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Metric and Bregman projections onto affine subspaces and their computation via sequential subspace optimization methods. (English) Zbl 1160.46048

Let \(X\) be a reflexive, smooth and strictly convex Banach space and let \(C\neq \emptyset\) be a closed convex subset of \(X.\) The metric projection of an element \(x\in X\) onto \(C\) is denoted by \(P_{C}(x)\) (the best approximation element of \(x\) in \(C\)). The Bregman projection of \(x\in X\) onto \(C\) with respect to the function \(f_{p}(x)=\left\| x\right\| ^{p}/p\) \((p>1)\) is the unique element \(\prod_{C}^{p}(x)\) in \(C\) such that \(\;\Delta_{p}(x,\prod_{C}^{p}(x))=\min \left\{\Delta_{p}(x,y):y\in C\right\} \), where \(\Delta_{p}(x,y)=\frac{1}{q}\left\| x\right\| ^{p}-\langle J_{p}(x)\mid y\rangle + \frac{1}{p}\left\| y\right\| ^{p}\;(\frac{1}{p}+ \frac{1}{q}=1,\;J_{p}:X\rightarrow 2^{X^{\ast }}\;\) is the duality mapping with gauge function \(t\mathcal{\rightarrow }t^{p-1})\) is the so-called Bregman distance between \(x\) and \(y.\) The connection between \(P_{C}(x)\) and \( \prod_{C}^{p}(x)\) is the following: \(P_{C}(x)-x=\prod_{C-x}^{p}(x).\)
The authors present some known as well as new interesting properties of these projections and obtain efficient iterative methods to compute Bregman projections onto affine subspaces of the form \(z+\mathcal{N(}A\mathcal{)}\) and \(z+\overline{\mathcal{R}(A)},\) where \(\mathcal{N(}A\mathcal{)}\) is the kernel and \(\overline{\mathcal{R}\mathbb{(}A\mathbb{)}}\) is the closure of the range of the continuous linear operator \(A:X\rightarrow Y\). Numerical experiments to compute \(p\)-norm solutions of matrix equations \(Ax=y\) for different values \(p\) and dimensions \(N\) of the search space are presented.

MSC:

46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
65K10 Numerical optimization and variational techniques

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[1] DOI: 10.1023/A:1022631928592 · Zbl 0886.90179 · doi:10.1023/A:1022631928592
[2] DOI: 10.1016/j.jmaa.2005.03.027 · Zbl 1095.46007 · doi:10.1016/j.jmaa.2005.03.027
[3] DOI: 10.1142/S0219199701000524 · Zbl 1032.49025 · doi:10.1142/S0219199701000524
[4] DOI: 10.1016/0041-5553(67)90040-7 · doi:10.1016/0041-5553(67)90040-7
[5] Butnariu D., Abstract and Applied Analysis pp 2006– (2006)
[6] DOI: 10.1016/j.acha.2007.02.002 · Zbl 1133.65022 · doi:10.1016/j.acha.2007.02.002
[7] Figiel T., Stud. Math. 56 pp 121– (1976)
[8] DOI: 10.1155/S1085337598000451 · Zbl 1030.90136 · doi:10.1155/S1085337598000451
[9] DOI: 10.1088/0266-5611/22/1/017 · Zbl 1088.65052 · doi:10.1088/0266-5611/22/1/017
[10] DOI: 10.1016/j.jat.2004.06.002 · Zbl 1067.46009 · doi:10.1016/j.jat.2004.06.002
[11] Stoer J., Math. Mech. 75 pp 69– (1995)
[12] DOI: 10.1016/0022-247X(91)90144-O · Zbl 0757.46034 · doi:10.1016/0022-247X(91)90144-O
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.