Abstract
The convex feasibility problem asks to find a point in the intersection of a collection of nonempty closed convex sets. This problem is of basic importance in mathematics and the physical sciences, and projection (or splitting) methods solve it by employing the projection operators associated with the individual sets to generate a sequence which converges to a solution. Motivated by an application in road design, we present the method of cyclic intrepid projections (CycIP) and provide a rigorous convergence analysis. We also report on very promising numerical experiments in which CycIP is compared to a commerical state-of-the-art optimization solver.
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Notes
- 1.
Given a nonempty subset \(S\) of \(X\) and \(x\in X\), we write \(d_S(x) := \inf _{s\in S}\Vert x-s\Vert \) for the distance from \(x\) to \(S\). If \(S\) is also closed and convex, then the infimum defining \(d_S(x)\) is attained at a unique vector called the projection of \(x\) onto \(S\) and denoted by \(P_S(x)\) or \(P_Sx\).
- 2.
This function is considered, e.g., in [7, Exercise 8.5].
- 3.
In [5], the authors compared CycIP with a Swiss Army Knife. The Wenger Swiss Army Knife version XXL, listed in the Guinness Book of World Records as the world’s most multi-functional penknife, contains 87 tools.
- 4.
Recall that if \(x=(\xi _1,\ldots ,\xi _n)\in X\), then \(\Vert x\Vert _\infty = \max \{|\xi _1|,\ldots ,|\xi _n|\}\).
- 5.
For further information on performance profiles, we refer the reader to [14].
- 6.
Recall that if \(x=(\xi _1,\ldots ,\xi _n)\in X\), then \(\Vert x\Vert _1 = |\xi _1|+\cdots +|\xi _n|\).
- 7.
To allow for a more fair comparison, we included in wall-clock time only the time required for running the solver’s software itself (and not the time for loading the problem data or for setting up the solver’s parameters).
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Acknowledgments
The authors thank the referee for very careful reading and constructive comments, Dr. Ramon Lawrence for the opportunity to run the numerical experiments on his server, and Scott Fazackerley and Wade Klaver for technical help. HHB also thanks Dr. Masato Wakayama and the Institute of Mathematics for Industry, Kyushu University, Fukuoka, Japan for their hospitality —some of this research benefited from the extremely stimulating environment during the “Math-for-Industry 2013” forum. HHB was partially supported by the Natural Sciences and Engineering Research Council of Canada (Discovery Grant and Accelerator Supplement) and by the Canada Research Chair Program.
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Bauschke, H.H., Iorio, F., Koch, V.R. (2014). The Method of Cyclic Intrepid Projections: Convergence Analysis and Numerical Experiments. In: Wakayama, M., et al. The Impact of Applications on Mathematics. Mathematics for Industry, vol 1. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54907-9_14
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