The Method of Cyclic Intrepid Projections: Convergence Analysis and Numerical Experiments
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.
KeywordsConvex set Feasibility problem Halfspace Intrepid projection Linear inequalities Projection Road design
AMS 2010 Subject ClassificationPrimary 65K05 90C25 Secondary 90C05
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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