Abstract
We introduce a new family of algorithms for computing fundamental operators arising from convex analysis. The new algorithms rely on the fact that the graph of the subdifferential of most convex operators depends linearly on the graph of the subdifferential of the function. By storing the subdifferential information, the computation of the conjugate is reduced to a matrix multiplication. We explain how other operators can be computed similarly, and present numerical experiments that compare graph-matrix calculus algorithms with piecewise-linear quadratic algorithms from computational convex analysis (CCA), and with a bundle method using warmstarting. Our results show that the new algorithms are an order of magnitude faster. They also add subdifferential calculus to our numerical library, and are very simple to implement.
AMS 2010 Subject Classification: 90C25, 26A51, 26B25, 47H05, 52A41
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Acknowledgements
The authors thank the two referees for their careful reading of the manuscripts and their multiple comments, which resulted in correcting an error in Lemma 12.4.
Yves Lucet was partially supported by a Discovery grant from the Natural Sciences and Engineering Research Council of Canada.
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Gardiner, B., Lucet, Y. (2011). Graph-Matrix Calculus for Computational Convex Analysis. In: Bauschke, H., Burachik, R., Combettes, P., Elser, V., Luke, D., Wolkowicz, H. (eds) Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer Optimization and Its Applications(), vol 49. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9569-8_12
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