Generalized monotone operators and their averaged resolvents

Bauschke, Heinz H.; Moursi, Walaa M.; Wang, Xianfu

doi:10.1007/s10107-020-01500-6

Cited by 41 publications

(56 citation statements)

References 34 publications

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“…Putting all this together, it is rather straightforward to see that the main results on the PPA and HPPA established in [12,13] generalize (in the case of Hilbert spaces) to ρ-comonotone operators which is the content of this short note. While the PPA has been considered for ρ-comonotone operators before (even for sequences of operators, error terms and relaxations, see [7]) our note shows that by the connection between the comonotonicity of A and the averagedness of J A as established in [5], many proofs for properties of the PPA and the HPPA for monotone operators can be easily adapted to cover the ρ-comonotone case. We also provide new quantitative results on the convergence.…”

Section: Introductionmentioning

confidence: 95%

“…it a fortiori is η-comonotone with η := ρ γ > − 1 2 . Hence by [5,Proposition 3.11(v)] applied to γ n A, the resolvent J γ n A : R(I + γ n A) → D(A) is α-averaged. The claim now follows from Proposition 2.7.…”

Section: Lemma 23 Ifmentioning

confidence: 99%

“…[7,8]). In the recent paper [5], this condition-called ρ-comonotonicity-is thoroughly investigated and related to properties of J A . One key result is that J A is an averaged mapping whenever (++) holds with ρ > − 1 2 .…”

Section: Introductionmentioning

confidence: 99%

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On the proximal point algorithm and its Halpern-type variant for generalized monotone operators in Hilbert space

Kohlenbach

2021

Optim Lett

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In a recent paper, Bauschke et al. study $$\rho $$ ρ -comonotonicity as a generalized notion of monotonicity of set-valued operators A in Hilbert space and characterize this condition on A in terms of the averagedness of its resolvent $$J_A.$$ J A . In this note we show that this result makes it possible to adapt many proofs of properties of the proximal point algorithm PPA and its strongly convergent Halpern-type variant HPPA to this more general class of operators. This also applies to quantitative results on the rates of convergence or metastability (in the sense of T. Tao). E.g. using this approach we get a simple proof for the convergence of the PPA in the boundedly compact case for $$\rho $$ ρ -comonotone operators and obtain an effective rate of metastability. If A has a modulus of regularity w.r.t. $$zer\, A$$ z e r A we also get a rate of convergence to some zero of A even without any compactness assumption. We also study a Halpern-type variant HPPA of the PPA for $$\rho $$ ρ -comonotone operators, prove its strong convergence (without any compactness or regularity assumption) and give a rate of metastability.

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Section: Introductionmentioning

confidence: 95%

Section: Lemma 23 Ifmentioning

confidence: 99%

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On the proximal point algorithm and its Halpern-type variant for generalized monotone operators in Hilbert space

Kohlenbach

2021

Optim Lett

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“…The operator T is said to be conically averaged with constant θ ∈ R ++ (see [4,7]) if there exists a nonexpansive operator N : X → X such that…”

Section: Proposition 21 (Fixed Points Of T Abc ) Let T Abc Be Defined By (3) Then Fix T Abc = ∅ If and Only If Zer(mentioning

confidence: 99%

An adaptive splitting algorithm for the sum of two generalized monotone operators and one cocoercive operator

Dao

Phan

2021

Fixed Point Theory Algorithms Sci Eng

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Splitting algorithms for finding a zero of sum of operators often involve multiple steps which are referred to as forward or backward steps. Forward steps are the explicit use of the operators and backward steps involve the operators implicitly via their resolvents. In this paper, we study an adaptive splitting algorithm for finding a zero of the sum of three operators. We assume that two of the operators are generalized monotone and their resolvents are computable, while the other operator is cocoercive but its resolvent is missing or costly to compute. Our splitting algorithm adapts new parameters to the generalized monotonicity of the operators and, at the same time, combines appropriate forward and backward steps to guarantee convergence to a solution of the problem.

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“…Using the concept of proto-differentiability of a multifunction and the notion of semi-differentiability of a single-valued map, the authors establish the differentiability of the solution of a parameterized monotone inclusion. • Bauschke et al [3] introduce the notion of conically nonexpansive operators which generalize nonexpansive mappings. Averaged operators are characterized as resolvents of comonotone operators under appropriate scaling.…”

mentioning

confidence: 99%

Special Issue: Continuous Optimization and Stability Analysis

2021

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Continuous optimization is a vital research area with many active branches such as linear and nonlinear optimization, stochastic optimization, vector optimization, semi-infinite optimization, etc. Stability analysis of optimization problems is of crucial interest in applications, and makes optimization an ideal testing bench for many disciplines such as convex analysis, set-valued analysis, variational analysis, and wellposedness, among others.The present special issue offers some recent developments in this wide area and aims to honor Prof. Marco Antonio López Cerdá on the occasion of his 70th birthday, as well as to show recognition by his colleagues to his contribution to this field. His achievements include some monographs, several surveys and more than 150 scientific articles on optimality, duality, stability and algorithms in semi-infinite programming, Lipschitz-type properties, error bounds, subdifferential calculus, and others such as game theory, robustness, stationarity and regularity.Marco A. López was born in Alcoy (Alicante, Spain) in 1949 and has spent most of his academic career (since 1985) as a full professor of statistics and operations research at the University of Alicante, where nowadays he is Emeritus Professor. Marco is a corresponding fellow of the Spanish Royal Academy of Sciences, an Honorary Research Fellow of the Federation University (Australia), and a Doctor Honoris Causa at the University of Limoges (France). He has also led important research projects, theoretical and applied. From 2008 to 2012 he was the coodinator of i-MATH Consolider B J.

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Generalized monotone operators and their averaged resolvents

Cited by 41 publications

References 34 publications

On the proximal point algorithm and its Halpern-type variant for generalized monotone operators in Hilbert space

On the proximal point algorithm and its Halpern-type variant for generalized monotone operators in Hilbert space

An adaptive splitting algorithm for the sum of two generalized monotone operators and one cocoercive operator

Special Issue: Continuous Optimization and Stability Analysis

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