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https://hdl.handle.net/2142/68176
Description
Title
Optimization, Convergence, and Duality
Author(s)
Bergstrom, Roy Clarence
Issue Date
1980
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
In the 1960's, a notion of convergence for a sequence of convex functions was studied by Wijsman, Mosco, and Joly. This convergence, not comparable to pointwise convergence, has several important properties: it is preserved under the Fenchel transform, and it is equivalent to a convergence which can be defined for the sequence of subdifferentials corresponding to the given convex functions. In this thesis simple conditions are established under which this convergence is preserved under the operations of addition, infimal convolution, and composition with linear transformations.
These technical facts are then applied in a variety of optimization settings, including generalized convex programming, the Fenchel-Rockafellar perturbational duality scheme, network optimization, certain approximation problems, and linear programming. As an example, for a sequence of convex programs, simple conditions are developed which ensure the convergence of the optimal values and also an upper semi-continuity property for convergence of the sets of optimal solutions. Under the same conditions, the sequence of dual programs is shown to have the same properties, and in addition, for programs far enough out in the sequence as well as in the limit, there is no duality gap.
The convergence is studied also in infinite dimensions, where some of the preservation properties are found to hold under suitable restrictions not necessarily requiring interiority. In another direction, the idea of convergence of a sequence of subdifferentials is studied for general maximal monotone operators, and preservation of such convergence under addition of operators is investigated.
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