Robert Sarkissian

Solving nonlinear multicommodity flow problems by the analytic center cutting plane method

The paper deals with nonlinear multicommodity flow problems with convex costs. A decomposition method is proposed to solve them. The approach applies a potential reduction algorithm to solve the master problem approximately and a column generation technique to define a sequence of primal linear programming problems. Each subproblem consists of finding a minimum cost flow between an origin and a destination node in an uncapacited network. It is thus formulated as a shortest path problem and solved with Dijkstra’s d-heap algorithm. An implementation is described that takes full advantage of the supersparsity of the network in the linear algebra operations. Computational results show the efficiency of this approach on well-known nondifferentiable problems and also large scale randomly generated problems (up to 1000 arcs and 5000 commodities).

Parallel Interior Point Solver for Structured Linear Programs

Abstract. Issues of implementation of an object-oriented library for parallel interior-point methods are addressed. The solver can easily exploit any special structure of the underlying optimization problem. In particular, it allows a nested embedding of structures and by this means very complicated real-life optimization problems can be modelled. The efficiency of the solver is illustrated on several problems arising in the optimization of networks. The sequential implementation outperforms the state-of-the-art commercial optimization software. The parallel implementation achieves speed-ups of about 3.1-3.9 on 4-processors parallel systems and speed-ups of about 10-12 on 16-processors parallel systems.

ACCPM — A library for convex optimization based on an analytic center cutting plane method☆

Hardware information: Any computer with C++ and FORTRAN 77 compilers. Software information: C++ and FORTRAN 77.

Using an interior point method for the master problem in a decomposition approach

Abstract We address some of the issues that arise when an interior point method is used to handle the master problem in a decomposition approach. The main points concem the efficient exploitation of the special structure of the master problem to reduce the cost of a single interior point iteration. The particular structure is the presence of GUB constraints and the natural partitioning of the constraint matrix into blocks built of cuts generated by different subproblems. The method can be used in a fairly general case, i.e., in any decomposition approach whenever the master is solved by an interior point method in which the normal equations are used to compute orthogonal projections. Computational results demonstrate its advantages for one particular decomposition approach: Analytic Center Cutting Plane Method (ACCPM) is applied to solve large scale nonlinear multicommodity network flow problems (up to 5000 arcs and 10000 commodities)

Parallel Implementation of a Central Decomposition Method for Solving Large-Scale Planning Problems

We use a decomposition approach to solve three types of realistic problems: block-angular linear programs arising in energy planning, Markov decision problems arising in production planning and multicommodity network problems arising in capacity planning for survivable telecommunication networks. Decomposition is an algorithmic device that breaks down computations into several independent subproblems. It is thus ideally suited to parallel implementation. To achieve robustness and greater reliability in the performance of the decomposition algorithm, we use the Analytic Center Cutting Plane Method (ACCPM) to handle the master program. We run the algorithm on two different parallel computing platforms: a network of PCs running under Linux and a genuine parallel machine, the IBM SP2. The approach is well adapted for this coarse grain parallelism and the results display good speed-ups for the classes of problems we have treated.

Efficient Management of Multiple Sets to Extract Complex Structures from Mathematical Programs

Most of the applied models written with an algebraic modeling language involve simultaneously several dimensions such as materials, location, time or uncertainty. The information about dimensions available in the algebraic formulation is usually sufficient to retrieve different block structures from mathematical programs. These structured problems can then be solved by adequate solution techniques. To illustrate this idea we focus on stochastic programming problems with recourse. Taking into account both time and uncertainty dimensions of these problems, we are able to retrieve different customized structures in their constraint matrices. We applied the Structure Exploiting Tool to retrieve the structure from models built with the GAMS modeling language. The underlying mathematical programs are solved with the decomposition algorithm that applies interior point methods. The optimization algorithm is run in a sequential and in a parallel computing environment.

Customized Block Structures in Algebraic Modeling Languages: The Stochastic Programming Case 1

Abstract Extracting complex block structures from an anonymous mathematical program is a difficult task. It is however a mandatory step to exploit them with adequate algorithmic techniques. Moreover, most economic models are usually built with an Algebraic Modeling Language (AML) which loose any structure. The recently developed concept SET (Structure Exploiting Tool) responds to these needs. This approach relates directly to the “semantic” of the original model that was used to generate the corresponding anonymous mathematical program. Examples from stochastic programming are presented.