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Graph-Matrix Calculus for Computational Convex Analysis

In: Fixed-Point Algorithms for Inverse Problems in Science and Engineering

Author

Listed:
  • Bryan Gardiner

    (University of British Columbia Okanagan)

  • Yves Lucet

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.

Suggested Citation

  • Bryan Gardiner & Yves Lucet, 2011. "Graph-Matrix Calculus for Computational Convex Analysis," Springer Optimization and Its Applications, in: Heinz H. Bauschke & Regina S. Burachik & Patrick L. Combettes & Veit Elser & D. Russell Luke & Henry (ed.), Fixed-Point Algorithms for Inverse Problems in Science and Engineering, chapter 0, pages 243-259, Springer.
    • Handle: RePEc:spr:spochp:978-1-4419-9569-8_12
    DOI: 10.1007/978-1-4419-9569-8_12
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    Cited by:

    1. Tasnuva Haque & Yves Lucet, 2018. "A linear-time algorithm to compute the conjugate of convex piecewise linear-quadratic bivariate functions," Computational Optimization and Applications, Springer, vol. 70(2), pages 593-613, June.
    2. Bryan Gardiner & Khan Jakee & Yves Lucet, 2014. "Computing the partial conjugate of convex piecewise linear-quadratic bivariate functions," Computational Optimization and Applications, Springer, vol. 58(1), pages 249-272, May.

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