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Fitzpatrick functions and continuous linear monotone operators

Borwein, Jonathan M. and Bauschke, Heinz H. and Wang, Shawn Xianfu (2007) Fitzpatrick functions and continuous linear monotone operators. SIAM Journal on Control and Optimization, 18 . pp. 257-276. ISSN 0363-0129

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    Abstract

    The notion of a maximal monotone operator is crucial in optimization as it captures both the subdifferential operator of a convex, lower semicontinuous, and proper function and any (not necessarily symmetric) continuous linear positive operator. It was recently discovered that most fundamental results on maximal monotone operators allow simpler proofs utilizing Fitzpatrick functions. In this paper, we study Fitzpatrick functions of continuous linear monotone operators defined on a Hilbert space. A novel characterization of skew operators is presented. A result by Br\'ezis and Haraux is reproved using the Fitzpatrick function. We investigate the Fitzpatrick function of the sum of two operators, and we show that a known upper bound is actually exact in finite-dimensional and more general settings. Cyclic monotonicity properties are also analyzed, and closed forms of the Fitzpatrick functions of all orders are provided for all rotators in the Euclidean plane.

    Item Type: Article
    Additional Information: pubdom TRUE
    Uncontrolled Keywords: Cyclic monotonicity, Fitzpatrick family, Fitzpatrick function, linear operator, maximal monotone operator, Moore-Penrose inverse, paramonotone operator, rotator.
    Subjects: 90-xx Economics, operations research, programming, games > 90Cxx Mathematical programming
    47-xx Operator theory > 47Hxx Nonlinear operators and their properties
    47-xx Operator theory > 47Bxx Special classes of linear operators
    Faculty: UNSPECIFIED
    Depositing User: Users 1 not found.
    Date Deposited: 31 Mar 2006
    Last Modified: 11 Jan 2015 19:01
    URI: https://docserver.carma.newcastle.edu.au/id/eprint/319

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