Parallel problem generation for structured problems in mathematical programming

Abstract

The aim of this research is to investigate parallel problem generation for structured optimization problems. The result of this research has produced a novel parallel model generator tool, namely the Parallel Structured Model Generator (PSMG). PSMG adopts the model syntax from SML to attain backward compatibility for the models already written in SML [1]. Unlike the proof-of-concept implementation for SML in [2], PSMG does not depend on AMPL [3]. In this thesis, we firstly explain what a structured problem is using concrete real-world problems modelled in SML. Presenting those example models allows us to exhibit PSMG’s modelling syntax and techniques in detail. PSMG provides an easy to use framework for modelling large scale nested structured problems including multi-stage stochastic problems. PSMG can be used for modelling linear programming (LP), quadratic programming (QP), and nonlinear programming (NLP) problems. The second part of this thesis describes considerable thoughts on logical calling sequence and dependencies in parallel operation and algorithms in PSMG. We explain the design concept for PSMG’s solver interface. The interface follows a solver driven work assignment approach that allows the solver to decide how to distribute problem parts to processors in order to obtain better data locality and load balancing for solving problems in parallel. PSMG adopts a delayed constraint expansion design. This allows the memory allocation for computed entities to only happen on a process when it is necessary. The computed entities can be the set expansions of the indexing expressions associated with the variable, parameter and constraint declarations, or temporary values used for set and parameter constructions. We also illustrate algorithms that are important for delivering efficient implementation of PSMG, such as routines for partitioning constraints according to blocks and automatic differentiation algorithms for evaluating Jacobian and Hessian matrices and their corresponding sparsity partterns. Furthermore, PSMG implements a generic solver interface which can be linked with different structure exploiting optimization solvers such as decomposition or interior point based solvers. The work required for linking with PSMG’s solver interface is also discussed. Finally, we evaluate PSMG’s run-time performance and memory usage by generating structured problems with various sizes. The results from both serial and parallel executions are discussed. The benchmark results show that PSMG achieve good parallel efficiency on up to 96 processes. PSMG distributes memory usage among parallel processors which enables the generation of problems that are too large to be processed on a single node due to memory restriction.