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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… (More)

- Heinz H. Bauschke, Xianfu Wang, Liangjin Yao
- Math. Program.
- 2010

Monotone operators are of central importance in modern optimization and nonlinear analysis. Their study has been revolutionized lately, due to the systematic use of the Fitzpatrick function. Pioneered by Penot and Svaiter, a topic of recent interest has been the representation of maximal monotone operators by so-called autoconjugate functions. Two explicit… (More)

In this paper, we give two explicit examples of unbounded linear maximal monotone operators. The first unbounded linear maximal monotone operator S on l 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… (More)

- Liangjin Yao
- 2011

The most important 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 A + ∂f provided that A is a maximally monotone linear relation, and f is a proper lower semicontinuous… (More)

- Liangjin Yao
- 2011

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 prove the maximal monotonicity of A+B provided that A and B are maximal monotone operators such that domA ∩ int domB ̸= ∅, A+NdomB is of type… (More)

- Liangjin Yao
- 2009

In this note, we study maximal monotonicity of linear relations (set-valued operators with linear graphs) on reflexive Banach spaces. We provide a new and simpler proof of a result due to BrézisBrowder which states that a monotone linear relation with closed graph is maximal monotone if and only if its adjoint is monotone. We also study Fitzpatrick… (More)

- Jonathan M. Borwein, Guoyin Li, Liangjin Yao
- SIAM Journal on Optimization
- 2014

In this paper, we study the rate of convergence of the cyclic projection algorithm applied to finitely many basic semi-algebraic convex sets. We establish an explicit convergence rate estimate which relies on the maximum degree of the polynomials that generate the basic semi-algebraic convex sets and the dimension of the underlying space. We achieve our… (More)

Monotone operators, especially in the form of subdifferential operators, are of basic importance in optimization. It is well known since Minty, Rockafellar, and Bertsekas-Eckstein that in Hilbert space, monotone operators can be understood and analyzed from the alternative viewpoint of firmly nonexpansive mappings, which were found to be precisely the… (More)

It is shown that, for maximally monotone linear relations defined on a general Banach space, the monotonicities of dense type, of negative-infimum type, and of FitzpatrickPhelps type are the same and equivalent to monotonicity of the adjoint. This result also provides affirmative answers to two problems: one posed by Phelps and Simons, and the other by… (More)

In this paper, we construct maximally monotone operators that are not of Gossez’s dense-type (D) in many nonreflexive spaces. Many of these operators also fail to possess the Brønsted-Rockafellar (BR) property. Using these operators, we show that the partial inf-convolution of two BC–functions will not always be a BC–function. This provides a negative… (More)