Heinz H. Bauschke
Rectangularity and paramonotonicity of maximally monotone operators.
Mon, 01/23/2012 - 15:01 — SomNambulAuthors: Heinz H. Bauschke, Xianfu Wang, Liangjin Yao
Subjects: Functional Analysis
AbstractMaximally 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.- Read more
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New Demiclosedness Principles for (firmly) nonexpansive operators.
Tue, 03/08/2011 - 16:01 — SomNambulAuthors: Heinz H. Bauschke
Subjects: Functional Analysis
AbstractThe demiclosedness principle is one of the key tools in nonlinear analysis
and fixed point theory. In this note, this principle is extended and made more
flexible by two mutually orthogonal affine subspaces. Versions for finitely
many (firmly) nonexpansive operators are presented. As an application, a simple
proof of the weak convergence of the Douglas-Rachford splitting algorithm is
provided.Firmly nonexpansive mappings and maximally monotone operators: correspondence and duality.
Wed, 01/26/2011 - 16:01 — SomNambulAuthors: Heinz H. Bauschke, Xianfu Wang, Sarah M. Moffat
Subjects: Functional Analysis
AbstractThe notion of a firmly nonexpansive mapping is central in fixed point theory
because of attractive convergence properties for iterates and the
correspondence with maximal monotone operators due to Minty. In this paper, we
systematically analyze the relationship between properties of firmly
nonexpansive mappings and associated maximal monotone operators. Dual and
self-dual properties are also identified. The results are illustrated through
several examples.Self-dual Smooth Approximations of Convex Functions via the Proximal Average.
Wed, 03/31/2010 - 15:01 — SomNambulAuthors: Heinz H. Bauschke, Xianfu Wang, Sarah M. Moffat
Subjects: Functional Analysis
AbstractThe proximal average of two convex functions has proven to be a useful tool
in convex analysis. In this note, we express Goebel's self-dual smoothing
operator in terms of the proximal average, which allows us to give a simple
proof of self duality. We also provide a novel self-dual smoothing operator.
Both operators are illustrated by smoothing the norm.Compositions and Averages of Two Resolvents: Relative Geometry of Fixed Points Sets and a Partial Answer to a Question by C. Byrne.
Fri, 03/26/2010 - 16:01 — SomNambulAuthors: Heinz H. Bauschke, Xianfu Wang
Subjects: Functional Analysis
AbstractWe show that the set of fixed points of the average of two resolvents can be
found from the set of fixed points for compositions of two resolvents
associated with scaled monotone operators. Recently, the proximal average has
attracted considerable attention in convex analysis. Our results imply that the
minimizers of proximal-average functions can be found from the set of fixed
points for compositions of two proximal mappings associated with scaled convex
functions.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 — SomNambulAuthors: Heinz H. Bauschke, Xianfu Wang, Liangjin Yao
Subjects: Functional Analysis
AbstractThe 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.On Borwein-Wiersma Decompositions of Monotone Linear Relations.
Wed, 12/16/2009 - 16:01 — SomNambulAuthors: Heinz H. Bauschke, Xianfu Wang, Liangjin Yao
Subjects: Functional Analysis
AbstractMonotone 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.The Resolvent Average for Positive Semidefinite Matrices.
Wed, 10/21/2009 - 14:15 — SomNambulAuthors: Heinz H. Bauschke, Xianfu Wang, Sarah M. Moffat
Subjects: Functional Analysis
AbstractWe define a new average - termed the resolvent average - for positive
semidefinite matrices. For positive definite matrices, the resolvent average
enjoys self-duality and it interpolates between the harmonic and the arithmetic
averages, which it approaches when taking appropriate limits. We compare the
resolvent average to the geometric mean. Some applications to matrix functions
are also given.The Resolvent Average for Positive Semidefinite Matrices.
Wed, 10/21/2009 - 14:15 — SomNambulAuthors: Heinz H. Bauschke, Xianfu Wang, Sarah M. Moffat
Subjects: Functional Analysis
AbstractWe define a new average - termed the resolvent average - for positive
semidefinite matrices. For positive definite matrices, the resolvent average
enjoys self-duality and it interpolates between the harmonic and the arithmetic
averages, which it approaches when taking appropriate limits. We compare the
resolvent average to the geometric mean. Some applications to matrix functions
are also given.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 — SomNambulAuthors: Heinz H. Bauschke, Xianfu Wang, Liangjin Yao
Subjects: Functional Analysis
AbstractIn 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 $L^{2}[0,1]$.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 — SomNambulAuthors: Heinz H. Bauschke, Xianfu Wang, Liangjin Yao
Subjects: Functional Analysis
AbstractIn 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 $L^{2}[0,1]$.Klee sets and Chebyshev centers for the right Bregman distance.
Mon, 08/17/2009 - 18:30 — SomNambulAuthors: Heinz H. Bauschke, Mason S. Macklem, Jason B. Sewell, Xianfu Wang
Subjects: Optimization and Control
AbstractWe systematically investigate the farthest distance function, farthest
points, Klee sets, and Chebyshev centers, with respect to Bregman distances
induced by Legendre functions. These objects are of considerable interest in
Information Geometry and Machine Learning; when the Legendre function is
specialized to the energy, one obtains classical notions from Approximation
Theory and Convex Analysis.- Read more
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