zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
The rate of convergence of Dykstra’s cyclic projections algorithm: The polyhedral case. (English) Zbl 0807.41019
Summary: Suppose $K$ is the intersection of a finite number of closed half-spaces in a Hilbert space $X$. Starting with any point $x\in X$, it is shown that the sequence of iterates $\left\{{x}_{n}\right\}$ generated by Dykstra’s cyclic projections algorithm satisfies the inequality $\parallel {x}_{n}-{P}_{K}\left(x\right)\parallel \le \rho {c}^{n}$ for all $n$, where ${P}_{K}\left(x\right)$ is the nearest point in $K$ to $x$, $\rho$ is a constant, and $0\le c<1$. In the case when $K$ is the intersection of just two closed half-spaces, a stronger result is established: the sequence of iterates is either finite or satisfies $\parallel {x}_{n}-{P}_{K}\left(x\right)\parallel \le {c}^{n-1}\parallel x-{P}_{K}\left(x\right)\parallel$ for all $n$, where $c$ is the cosine of the angle between the two functionals which define the half-spaces. Moreover, the constant $c$ is the best possible. Applications are made to isotone and convex regression, and linear and quadratic programming.
MSC:
 41A65 Abstract approximation theory 47N10 Applications of operator theory in optimization, convex analysis, programming, economics 49M30 Other numerical methods in calculus of variations