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Keywords
(9)
Fitzpatrick Function
Hilbert Space
Linear Operator
Linear Positive Operator
lower semicontinuity
Maximal Monotone Operator
Monotone Operator
moorepenrose inverse
Upper Bound
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(3)
Fitzpatrick functions, cyclic monotonicity and Rockafellar's antiderivative
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The relevance of convex analysis for the study of monotonicity
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Fitzpatrick Functions and Continuous Linear Monotone Operators
Fitzpatrick Functions and Continuous Linear Monotone Operators,10.1137/060655468,Siam Journal on Optimization,Heinz H. Bauschke,Jonathan M. Borwein,Xi
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Fitzpatrick Functions and Continuous Linear Monotone Operators
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Citations: 21
)
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Heinz H. Bauschke
,
Jonathan M. Borwein
,
Xianfu Wang
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 Brezis 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 finitedimensional 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.
Journal:
Siam Journal on Optimization  SIAMJO
, vol. 18, no. 3, pp. 789809, 2007
DOI:
10.1137/060655468
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Citation Context
(13)
...Besides these results for subdifferential operators, it was known that rectangularity and paramonotonicity coincide for certain matrices
2
, Remark 4...
Heinz H. Bauschke
,
et al.
Rectangularity and paramonotonicity of maximally monotone operators
...Moore–Penrose inverse (i.e., pseudoinverse) is considered to be one of the basic problems widely encountered in various science and engineering fields, e.g., robotics [1], signal processing [2], pattern recognition [3], optimization [4,
5
] and biology [6]...
Yunong Zhang
,
et al.
Zhang neural network solving for timevarying fullrank matrix MooreP...
...Monotone Operator Theory has been revolutionized through the systematic use of the Fitzpatrick function; new results have been obtained and previously known result have been reproved in a simpler fashion — see, e.g., [1, 2,
3
, 6, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 22, 23, 26, 27, 28, 29, 30, 33, 37, 38, 39, 41, 42, 43, 44, 46]...
Heinz H. Bauschke
,
et al.
Autoconjugate representers for linear monotone operators
...discontinuous linear operators — and lately even linear relations — have received some attention in Monotone Operator Theory [1,
2
, 4, 5, 11, 17, 18] because they provide additional classes of examples apart from the well known and well understood subdifferential operators in the sense of Convex Analysis...
Heinz H. Bauschke
,
et al.
An Answer to S. Simons’ Question on the Maximal Monotonicity of the Su...
...Recently, Fitzpatrick functions of order n [1] have turned out to be a useful tool in the study of ncyclic monotonicity (see [1,
3
, 4, 13])...
...ric), it will be convenient to define (as in, e.g., [
3
])...
...Remark 18.20. Theorem 18.19 generalizes [
3, Theorem 5.4
]...
Liangjin Yao
.
The BrézisBrowder Theorem Revisited and Properties of Fitzpatrick Fun...
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,
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(
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Sort by:
Citations
(21)
Rectangularity and paramonotonicity of maximally monotone operators
Heinz H. Bauschke
,
Xianfu Wang
,
Liangjin Yao
Journal:
Optimization
, vol. aheadofp, no. aheadofp, pp. 118, 2012
Zhang neural network solving for timevarying fullrank matrix MoorePenrose inverse
(
Citations: 1
)
Yunong Zhang
,
Yiwen Yang
,
Ning Tan
,
Binghuang Cai
Journal:
Computing
, vol. 92, no. 2, pp. 97121, 2011
Autoconjugate representers for linear monotone operators
(
Citations: 10
)
Heinz H. Bauschke
,
Xianfu Wang
,
Liangjin Yao
Journal:
Mathematical Programming
, vol. 123, no. 1, pp. 524, 2010
Examples of discontinuous maximal monotone linear operators and the solution to a recent problem posed by B.F. Svaiter
(
Citations: 7
)
Heinz H. Bauschke
,
Xianfu Wang
,
Liangjin Yao
Journal:
Journal of Mathematical Analysis and Applications  J MATH ANAL APPL
, vol. 370, no. 1, pp. 224241, 2010
On the maximal monotonicity of the sum of a maximal monotone linear relation and the subdifferential operator of a sublinear function
(
Citations: 2
)
Heinz H. Bauschke
,
Xianfu Wang
,
Liangjin Yao
Published in 2010.