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Liangjin Yao

  1. Rectangularity and paramonotonicity of maximally monotone operators.

    Mon, 01/23/2012 - 15:01 — SomNambul
    Authors: Heinz H. Bauschke, Xianfu Wang, Liangjin Yao
    Subjects: Functional Analysis
    Abstract

    Maximally monotone operators play a key role in modern optimization and
    variational analysis. Two useful subclasses are rectangular (also known as star
    monotone) and paramonotone operators, which were introduced by Brezis and
    Haraux, and by Censor, Iusem and Zenios, respectively. The former class has
    useful range properties while the latter class is of importance for interior
    point methods and duality theory.

  2. Maximally Monotone Linear Subspace Extensions of Monotone Subspaces: Explicit Constructions and Characterizations.

    Wed, 03/09/2011 - 16:01 — SomNambul
    Authors: Xianfu Wang, Liangjin Yao
    Subjects: Functional Analysis
    Abstract

    Monotone linear relations play important roles in variational inequality
    problems and quadratic optimizations. In this paper, we give explicit maximally
    monotone linear subspace extensions of a monotone linear relation in finite
    dimensional spaces. Examples are provided to illustrate our extensions. Our
    results generalize a recent result by Crouzeix and Anaya.

  3. The sum of a maximal monotone operator of type (FPV) and a maximal monotone operator with full domain is maximal monotone.

    Fri, 05/14/2010 - 15:01 — SomNambul
    Authors: Liangjin Yao
    Subjects: Functional Analysis
    Abstract

    The most important open problem in Monotone Operator Theory concerns the
    maximal monotonicity of the sum of two maximal monotone operators provided that
    Rockafellar's constraint qualification holds. In this paper, we provide a new
    maximal monotonicity result for the sum of two maximal monotone operators $A$
    and $B$ in this setting satisfying that $A+N_{\bar{\dom B}}$ is of type (FPV)
    and $\dom A\cap\bar{\dom B}\subseteq\dom B$. The proof relies on some results
    on the Fitzpatrick function.

  4. On the maximal monotonicity of the sum of a maximal monotone linear relation and the subdifferential operator of a sublinear function.

    Tue, 01/05/2010 - 16:01 — SomNambul
    Authors: Heinz H. Bauschke, Xianfu Wang, Liangjin Yao
    Subjects: Functional Analysis
    Abstract

    The most important open problem in Monotone Operator Theory concerns the
    maximal monotonicity of the sum of two maximal monotone operators provided that
    Rockafellar's constraint qualification holds.

    In this note, we provide a new maximal monotonicity result for the sum of a
    maximal monotone relation and the subdifferential operator of a proper, lower
    semicontinuous, sublinear function. The proof relies on Rockafellar's formula
    for the Fenchel conjugate of the sum as well as some results on the Fitzpatrick
    function.

  5. On Borwein-Wiersma Decompositions of Monotone Linear Relations.

    Wed, 12/16/2009 - 16:01 — SomNambul
    Authors: Heinz H. Bauschke, Xianfu Wang, Liangjin Yao
    Subjects: Functional Analysis
    Abstract

    Monotone operators are of basic importance in optimization as they generalize
    simultaneously subdifferential operators of convex functions and positive
    semidefinite (not necessarily symmetric) matrices. In 1970, Asplund studied the
    additive decomposition of a maximal monotone operator as the sum of a
    subdifferential operator and an "irreducible" monotone operator. In 2007,
    Borwein and Wiersma [SIAM J. Optim. 18 (2007), pp.

  6. Examples of discontinuous maximal monotone linear operators and the solution to a recent problem posed by B.F. Svaiter.

    Wed, 09/16/2009 - 13:15 — SomNambul
    Authors: Heinz H. Bauschke, Xianfu Wang, Liangjin Yao
    Subjects: Functional Analysis
    Abstract

    In this paper, we give two explicit examples of unbounded linear maximal
    monotone operators. The first unbounded linear maximal monotone operator $S$ on
    $\ell^{2}$ is skew. We show its domain is a proper subset of the domain of its
    adjoint $S^*$, and $-S^*$ is not maximal monotone. This gives a negative answer
    to a recent question posed by Svaiter. The second unbounded linear maximal
    monotone operator is the inverse Volterra operator $T$ on $L^{2}[0,1]$.

  7. Examples of discontinuous maximal monotone linear operators and the solution to a recent problem posed by B.F. Svaiter.

    Wed, 09/16/2009 - 13:15 — SomNambul
    Authors: Heinz H. Bauschke, Xianfu Wang, Liangjin Yao
    Subjects: Functional Analysis
    Abstract

    In this paper, we give two explicit examples of unbounded linear maximal
    monotone operators. The first unbounded linear maximal monotone operator $S$ on
    $\ell^{2}$ is skew. We show its domain is a proper subset of the domain of its
    adjoint $S^*$, and $-S^*$ is not maximal monotone. This gives a negative answer
    to a recent question posed by Svaiter. The second unbounded linear maximal
    monotone operator is the inverse Volterra operator $T$ on $L^{2}[0,1]$.

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