Interior Point Techniques in Optimization. Complementarity, Sensitivity, andAlgorithms (Inwendige Punt Technieken in Optimalisatie. Complementariteit, Gevoeligheid en Algoritmen)

摘要

In this thesis, interior point methodology is used to derive new results, developand analyze some new primal-dual interior point algorithms, and unify (and extend) existing literature in interior point methods. Specifically, it is investigated how initial results and methods for linear programming (LP) can be extended to nonlinear programming (NLP). As a result, this thesis contains results in various areas of mathematical programming, viz., linear, quadratic, nonlinear, combinational and semidefinite optimization. In this thesis we use (Strict) complementarity to derive results in the theory and sensitivity analysis for LP (Chapter 2). The v-space is used in the development of a new class of primal-dual algorithms for (non)linear complementarity problems (Chapter 3) and in developing a unifying, called target-following, framework for analyzing interior point methods (Chapter 4). Finally, we use complementarity in three other subjects: we investigate sensitivity analysis in quadratic programming, analyze two applications of semidefinite programming, and consider interior point based decomposition methods (Chapter 5). The thesis is concluded with a section containing a summary, conclusions and directions for further research.