Dykstra's algorithm for constrained least-squares rectangular matrix problems

Escalante, René; Raydan, Marcos

doi:10.1016/s0898-1221(98)00020-0

Cited by 20 publications

(16 citation statements)

References 13 publications

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“…Unfortunately, the only known approach for using alternating projection methods on the general problem (41) is based on the use of the singular value decomposition (SVD) of the matrix A (see for instance [15]), and this could lead to a prohibitive amount of computational work in the large scale case. However, problem (41) can be viewed as a particular case of (1), in which f : IR ncols×(ncols+1)/2 → IR, is given by…”

Section: Test Problemmentioning

confidence: 99%

Inexact spectral projected gradient methods on convex sets

Birgin¹

2003

IMA Journal of Numerical Analysis

166

178

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A new method is introduced for large scale convex constrained optimization. The general model algorithm involves, at each iteration, the approximate minimization of a convex quadratic on the feasible set of the original problem and global convergence is obtained by means of nonmonotone line searches. A specific algorithm, the Inexact Spectral Projected Gradient method (ISPG), is implemented using inexact projections computed by Dykstra's alternating projection method and generates interior iterates. The ISPG method is a generalization of the Spectral Projected Gradient method (SPG), but can be used when projections are difficult to compute. Numerical results for constrained least-squares rectangular matrix problems are presented.

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Section: Test Problemmentioning

confidence: 99%

Inexact spectral projected gradient methods on convex sets

Birgin¹

2003

IMA Journal of Numerical Analysis

166

178

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“…The solution x * is called the projection of x 0 onto Ω and is denoted by P Ω (x 0 ). Dykstra's algorithm for solving (1) has been extensively studied since it fits in many different applications (see [1,2,4,8,9,11,12,13,18,21,23,24,26,28,29]). Here, we consider the case…”

Section: Introductionmentioning

confidence: 99%

Robust Stopping Criteria for Dykstra's Algorithm

Birgin¹,

Raydan²

2005

SIAM J. Sci. Comput.

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Dykstra's algorithm is a suitable alternating projection scheme for solving the optimization problem of finding the closest point to a given one in the intersection of a finite number of closed and convex sets. It has been recently used in a wide variety of applications. However, in practice, the commonly used stopping criteria are not robust and could stop the iterative process prematurely at a point that does not solve the optimization problem. In this work we present a counter-example to illustrate the weakness of the commonly used criteria, and then we develop robust stopping rules. Additional experimental results are shown to illustrate the advantages of this new stopping criteria, including their associated computational cost.

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“…Among all extensions and variants of APM, it is worth mentioning that Dykstra and Boyle [21], [5] found a suitable modification of von Neumann's scheme for closed and convex sets. APM and their variants have been used by many researches to solve problems on a wide variety of applications [4,6,10,22,23,27,29,32,35,36,38].…”

Section: Alternating Projection Methods (Apm)mentioning

confidence: 99%