Decomposition and diet problems
The purpose of this thesis is to efficiently solve real life problems. We study LPs. We study an NLP and an MINLP based on what is known as the generalised pooling problem (GPP), and we study an MIP that we call the cattle mating problem. These problems are often very large or otherwise difficult to solve by direct methods, and are best solved by decomposition methods. During the thesis we introduce algorithms that exploit the structure of the problems to decompose them. We are able to solve row-linked, column-linked and general LPs efficiently by modifying the tableau simplex method, and suggest how this work could be applied to the revised simplex method. We modify an existing sequential linear programming solver that is currently used by Format International to solve GPPs, and show the modified solver takes less time and is at least as likely to find the global minimum as the old solver. We solve multifactory versions of the GPP by augmented Lagrangian decomposition, and show this is more efficient than solving the problems directly. We introduce a decomposition algorithm to solve a MINLP version of the GPP by decomposing it into NLP and ILP subproblems. This is able to solve large problems that could not be solved directly. We introduce an efficient decomposition algorithm to solve the MIP cattle mating problem, which has been adopted for use by the Irish Cattle Breeding Federation. Most of the solve methods we introduce are designed only to find local minima. However, for the multifactory version of the GPP we introduce two methods that give a good chance of finding the global minimum, both of which succeed in finding the global minimum on test problems.