Liberating the Subgradient Optimality Conditions from Constraint Qualifications

Authors:
V. Jeyakumar;Z. Y. Wu;G. M. Lee;N. Dinh
Affiliations:
School of Mathematics, University of New South Wales, Sydney, Australia 2052;Aff2 Aff5;Department of Applied Mathematics, Pukyong National University, Pusan, Korea 608---737;Department of Mathematics-Informatics, Hochiminh City University of Pedagogy, HCM city, Vietnam
Venue:
Journal of Global Optimization
Year:
2006

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Abstract

In convex optimization the significance of constraint qualifications is evidenced by the simple duality theory, and the elegant subgradient optimality conditions which completely characterize a minimizer. However, the constraint qualifications do not always hold even for finite dimensional optimization problems and frequently fail for infinite dimensional problems. In the present work we take a broader view of the subgradient optimality conditions by allowing them to depend on a sequence of 驴-subgradients at a minimizer and then by letting them to hold in the limit. Liberating the optimality conditions in this way permits us to obtain a complete characterization of optimality without a constraint qualification. As an easy consequence of these results we obtain optimality conditions for conic convex optimization problems without a constraint qualification. We derive these conditions by applying a powerful combination of conjugate analysis and 驴-subdifferential calculus. Numerical examples are discussed to illustrate the significance of the sequential conditions.