Quantitative inconsistent feasibility for averaged mappings

Sipoş, Andrei

doi:10.1007/s11590-021-01812-2

Cited by 3 publications

(20 citation statements)

References 13 publications

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“…This situation is given in the implicit uses of instances of proposition 3.1 in [9,10] discussed in §3b below, where—in [9] - B:=M is single-valued, domfalse(Mfalse)=X and M is 2-Lipschitzian while in [10] both A and B are single-valued and Lipschitz continuous.…”

Section: Logical Systems and Bound Extraction Theorems For Monotone O...mentioning

confidence: 99%

“…This in particular covers the situation of proposition 2.20 for the class of 3-cyclically monotone operators which can in that way be seen as an application of corollary 2.29. Also the extraction of a modulus of uniform rectangularity from the proof that β-cocoercive operators are rectangular carried out in [10] can be viewed as an instance of corollary 2.29 using the fact that being β-cocoercive is a purely universal condition with β>0 as a new input which we may assume without loss of generality to be a rational number, namely a positive rational lower bound of β.…”

Section: Logical Systems and Bound Extraction Theorems For Monotone O...mentioning

confidence: 99%

“…We now show that the (part of the) Brezis–Haraux theorem [29] used in Bauschke’s solution to the zero displacement conjecture [30] as well as in its generalizations to compositions of firmly non-expansive mappings in [31] and of averaged mappings in [32] can be formulated in an appropriate formal system to which logical bound extraction metatheorems apply. These then also guarantee a quantitative version of this theorem as given below (which—in more special forms—is used in analysing quantitatively the specific situations mentioned above in [9,10]).…”

Section: The Brezis–haraux Theorem and Its Formalization In Logical S...mentioning

confidence: 99%

“…For convenience, we use ε>0 instead of 1/false(n+1false) and K:false(0,normal∞false)false→double-struckN∖falsefalse{0falsefalse}. Analysing the proof of (3.1) from [29] one obtains the following (compare also [9,10]):…”

Section: The Brezis–haraux Theorem and Its Formalization In Logical S...mentioning

confidence: 99%

“…We then establish two quantitative versions of the Brezis–Haraux theorem. The first one uses that one has a modulus for A being uniformly rectangular, which is the case in the recent applications [9,10] and explains these applications as instances of a general logic metatheorem. The second quantitative version only uses that one has a modulus of A being weakly uniformly rectangular on the expense of a more complicated bound but is applicable even in situations where the property of A being rectangular is only established non-effectively and may pave the way towards new applications.…”

Section: Introductionmentioning

confidence: 99%

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Proof theory and non-smooth analysis

Kohlenbach

Pischke

2023

Phil. Trans. R. Soc. A.

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We develop a general proof-theoretic framework for various classes of set-valued operators, including maximally as well as cyclically monotone and rectangular operators and we discuss a treatment for sums of set-valued operators A , B in that context such that all of the previous fits into logical metatheorems on bound extractions. In particular, we introduce quantitative forms for A being (weakly) uniformly rectangular with witnessing moduli. Based on this, we give quantitative forms of the Brezis–Haraux theorem that use such moduli as input. It turns out that a modulus for weak uniform rectangularity, which can be extracted even from non-effective proofs of rectangularity, is sufficient while the bound gets simpler in the case of a modulus for A being uniform rectangular which can be extracted from semi-constructive proofs. We use our results to explain recent proof minings in the context of Bauschke’s solution to the zero displacement conjecture and its extensions to other classes of functions than metric projections as instances of logical metatheorems. This article is part of the theme issue ‘Modern perspectives in Proof Theory’.

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Section: Logical Systems and Bound Extraction Theorems For Monotone O...mentioning

confidence: 99%

Section: Logical Systems and Bound Extraction Theorems For Monotone O...mentioning

confidence: 99%

Section: The Brezis–haraux Theorem and Its Formalization In Logical S...mentioning

confidence: 99%

Section: The Brezis–haraux Theorem and Its Formalization In Logical S...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Proof theory and non-smooth analysis

Kohlenbach

Pischke

2023

Phil. Trans. R. Soc. A.

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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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The computational content of super strongly nonexpansive mappings and uniformly monotone operators

Sipoş¹

2023

Preprint

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Recently, Liu, Moursi and Vanderwerff have introduced the class of super strongly nonexpansive mappings as a counterpart to operators which are maximally monotone and uniformly monotone. We give a quantitative study of these notions in the style of proof mining, providing a modulus of super strong nonexpansiveness, giving concrete examples of it and connecting it to moduli associated to uniform monotonicity. For the supercoercive case, we analyze the situation further, yielding a quantitative inconsistent feasibility result for this class (obtaining effective uniform rates of asymptotic regularity), a result which is also qualitatively new.

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Quantitative inconsistent feasibility for averaged mappings

Cited by 3 publications

References 13 publications

Proof theory and non-smooth analysis

Proof theory and non-smooth analysis

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

The computational content of super strongly nonexpansive mappings and uniformly monotone operators

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