Pseudonormality and a language multiplier theory for constrained optimization

Lagrange multipliers are central to analytical and computational studies in linear and non-linear optimization and have applications in a wide variety of fields, including communication, networking, economics, and manufacturing. In the past, the main research in Lagrange multiplier theory has focused on developing general and easily verifiable conditions on the constraint set, called constraint qualifications, that guarantee the existence of Lagrange multipliers for the optimization problem of interest. In this thesis, we present a new development of Lagrange multiplier theory that significantly differs from the classical treatments. Our objective is to generalize, unify, and streamline the theory of constraint qualifications. As a starting point, we derive an enahanced set of necessary optimality conditions of the Fritz John-type, which are stronger than the classical Karush-Kuhn-Tucker conditions. They are also more general in that they apply even when there is a possibly nonconvex abstract set constraint, in addition to smooth equality and inequality constraints. These optimality conditions motivate the introduction of a new condition, called pseudonormality, which emerges as central within the taxonomy of significant characteristics of a constraint set. In particular, pseudonormality unifies and extends the major constraint qualifications. In addition, pseudonormality provides the connecting link between constraint qualifications and exact penalty functions. Our analysis also yields identification of different types of Lagrange multipliers. Under some convexity assumptions, we show that there exists a special Lagrange multiplier vector, called informative, which carries significant sensitivity information regarding the constraints that directly affect the optimal cost change.

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

2003

Author dc:creator

• Ozdaglar, Asuman E

Advisor dc:contributor.advisor

• Dimitri P. Bertsekas.

• 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/1721.1/29319

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/29319