a loosely coupled level with the network subproblems. Data-level parallel computing is explored on a massively parallel SIMD system, the Connection Machine CM-2. As alternatives to simplicial decomposition, two decomposition techniques (1) based on the truncated Newton algorithm (2) on a cyclic decomposition are also considered. The truncated Newton based decomposition is developed for the single commodity case. We conclude the thesis with an application from Naval Personnel Assignment formulated as a large network model with side constraints and solved using the penalty algorithm.

#### Subject Area

Operations research|Computer science

#### Recommended Citation

Pinar, Mustafa Celebi, "Decomposition and parallel solution of network-structured optimization problems" (1992). *Dissertations available from ProQuest*. AAI9227743.

https://repository.upenn.edu/dissertations/AAI9227743