Best approximation in inner product spaces

By Frank Deutsch. 2001

244 citations recorded by the source corpus.

Subfields: Computer Graphics and Computer-Aided Design, Computer Networks and Communications, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/07656cff6f4157a9e9be17e729955bee3481c066

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In Computer Graphics and Computer-Aided Design

In Computer Networks and Communications

In Mathematical Physics

Citations in the corpus

Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Lojasiewicz Inequality (2008, unknown): “s proved of fundamental importance in the applications of the method to the numerical analysis of ill-posed inverse problems. A rich litterature has been devoted to this subject, one may consult [7], [16], [14]. Indeed, in recent papers [5], [4] a new class of alternating minimization algorithms wit”
Randomized Methods for Linear Constraints: Convergence Rates and Conditioning (2008, background): “ulus of regularity of aects the convergence rate of iterated projection algorithms. We remark that linear convergence for iterated projection methods on convex sets has been very widely studied: see [5], for example. Our aim here is to observe, by analogy with previous sections, how randomization ma”
Linear convergence of the Douglas–Rachford method for two closed sets (2014, unknown): “, [5, 12, 18, 26, 27] and the references therein.”
The asymptotic behavior of the composition of two resolvents (2005, result): “We now provide a novel proof of his result (see [37] and [ 21, Chapter 9 ] for further information, and [13, section 3] for a recent elementary proof).”
A new proximal point iteration that converges weakly but not in norm (2005, background): “Fix n 2 N. The well-known formula for the projection onto a hyperplane [ 8, Section 6.17 ] now yields yn 6= PHyn = yn h f,ynif = yn + |hf,yni|f and 0 < kyn PHynk = |hf,yni| = n (recall kfk = 1 by Example 4.1.(i) and hf,yni < 0 by Example 4.1.(iii)&(iv)).”
A Quadratically Convergent Newton Method for Computing the Nearest Correlation Matrix (2006, background): “The property CHIP was initially characterized by Chui, Deutsch, and Ward [9] and was refined by Deutsch, Li, and Ward [14] to strong CHIP, which turns out to be a necessary and sufficient condition for the solution of (2) to have representation (3).”
Convex Functions: Constructions, Characterizations and Counterexamples (2010, methodology): “Aproof based on the Brouwer fixed point theorem is given in [175] and sketched in [72, Proposition 5].”
Restricted Normal Cones and the Method of Alternating Projections: Theory (2012, background): “We point readers interested in reviewing the rich history of this algorithm to [2], [6], [9], and the references therein.”
Dykstra's Algorithm, ADMM, and Coordinate Descent: Connections, Insights, and Extensions (2017, methodology): “We will reference many of these works in the coming sections, when we discuss connections between Dykstra’s algorithm and other methods; for other developments, see the comprehensive books Deutsch (2001); Bauschke and Combettes (2011) and review article Bauschke and Koch (2013).”