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Set intersection problems: supporting hyperplanes and quadratic programming

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Abstract

We study how the supporting hyperplanes produced by the projection process can complement the method of alternating projections and its variants for the convex set intersection problem. For the problem of finding the closest point in the intersection of closed convex sets, we propose an algorithm that, like Dykstra’s algorithm, converges strongly in a Hilbert space. Moreover, this algorithm converges in finitely many iterations when the closed convex sets are cones in \({\mathbb {R}}^{n}\) satisfying an alignment condition. Next, we propose modifications of the alternating projection algorithm, and prove its convergence. The algorithm converges superlinearly in \({\mathbb {R}}^{n}\) under some nice conditions. Under a conical condition, the convergence can be finite. Lastly, we discuss the case where the intersection of the sets is empty.

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Acknowledgments

We thank Boris Mordukhovich for asking the question of whether the alternating projection algorithm can achieve superlinear convergence at the ISMP in 2012 during Russell Luke’s talk, which led to the idea of considering supporting hyperplanes. We also thank the referees and associate editor for their helpful suggestions and additional references. The author gratefully acknowledges his startup grant at NUS.

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  1. Department of Mathematics, National University of Singapore, Block S17 08-11, 10 Lower Kent Ridge Road, Singapore, 119076, Singapore

    C. H. Jeffrey Pang

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  1. C. H. Jeffrey Pang
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Pang, C.H.J. Set intersection problems: supporting hyperplanes and quadratic programming. Math. Program. 149, 329–359 (2015). https://doi.org/10.1007/s10107-014-0759-z

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