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Strong rotundity and optimization

Borwein, Jonathan M. and Lewis, Adrian (1994) Strong rotundity and optimization. SIAM Journal on Control and Optimization, 4 . pp. 146-158. ISSN 0363-0129

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    Abstract

    Standard techniques from the study of well-posedness show that if a fixed convex objective function is minimized in turn over a sequence of convex feasible regions converging Mosco to a limiting feasible region, then the optimal solutions converge in norm to the optimal solution of the limiting problem. Certain conditions on the objective function are needed as is a constraint qualification. If, as may easily occur in practice, the constraint qualification fails, stronger set convergence is required, together with stronger analytic/geometric properties of the objective function: strict convexity (to ensure uniqueness), weakly compact level sets (to ensure existence and weak convergence), and the Kadec property (to deduce norm convergence). By analogy with the Lp norms, such properties are termed "strong rotundity." A very simple characterization of strongly rotund integral functionals on L1 is presented that shows, for example, that the Boltzmann-Shannon entropy ∫ x log x is strongly rotund. Examples are discussed, and the existence of everywhere- and densely-defined strongly rotund functions is investigated.

    Item Type: Article
    Subjects: UNSPECIFIED
    Faculty: UNSPECIFIED
    Depositing User: Mrs Naghmana Tehseen
    Date Deposited: 14 Jan 2015 11:54
    Last Modified: 14 Jan 2015 11:54
    URI: https://docserver.carma.newcastle.edu.au/id/eprint/1550

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