55 research outputs found
Maximality of the sum of a maximally monotone linear relation and a maximally monotone operator
The most famous open problem in Monotone Operator Theory concerns the maximal
monotonicity of the sum of two maximally monotone operators provided that
Rockafellar's constraint qualification holds.
In this paper, we prove the maximal monotonicity of provided that are maximally monotone and is a linear relation, as soon as
Rockafellar's constraint qualification holds: \dom A\cap\inte\dom
B\neq\varnothing. Moreover, is of type (FPV).Comment: 16 pages. arXiv admin note: substantial text overlap with
arXiv:1010.4346, arXiv:1005.224
Monotone Linear Relations: Maximality and Fitzpatrick Functions
We analyze and characterize maximal monotonicity of linear relations
(set-valued operators with linear graphs). An important tool in our study are
Fitzpatrick functions. The results obtained partially extend work on linear and
at most single-valued operators by Phelps and Simons and by Bauschke, Borwein
and Wang. Furthermore, a description of skew linear relations in terms of the
Fitzpatrick family is obtained. We also answer one of Simons problems by
showing that if a maximal monotone operator has a convex graph, then this graph
must actually be affine
Rectangularity and paramonotonicity of maximally monotone operators
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. Both notions are automatic for
subdifferential operators and known to coincide for certain matrices; however,
more precise relationships between rectangularity and paramonotonicity were not
known.
Our aim is to provide new results and examples concerning these notions. It
is shown that rectangularity and paramonotonicity are actually independent.
Moreover, for linear relations, rectangularity implies paramonotonicity but the
converse implication requires additional assumptions. We also consider
continuous linear monotone operators, and we point out that in Hilbert space
both notions are automatic for certain displacement mappings
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