scholarly journals Duality Mapping for Schatten Matrix Norms

2021 ◽  
pp. 1-17
Author(s):  
Shayan Aziznejad ◽  
Michael Unser
Keyword(s):  
Duality Mapping ◽  
Matrix Norms

scholarly journals The Global Convergence of the Nonlinear Power Method for Mixed-Subordinate Matrix Norms

2021 ◽  
Vol 88 (1) ◽  
Author(s):  
Antoine Gautier ◽  
Matthias Hein ◽  
Francesco Tudisco
Keyword(s):  
Global Convergence ◽  
Convergence Theorem ◽  
Matrix Norm ◽  
Nonnegative Matrices ◽  
Np Hard ◽  
Power Method ◽  
Contraction Ratio ◽  
Matrix Norms ◽  
Vector Norms

AbstractWe analyze the global convergence of the power iterates for the computation of a general mixed-subordinate matrix norm. We prove a new global convergence theorem for a class of entrywise nonnegative matrices that generalizes and improves a well-known results for mixed-subordinate $$\ell ^p$$ ℓ p matrix norms. In particular, exploiting the Birkoff–Hopf contraction ratio of nonnegative matrices, we obtain novel and explicit global convergence guarantees for a range of matrix norms whose computation has been recently proven to be NP-hard in the general case, including the case of mixed-subordinate norms induced by the vector norms made by the sum of different $$\ell ^p$$ ℓ p -norms of subsets of entries.

On certain classes of matrix norms

1983 ◽  
Vol 14 (4) ◽  
pp. 357-363
Author(s):  
Massoud Malek-Shahmirzadi ◽  
Kunio Oshima
Keyword(s):  
Matrix Norms

scholarly journals Iterative Algorithm for Common Fixed Points of Infinite Family of Nonexpansive Mappings in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Songnian He ◽  
Jun Guo
Keyword(s):  
Fixed Points ◽  
Iterative Algorithm ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Common Fixed Points ◽  
Nonempty Closed Convex Subset ◽  
Duality Mapping ◽  
Normalized Duality Mapping ◽  
Uniformly Smooth ◽  
Computational Work

LetCbe a nonempty closed convex subset of a real uniformly smooth Banach spaceX,{Tk}k=1∞:C→Can infinite family of nonexpansive mappings with the nonempty set of common fixed points⋂k=1∞Fix⁡(Tk), andf:C→Ca contraction. We introduce an explicit iterative algorithmxn+1=αnf(xn)+(1-αn)Lnxn, whereLn=∑k=1n(ωk/sn)Tk,Sn=∑k=1nωk,  andwk>0with∑k=1∞ωk=1. Under certain appropriate conditions on{αn}, we prove that{xn}converges strongly to a common fixed pointx*of{Tk}k=1∞, which solves the following variational inequality:〈x*-f(x*),J(x*-p)〉≤0,    p∈⋂k=1∞Fix(Tk), whereJis the (normalized) duality mapping ofX. This algorithm is brief and needs less computational work, since it does not involveW-mapping.

Explicit algorithms for J-fixed points of some non linear mappings in certain Banach spaces

2020 ◽  
Vol 29 (1) ◽  
pp. 27-36
Author(s):  
M. M. GUEYE ◽  
M. SENE ◽  
M. NDIAYE ◽  
N. DJITTE
Keyword(s):  
Fixed Points ◽  
Banach Spaces ◽  
Fixed Point Theory ◽  
Maximal Monotone ◽  
Point Theory ◽  
Duality Mapping ◽  
Monotone Mappings ◽  
Normalized Duality Mapping ◽  
Maximal Monotone Mappings ◽  
Pseudocontractive Mappings

Let E be a real normed linear space and E∗ its dual. In a recent work, Chidume et al. [Chidume, C. E. and Idu, K. O., Approximation of zeros of bounded maximal monotone mappings, solutions of hammerstein integral equations and convex minimizations problems, Fixed Point Theory and Applications, 97 (2016)] introduced the new concepts of J-fixed points and J-pseudocontractive mappings and they shown that a mapping A : E → 2 E∗ is monotone if and only if the map T := (J −A) : E → 2 E∗ is J-pseudocontractive, where J is the normalized duality mapping of E. It is our purpose in this work to introduce an algorithm for approximating J-fixed points of J-pseudocontractive mappings. Our results are applied to approximate zeros of monotone mappings in certain Banach spaces. The results obtained here, extend and unify some recent results in this direction for the class of maximal monotone mappings in uniformly smooth and strictly convex real Banach spaces. Our proof is of independent interest.

scholarly journals Lower bounds for tau coefficients and operator norms using composite matrix norms

1987 ◽  
Vol 35 (1) ◽  
pp. 49-57
Author(s):  
Choon Peng Tan
Keyword(s):  
Lower Bounds ◽  
General Class ◽  
Special Class ◽  
Operator Norm ◽  
Composite Matrix ◽  
Operator Norms ◽  
Class Of Matrices ◽  
Matrix Norms

Lower bounds for the tau coefficients and operator norms are derived by using composite matrix norms. For a special class of matrices B, our bounds on ‖B‖p (the operator norm of B induced by the ℓp norm) improve upon a general class of Maitre (1967) bounds for p ≥ 2.

scholarly journals Nonlinear potentials in function spaces

2002 ◽  
Vol 165 ◽  
pp. 91-116 ◽  
Author(s):  
Murali Rao ◽  
Zoran Vondraćek
Keyword(s):  
Banach Space ◽  
Potential Theory ◽  
Finite Energy ◽  
Duality Mapping ◽  
Quadratic Functional ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Nonlinear Potential Theory ◽  
Nonlinear Potential ◽  
Nonlinear Potentials

We introduce a framework for a nonlinear potential theory without a kernel on a reflexive, strictly convex and smooth Banach space of functions. Nonlinear potentials are defined as images of nonnegative continuous linear functionals on that space under the duality mapping. We study potentials and reduced functions by using a variant of the Gauss-Frostman quadratic functional. The framework allows a development of other main concepts of nonlinear potential theory such as capacities, equilibrium potentials and measures of finite energy.

scholarly journals Path Convergence and Approximation of Common Zeroes of a Finite Family ofm-Accretive Mappings in Banach Spaces

2010 ◽  
Vol 2010 ◽  
pp. 1-14 ◽  
Author(s):  
Yekini Shehu ◽  
Jerry N. Ezeora
Keyword(s):  
Real Banach Space ◽  
Iterative Scheme ◽  
Finite Family ◽  
Duality Mapping ◽  
Strong Convergence Theorem ◽  
Sunny Nonexpansive Retraction ◽  
Mild Conditions ◽  
Pseudocontractive Mappings ◽  
Uniformly Smooth ◽  
Accretive Mappings

LetEbe a real Banach space which is uniformly smooth and uniformly convex. LetKbe a nonempty, closed, and convex sunny nonexpansive retract ofE, whereQis the sunny nonexpansive retraction. IfEadmits weakly sequentially continuous duality mappingj, path convergence is proved for a nonexpansive mappingT:K→K. As an application, we prove strong convergence theorem for common zeroes of a finite family ofm-accretive mappings ofKtoE. As a consequence, an iterative scheme is constructed to converge to a common fixed point (assuming existence) of a finite family of pseudocontractive mappings fromKtoEunder certain mild conditions.

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