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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Convex optimization and the epi-distance topology


Authors: Gerald Beer and Roberto Lucchetti
Journal: Trans. Amer. Math. Soc. 327 (1991), 795-813
MSC: Primary 49J45; Secondary 41A50, 90C48
MathSciNet review: 1012526
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Abstract: Let $ \Gamma (X)$ denote the proper, lower semicontinuous, convex functions on a Banach space $ X$, equipped with the completely metrizable topology $ \tau $ of uniform convergence of distance functions on bounded sets. A function $ f$ in $ \Gamma (X)$ is called well-posed provided it has a unique minimizer, and each minimizing sequence converges to this minimizer. We show that well-posedness of $ f \in \Gamma (X)$ is the minimal condition that guarantees strong convergence of approximate minima of $ \tau $-approximating functions to the minimum of $ f$. Moreover, we show that most functions in $ \langle \Gamma (X),{\tau _{aw}}\rangle $ are well-posed, and that this fails if $ \Gamma (X)$ is topologized by the weaker topology of Mosco convergence, whenever $ X$ is infinite dimensional. Applications to metric projections are also given, including a fundamental characterization of approximative compactness.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1991-1012526-X
PII: S 0002-9947(1991)1012526-X
Keywords: Convex optimization, convex function, well-posed minimization problem, epi-distance topology, Mosco convergence, metric projection, approximative compactness, Chebyshev set
Article copyright: © Copyright 1991 American Mathematical Society