An efficient algorithm for nonlinear integer programming

M.Sc., Faculty of Sciences, University of the Witwatersrand, 2011Abstract This dissertation is concerned with discrete global optimization of nonlinear problems. These problems are constrained and unconstrained and are not easily solvable since there exists multiplicity of local and global minima. In this dissertation, we study the current methods for solving such problems and highlight their ine ciencies. We introduce a new local search procedure. We study the rapidly-exploring random tree (RRT) method, found mostly in the research area of robotics. We then design two global optimization algorithms based on RRT. RRT has never been used in the eld of global optimization. We exploit its attractive properties to develop two new algorithms for solving the discrete nonlinear optimization problems. The rst method is called RRT-Optimizer and is denoted as RRTOpt. RRTOpt is then modi ed to include probabilistic elements within the RRT. We have denoted this method by RRTOptv1. Results are generated for both methods and numerical comparisons are made with a number of recent methods

Trajectory-based methods for solving nonlinear and mixed integer nonlinear programming problems

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2015.I would like to acknowledge a number of people who contributed towards the completion of this thesis. Firstly, I thank my supervisor Professor Montaz Ali for his patience, enthusiasm, guidance and teachings. The skills I have acquired during this process have infiltrated every aspect of my life. I remain forever grateful. Secondly, I would like to say a special thank you to Professor Jan Snyman for his assistance, which contributed immensely towards this thesis. I would also like to thank Professor Dominque Orban for his willingness to assist me for countless hours with the installation of CUTEr, as well as Professor Jose Mario Martinez for his email correspondence. A heartfelt thanks goes out to my family and friends at large, for their prayers, support and faith in me when I had little faith in myself. Thank you also to my colleagues who kept me sane and motivated, as well as all the support staff who played a pivotal roll in this process. Above all, I would like to thank God, without whom none of this would have been possible