First Advisor
Nam Nguyen
Term of Graduation
January 2026
Date of Publication
9-1-2026
Document Type
Dissertation
Language
English
Subjects
Convex Analysis, Fenchel Conjugate, Lagrangian Duality, Machine Learning, Quasi-Relative Interior, Set-Valued Mappings
Physical Description
1 online resource ( pages)
Abstract
Convex analysis, optimization, and duality theory are broadly applicable to a multitude of real-world problems. Theoretical frameworks ensuring strong duality and optimality are fundamentally tied to convex separation under qualification conditions frequently involving topological or relative interiors. These can be empty in convex sets that appear naturally in infinite-dimensional models, yet guarantees of optimality and strong duality are often attainable. This dissertation presents frameworks for applying convex separation via generalized relative interiors to develop generalized calculus rules as well as conditions for strong duality and optimality for infinite-dimensional constrained convex optimization problems.
The first major contribution is the introduction of an extended-real-valued Fenchel conjugate for convex set-valued mappings. This new conjugate has relationships with the coderivative that parallel those between the Fenchel conjugate and the subdifferential of a convex function. We apply convex separation via generalized relative interiors and derive a rich set of calculus rules for this new Fenchel conjugate, which, together with the relationships between it and the coderivative as well as the classical Fenchel conjugate, allow for a holistic derivation of known coderivative, conjugate, and subdifferential calculus rules.
The second major contribution is an application of generalized relative interiors to derive conditions for optimality and strong duality for infinite-dimensional constrained support vector machine models. We derive a solution using the Lagrangian dual formulation, which demonstrates that a large class of primal convex constraints can be absorbed into the dual cost function due to the differentiability of the squared distance function. This provides a principled framework for extending finite-dimensional optimization theory to infinite dimensions, thereby expanding the scope of finite-dimensional theoretical frameworks to problems best modeled in infinite-dimensional settings.
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Recommended Citation
Sandine, Gary Lee, "Set-Valued Conjugate Calculus and Duality in Optimization and Machine Learning via Generalized Relative Interiors" (2026). Dissertations and Theses. Paper 7201.