Interior-point methods have not only shown their efficiency for linear and some nonlinear programming problems, but also for cutting plane methods and large scale optimization. The analytic center cutting plane method uses the analytic center of the current polyhedral approximation of the feasible region to add a new cutting plane. In this thesis, analytic center cutting plane and path-following interior-point methodologies are used to solve the following problems: (1) convex feasibility problems defined by a deep cut separation oracle; (2) convex optimization problems involving a nonlinear objective and a constraint set defined implicitly by a separation oracle; (3) variational inequalities involving a nonlinear operator and a convex set explicitly defined; (4) variational inequalities involving an affine operator and a constraint set defined implicitly by a deep cut separation oracle; and (5) variational inequalities involving a nonlinear operator and a constraint set defined implicitly by a deep cut separation oracle. Here, the oracle is a routine that takes as input a test point. If the point belongs to the feasible region, it answers "yes", otherwise it answers "no" and returns a cut separating the point from the feasible region. Complexity bounds are established for algorithms developed for Cases 1, 2 and 4. The algorithm developed for Case 3 will be proven to be convergent, whereas, in Case 5, the developed algorithm will be shown to find an approximate solution in finite time.
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:QMM.35615 |
| Date | January 1997 |
| Creators | Sharifi Mokhtarian, Faranak. |
| Contributors | Goffin, Jean-Louis (advisor) |
| Publisher | McGill University |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Format | application/pdf |
| Coverage | Doctor of Philosophy (Department of Mathematics and Statistics.) |
| Rights | All items in eScholarship@McGill are protected by copyright with all rights reserved unless otherwise indicated. |
| Relation | alephsysno: 001617516, proquestno: NQ44580, Theses scanned by UMI/ProQuest. |
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