Analysis of critical points for nonconvex optimization

In this thesis, we establish sufficient conditions under which an optimization problem has a unique local optimum. Motivated by the practical need for establishing the uniqueness of the optimum in an optimization problem in fields such as global optimization, equilibrium analysis, and efficient algorithm design, we provide sufficient conditions that are not merely theoretical characterizations of uniqueness, but rather, given an optimization problem, can be checked algebraically. In our analysis we use results from two major mathematical disciplines. Using the mountain pass theory of variational analysis, we are able to establish the uniqueness of the local optimum for problems in which every stationary point of the objective function is a strict local minimum and the function satisfies certain boundary conditions on the constraint region. Using the index theory of differential topology, we are able to establish the uniqueness of the local optimum for problems in which every generalized stationary point (Karush-Kuhn-Tucker point) of the objective function is a strict local minimum and the function satisfies some non-degeneracy assumptions. The uniqueness results we establish using the mountain pass theory and the topological index theory are comparable but not identical.

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2005

Author dc:creator

• Å imÅ ek, Alp

Advisor dc:contributor.advisor

• Asuman Ozdaglar.

• Electrical Engineering and Computer Science.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/32108

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/32108