Successive approximations

Krasnosel’skii, M. A.; Vainikko, G. M.; Zabreiko, P. P.; Rutitskii, Ya. B.; Stetsenko, V. Ya.

doi:10.1007/978-94-010-2715-1_1

Cited by 5 publications

(8 citation statements)

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“…where {a n } ⊂ (0, 1). The scheme defined by Mann [21] fails to converge to a fixed point of pseudo-contractive mappings. In 1974 [18] Ishikawa introduced a two-step iterative scheme to approximate the fixed points of pseudo-contractive mappings as follows…”

Section: Definition 4 [19]mentioning

confidence: 99%

“…The generalizations of Banach contraction mapping principle are attained by weakening the contractive conditions and to compensate that the structure of the metric space is enriched by endowing it with some geometrical properties. In 1955 Kranoselekii [21] proved that for non-expansive mapping Picard iteration scheme may fail to converge to a fixed point even if the map T has a unique fixed point. Browder [8], Gohde [14], Kirk [20] studied non expansive maps independently.…”

Section: Introductionmentioning

confidence: 99%

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Approximating the fixed points of Suzuki's generalized non-expansive map via an efficient iterative scheme with an application

Gautam,

Kaur

2024

Tamkang J. Math.

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This paper is aimed at proving the efficiency of a faster iterative scheme called $PC^*$-iterative scheme to approximate the fixed points for the class of Suzuki's Generalized non-expansive mapping in a uniformly convex Banach space. We will prove some weak and strong convergence results. It is justified numerically that the $PC^*$-iterative scheme converges faster than many other remarkable iterative schemes. We will also provide numerical illustrations with graphical representations to prove the efficiency of $PC^*$ iterative scheme. As an application of the solution of a fractional differential equation is obtained by using $PC^*$ iterative scheme.

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Section: Definition 4 [19]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Approximating the fixed points of Suzuki's generalized non-expansive map via an efficient iterative scheme with an application

Gautam,

Kaur

2024

Tamkang J. Math.

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“…Evaluation of the model is well-defined and tractable under a simple assumption. By a classic result 29 , it suffices to ensure, for all d , is averaged , i.e. there is and Q such that , where Q is 1-Lipschitz.…”

Section: Explainability Via Optimizationmentioning

confidence: 99%

Explainable AI via learning to optimize

Heaton¹,

Fung

2023

Sci Rep

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Indecipherable black boxes are common in machine learning (ML), but applications increasingly require explainable artificial intelligence (XAI). The core of XAI is to establish transparent and interpretable data-driven algorithms. This work provides concrete tools for XAI in situations where prior knowledge must be encoded and untrustworthy inferences flagged. We use the “learn to optimize” (L2O) methodology wherein each inference solves a data-driven optimization problem. Our L2O models are straightforward to implement, directly encode prior knowledge, and yield theoretical guarantees (e.g. satisfaction of constraints). We also propose use of interpretable certificates to verify whether model inferences are trustworthy. Numerical examples are provided in the applications of dictionary-based signal recovery, CT imaging, and arbitrage trading of cryptoassets. Code and additional documentation can be found at https://xai-l2o.research.typal.academy.

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“…Krasnoselskii-Mann (KM) iterations [35,40] are at the core of numerical methods used in optimization, fixed point theory and variational analysis, since they include many fundamental splitting algorithms whose convergence can be analyzed in a unified manner. These include the forward-backward [37,46] to approximate a zero of the sum of two maximally monotone operators, and its various particular instances: on the one hand, we have the gradient projection algorithm [31,36], the gradient method [14] and the proximal point algorithm [11,32,41,50], to cite some abstract methods, as well as the Iterative Shrinkage-Thresholding Algorithm (ISTA) [20,23], to speak more concretely.…”

Section: Introductionmentioning

confidence: 99%

Inertial Krasnoselskii-Mann Iterations

Maulén,

Fierro,

Peypouquet

2024

Set-Valued Var. Anal

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We establish the weak convergence of inertial Krasnoselskii-Mann iterations towards a common fixed point of a family of quasi-nonexpansive operators, along with estimates for the non-asymptotic rate at which the residuals vanish. Strong and linear convergence are obtained in the quasi-contractive setting. In both cases, we highlight the relationship with the non-inertial case, and show that passing from one regime to the other is a continuous process in terms of the hypotheses on the parameters. Numerical illustrations are provided for an inertial primal-dual method and an inertial three-operator splitting algorithm, whose performance is superior to that of their non-inertial counterparts.

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Successive approximations

Cited by 5 publications

References 0 publications

Approximating the fixed points of Suzuki's generalized non-expansive map via an efficient iterative scheme with an application

Approximating the fixed points of Suzuki's generalized non-expansive map via an efficient iterative scheme with an application

Explainable AI via learning to optimize

Inertial Krasnoselskii-Mann Iterations

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