[1] H. H. Bauschke, P. L. Combettes, and D. R. Luke. Phase retrieval, error reduction algorithm and Fienup variants : a view from convex feasibility. J. Opt. Soc.
´rence de la SMAI sur l’Optimisation et la D´ CODE–2007 : Confe ecision Institut Henri Poincar´e, Paris 18–20 avril 2007
BEST APPROXIMATION IN A NONCONVEX SETTING D. Russell Luke1 1 Department
of Mathematical Sciences University of Delaware
Projection algorithms are powerful yet simple iterative techniques for finding the intersections of sets. Perhaps the most prevalent example of a projection algorithm is the alternating Projections Onto Convex Sets (POCS) dating back to von Neumann [12]. This and algorithms like it have been applied in image processing [7], medical imaging, economics and optimal control [5]. The theory for these algorithms is limited mainly to convex setting and to consistent feasibility problems, that is problems where the set intersection is nonempty ; if the intersection is empty the problem is referred to as an inconsistent feasibility problem. Examples abound of practitioners using these methods for nonconvex and/or inconsistent problems. We have in recent years been particularly interested in projection algorithms in crystallography and astronomy [1,10], and more recently in inverse scattering [4, 6]. Until now, we have been forced to rely on convex heuristics to justify certain strategies [2,11] for want of an adequate nonconvex theory. In the absence of a nonconvex theory, practitioners resort to ad hoc stopping criteria and other strategies to get their algorithms to work according to user-defined criteria. A common phenomenon is instability that requires the user to stop the algorithm before iterates “blow up”. Using convex heuristics we were able to provide plausible explanations [3] and remedies [11] for algorithmic instabilities, however a general theory was not pursued. Our goal in this paper is two-fold : first to prove the convergence in the convex setting of an algorithm that we have proposed to solve inconsistent feasibility problems [11], and second to modify the theory to accommodate nonconvexity. Our algorithm is a relaxation of a fixed point operator used by Lions and Mercier to solve differential inclusions [9]. Our task with regard to the first goal is to characterize the fixed point set and the range of the operator, as well as to verify the assumptions of classical theorems. The novelty of our operator is that it addresses the crucial instance of inconsistent feasibility problems. Inconsistency is a source of instability for more
conventional strategies. For the second objective, in addition to characterizing the fixed point set and the range of the operator, we formulate local, nonconvex versions of convex theorems. Since our ultimate aim is to prove convergence of projection algorithms for nonconvex phase retrieval problems in crystallography, our focus will be on our particular operator, however it is hoped that the techniques used here are transferable to other cases. Our results complement other recent results on the rate of convergence of alternating projections for consistent nonconvex problems [8]. Bibliographie [1] H. H. Bauschke, P. L. Combettes, and D. R. Luke. Phase retrieval, error reduction algorithm and Fienup variants : a view from convex feasibility. J. Opt. Soc. Am. A., 19, (2002), 1334–45. [2] H. H. Bauschke, P. L. Combettes, and D. R. Luke. A hybrid projection reflection method for phase retrieval. J. Opt. Soc. Am. A., 20, (2003) ,1025–34. [3] H. H. Bauschke, P. L. Combettes, and D. R. Luke. Finding best approximation pairs relative to two closed convex sets in Hilbert spaces. J. Approx. Theory, 127, (2004), 178–92. [4] M Brigonne and M. Piana. The use of constraints for solving inverse scattering problems : physical optics and the linear sampling method. Inverse Problems, 21, (2005), 207–222. [5] Y. Censor and S. A. Zenios. Parallel Optimization : Theory Algorithms and Applications. Oxford University Press, 1997. [6] R. Chapko and R. Kress. A hybrid method for inverse boundary value problems in potential theory. J. Inv. Il l-Posed Problems, 13, (2005), 27–40. [7] P. L. Combettes. The convex feasibility problem in image recovery. In P. W. Hawkes, editor, Advances in Imaging and Electron Physics, 95, pp. 155–270. Academic Press, New York, 1996. [8] A. S. Lewis and J. Malick. Alternating projections on manifolds. Preprint, July 2006. [9] P. L. Lions and B. Mercier. Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal., 16, (1979), 964–979. [10] D. R. Luke, J. V. Burke, and R. G. Lyon. Optical wavefront reconstruction : Theory and numerical methods. SIAM Rev., 44, (2002), 169–224. [11] D. R. Luke. Relaxed averaged alternating reflections for diffraction imaging. Inv. Prob., 21, (2005), 37–50. [12] J. von Neumann. On rings of operators, reduction theory. Ann. Math., 50, (1949), 401–485.
