Due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences (most notably, computerized tomography), algorithms for solving convex feasibility problems continue to receive great attention. To unify, generalize, and review some of these algorithms, a very broad and flexible framework is investigated . Several crucial new concepts which allow a systematic discussion of questions on behaviour in general Hilbert spaces and on the quality of convergence are brought out. Numerous examples are given. 1991 M.R. Subject Classification. Primary 47H09, 49M45, 65-02, 65J05, 90C25; Secondary 26B25, 41A65, 46C99, 46N10, 47N10, 52A05, 52A41, 65F10, 65K05, 90C90, 92C55. Key words and phrases. Angle between two subspaces, averaged mapping, Cimmino's method, computerized tomography, convex feasibility problem, convex function, convex inequalities, convex programming, convex set, Fej'er monotone sequence, firmly nonexpansive mapping, H...

A unified fixed point theoretic framework is proposed to investigate the asymptotic behavior of algorithms for finding solutions to monotone inclusion problems. The basic iterative scheme under consideration involves nonstationary compositions of perturbed averaged nonexpansive operators. The analysis covers proximal methods for common zero problems as well as various splitting methods for finding a zero of the sum of monotone operators.

We consider a wide class of iterative methods arising in numerical mathematics and optimization which are known to converge only weakly. Exploiting an idea originally proposed by Haugazeau, we present a simple modification of these methods which makes them strongly convergent without additional assumptions. Several applications are discussed. AMS 1991 subject classification. Primary: 65J15, 47N10; secondary 41A29, 47H05, 47H09, 65K10, 90C25. Key words. Convex feasibility, Fej'er-monotonicity, firmly nonexpansive mapping, fixed point, Haugazeau, maximal monotone operator, projection, proximal point algorithm, resolvent, subgradient algorithm. 1 Introduction Let H be a real Hilbert space with scalar product h\Delta j \Deltai, norm k \Delta k, and distance d. In 1965, Bregman [5] proposed a simple iterative method for finding a common point of m intersecting closed convex sets (S i ) 1im in H. He showed that, given an arbitrary starting point x 0 2 H, the sequence (x n ) n0 gene...

The phase retrieval problem is of paramount importance in various areas of applied physics and engineering. The state of the art for solving this problem in two dimensions relies heavily on the pioneering work of Gerchberg, Saxton, and Fienup. Despite the widespread use of the algorithms proposed by these three researchers, current mathematical theory cannot explain their remarkable success. Nevertheless, great insight can be gained into the behavior, the shortcomings, and the performance of these algorithms from their possible counterparts in convex optimization theory. An important step in this direction was made two decades ago when the error reduction algorithm was identified as a nonconvex alternating projection algorithm. The purpose of this paper is to formulate the phase retrieval problem with mathematical care and to establish new connections between well established numerical phase retrieval schemes and classical convex optimization methods. Specifically, it is shown that Fienup’s basic inputoutput algorithm corresponds to Dykstra’s algorithm, and that Fienup’s hybrid input-output algorithm can be viewed as an instance of the Douglas-Rachford algorithm. This work provides a theoretical framework to better understand and, potentially, improve existing phase recovery algorithms. 1 1

We consider the problem of nding a best approximation pair, i.e., two points which achieve the minimum distance between two closed convex sets in a Hilbert space. When the sets intersect, the method under consideration, termed ASR for averaged successive reections, is a special instance of an algorithm due to Lions and Mercier for nding a zero of the sum of two maximal monotone operators. We investigate systematically the asymptotic behavior of ASR in the general case when the sets do not necessarily intersect and show that the method produces best approximation pairs provided they exist. Finitely many sets are handled in a product space, in which case the ASR method is shown to coincide with a special case of Spingarn's method of partial inverses.

The strong conical hull intersection property and bounded linear regularity are properties of a collection of finitely many closed convex intersecting sets in Euclidean space. These fundamental notions occur in various branches of convex optimization (constrained approximation, convex feasibility problems, linear inequalities, for instance). It is shown that the standard constraint qualification from convex analysis implies bounded linear regularity, which in turn yields the strong conical hull intersection property. Jameson's duality for two cones, which relates bounded linear regularity to property (G), is re-derived and refined. For polyhedral cones, a statement dual to Hoffman's error bound result is obtained. A sharpening of a result on error bounds for convex inequalities by Auslender and Crouzeix is presented. Finally, for two subspaces, property (G) is quantified by the angle between the subspaces. 1991 M.R. Subject Classification. Primary 90C25; Secondary 15A39, 41A29, 46A40...

Abstract — We consider the problem of synthesizing feasible signals in a Hilbert space in the presence of inconsistent convex constraints, some of which must imperatively be satisfied. This problem is formalized as that of minimizing a convex objective measuring the amount of violation of the soft constraints over the intersection of the sets associated with the hard ones. The resulting convex optimization problem is analyzed, and numerical solution schemes are presented along with convergence results. The proposed formalism and its algorithmic framework unify and extend existing approaches to inconsistent signal feasibility problems. An application to signal synthesis is demonstrated. Index Terms—Convex feasibility problem, fixed point, Hilbert space, inconsistent constraints, monotone operator, optimization,

Dykstra's algorithm and the method of cyclic Bregman projections are often employed to solve best approximation and convex feasiblity problems, which are fundamental in mathematics and the physical sciences. Censor and Reich very recently suggested a synthesis of these methods, Dykstra's algorithm with Bregman projections, to tackle a non-orthogonal best approximation problem. They obtained convergence when each constraint is a halfspace. It is shown here that this new algorithm works for general closed convex constraints; this complements Censor and Reich's result and relates to a framework by Tseng. The proof rests on Boyle and Dykstra's original work and on strong properties of Bregman distances corresponding to Legendre functions. Special cases and observations simplifying the implementation of the algorithm are also discussed. 1991 M.R. Subject Classication. Primary 49M; Secondary 41A29, 65J05, 90C25. Key words and phrases. Best approximation, Bregman distance, Bregman projecti...

. The powerful von Neumann-Halperin method of alternating projections (MAP) is an algorithm for determining the best approximation to any given point in a Hilbert space from the intersection of a nite number of subspaces. It achieves this by reducing the problem to an iterative scheme which involves only computing best approximations from the individual subspaces which make up the intersection. The main practical drawback of this algorithm, at least for some applications, is that the method is slowly convergent. In this paper, we consider a general class of iterative methods which includes the MAP as a special case. For such methods, we study an \accelerated" version of this algorithm which was considered earlier by Gubin, Polyak, and Raik [14] and by Gearhart and Koshy [13]. We show that the accelerated algorithm converges faster than the MAP in the case of two subspaces, but is in general not faster than the MAP for more than two subspaces! However, for a \symmetric" version of the ...

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