Dykstra’s Algorithm and Robust Stopping Criteria

Birgin, Ernesto G.; Raydan, Marcos

doi:10.1007/978-0-387-74759-0_143

Cited by 4 publications

(5 citation statements)

References 40 publications

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“…1b. Dykstra's alternating projection algorithm is a superior alternative, which we choose because it achieves projection CPT P(C E ) onto the closest point in the intersection of the sets [24]. A similar method was used in Ref [25] to solve a feasibility problem where input and output states are given and an exact but non-unique solution is sought.…”

Section: Composite Projectionmentioning

confidence: 99%

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Quantum process tomography via completely positive and trace-preserving projection

et al. 2018

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We present an algorithm for projecting superoperators onto the set of completely positive, tracepreserving maps. When combined with gradient descent of a cost function, the procedure results in an algorithm for quantum process tomography: finding the quantum process that best fits a set of sufficient observations. We compare the performance of our algorithm to the diluted iterative algorithm as well as second-order solvers interfaced with the popular cvx package for matlab, and find it to be significantly faster and more accurate while guaranteeing a physical estimate.

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Section: Composite Projectionmentioning

confidence: 99%

“…By contrast the ML approach (which we adopt) finds the best possible fit to some given, noisy data and treats the more realistic situation where an exact solution cannot be expected. Pseudo-code for this subroutine (showing our use of a robust stopping criterion due to Birgin and Raydan [24]) is given below.…”

Section: Composite Projectionmentioning

confidence: 99%

Quantum process tomography via completely positive and trace-preserving projection

et al. 2018

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“…, m} such that m divides k − k ′ . We refer the reader to [BB96,BR09,ER11], as well as [Deu01,Chapter 9] and [BZ05, Subsubsection 4.5.4], for more on the literature of using projection methods to solve the SIP.…”

Section: Introductionmentioning

confidence: 99%

First order constrained optimization algorithms with feasibility updates

Pang¹

2015

Preprint

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We propose first order algorithms for convex optimization problems where the feasible set is described by a large number of convex inequalities that is to be explored by subgradient projections. The first algorithm is an adaptation of a subgradient algorithm, and has convergence rate 1/ √ k. The second algorithm has convergence rate 1/k when (1) one has linear metric inequality in the feasible set, (2) the objective function is strongly convex, differentiable and has Lipschitz gradient, and (3) it is easy to optimize the objective function on the intersection of two halfspaces. This second algorithm generalizes Haugazeau's algorithm. The third algorithm adapts the second algorithm when condition (3) is dropped. We give examples to show that the second algorithm performs poorly when the objective function is not strongly convex, or when the linear metric inequality is absent.

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“…Acceleration techniques include [GPR67, GK89,BDHP03]. For more on the background and recent developments of the MAP and its variants, we refer the reader to [BB96,BR09,ER11], as well as [Deu01b, Chapter 9], [BZ05, Subsubsection 4.5.4], and [Deu01a,Deu95]. Superlinear convergence can be achieved if the smoothness of the sets K i are exploited [GP98,GP01].…”

Section: Introductionmentioning

confidence: 99%

Set intersection problems: supporting hyperplanes and quadratic programming

Pang

2014

Math. Program.

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The Set Intersection Problem (SIP) is the problem of finding a point in the intersection of convex sets. This problem is typically solved by the method of alternating projections. To accelerate the convergence, the idea of using Quadratic Programming (QP) to project a point onto the intersection of halfspaces generated by the projection process was discussed in earlier papers. This paper looks at how one can integrate projection algorithms together with an active set QP algorithm. As a byproduct of our analysis, we show how to accelerate an SIP algorithm involving box constraints, and how to extend a version of the Algebraic Reconstruction Technique (ART) while preserving finite convergence. Lastly, the warmstart property of active set QP algorithms is a valuable property for the problem of projecting onto the intersection of convex sets.

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Dykstra’s Algorithm and Robust Stopping Criteria

Cited by 4 publications

References 40 publications

Quantum process tomography via completely positive and trace-preserving projection

Quantum process tomography via completely positive and trace-preserving projection

First order constrained optimization algorithms with feasibility updates

Set intersection problems: supporting hyperplanes and quadratic programming

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