Journal of Convex Analysis 21 (2014), No. 1, 219--235
Copyright Heldermann Verlag 2014
On Moving Averages
Heinz H. Bauschke
Dept. of Mathematics, University of British Columbia, Kelowna, B.C. V1V 1V7, Canada
heinz.bauschke@ubc.ca
Joshua Sarada
59 Tuscany Valley Hts NW, Calgary, Alberta T3L 2E7, Canada
jshsarada@gmail.com
Xianfu Wang
Dept. of Mathematics, University of British Columbia, Kelowna, B.C. V1V 1V7, Canada
shawn.wang@ubc.ca
We show that the moving arithmetic average is closely connected to a Gauss-Seidel type fixed point method studied by H. H. Bauschke, X. Wang and C. J. S. Wylie [Fixed points of averages of resolvents: geometry and algorithms, SIAM J. Optimization 22 (2012) 24--40] and which was observed to converge only numerically. Our analysis establishes a rigorous proof of convergence of their algorithm in a special case; moreover, the limit is explicitly identified. Moving averages in Banach spaces and Kolmogorov means are also studied. Furthermore, we consider moving proximal averages and epi-averages of convex functions.
Keywords: Arithmetic mean, difference equation, epi-average, Kolmogorov mean, linear recurrence relation, means, moving average, proximal average, stochastic matrix.
MSC: 15B51, 26E60, 47H10; 39A06, 65H04, 47J25, 49J53
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