On the Asymptotic Directions of the S-Dimensional Optimum Gradient Method

Forsythe, George E.

doi:10.21236/ad0650619

Cited by 27 publications

(57 citation statements)

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“…The above definition is important and is used for some other gradient methods (see Forsythe, 1968;Dai & Yang, 2001). For the case m = 2, we can obtain by (2.1), (2.2) and (3.4) and the definition of u k that…”

Section: Analysis For the Case M = 2 And N =mentioning

confidence: 99%

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The cyclic Barzilai-–Borwein method for unconstrained optimization

Dai¹,

Hager²,

Schittkowski³

et al. 2006

IMA Journal of Numerical Analysis

172

163

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In the cyclic Barzilai-Borwein (CBB) method, the same Barzilai-Borwein (BB) stepsize is reused for m consecutive iterations. It is proved that CBB is locally linearly convergent at a local minimizer with positive definite Hessian. Numerical evidence indicates that when m > n/2 3, where n is the problem dimension, CBB is locally superlinearly convergent. In the special case m = 3 and n = 2, it is proved that the convergence rate is no better than linear, in general. An implementation of the CBB method, called adaptive cyclic Barzilai-Borwein (ACBB), combines a non-monotone line search and an adaptive choice for the cycle length m. In numerical experiments using the CUTEr test problem library, ACBB performs better than the existing BB gradient algorithm, while it is competitive with the well-known PRP+ conjugate gradient algorithm.

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Section: Analysis For the Case M = 2 And N =mentioning

confidence: 99%

“…It is well known that SD can be very slow when the Hessian of f is ill-conditioned at a local minimum (see Akaike, 1959;Forsythe, 1968). In this case, the iterates slowly approach the minimum in a zigzag fashion.…”

Section: Introductionmentioning

confidence: 99%

The cyclic Barzilai-–Borwein method for unconstrained optimization

Dai¹,

Hager²,

Schittkowski³

et al. 2006

IMA Journal of Numerical Analysis

172

163

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“…Richardson-like methods for solving (3) correspond to gradient algorithms for minimizing (4) and obey to the following iterations…”

mentioning

confidence: 99%

“…see [4,15]. Hence, although one regularly observes values of R CR k that are significantly smaller than R * k for k < n (and although R CR n = 0, that is, the solution is found exactly in n iterations in the exact arithmetic), for any k < n one has max x 0 ,A R CR k = R * k , with (R * k ) 1/k decreasing monotonically to R ∞ as k → ∞.…”

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confidence: 99%

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Estimation of Spectral Bounds in Gradient Algorithms

2012

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We consider the solution of linear systems of equations Ax = b, with A a symmetric positive-definite matrix in R n×n , through Richardson-type iterations or, equivalently, the minimization of convex quadratic functions (1/2)(Ax, x) − (b, x) with a gradient algorithm. The use of step-sizes asymptotically distributed with the arcsine distribution on the spectrum of A then yields an asymptotic rate of convergence after k < n iterations, k → ∞, that coincides with that of the conjugate-gradient algorithm in the worst case. However, the spectral bounds m and M are generally unknown and thus need to be estimated to allow the construction of simple and cost-effective gradient algorithms with fast convergence. It is the purpose of this paper to analyse the properties of estimators of m and M based on moments of probability measures ν k defined on the spectrum of A and generated by the algorithm on its way towards the optimal solution. A precise analysis of the behavior of the rate of convergence of the algorithm is also given. Two situations are considered: (i) the sequence of step-sizes corresponds to i.i.d. random variables, (ii) they are generated through a dynamical system (fractional parts of the golden ratio) producing a low-discrepancy sequence. In the first case, properties of random walk can be used to prove the convergence of simple spectral bound estimators based on the first moment of ν k . The second option requires a more careful choice of spectral bounds estimators but is shown to produce much less fluctuations for the rate of convergence of the algorithm.

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A new fast‐lock, low‐jitter, and all‐digital frequency synthesizer for DVB‐T receivers

Gholami

Rahimpour

Ardeshir

et al. 2013

Circuit Theory & Apps

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SUMMARYLock time and convergence time are the most important challenges in delay-locked loops (DLLs). In this paper we cover French very high frequency band with a novel all-digital fast-lock DLL-based frequency synthesizer. Because this new architecture uses a digital signal processing unit instead of using phase frequency detector, charge pump, and loop filter in conventional DLL, therefore, it shows better jitter performance, lock time, and convergence speed than previous related works. Optimization methods are used to make input and output signals of the proposed DLL in phase. The proposed architecture is designed to cover all channels of French very high frequency band by choosing number of delay cells in signal path. Simulation has been done for 22-27 delay cells, and f REF = 16 MHz, which can produce output frequency in range of 176-216 MHz. Locking time is approximately 0.3 μs, which is equal to five clock cycles of reference clock. All of the simulation results show superiority of the proposed structure.

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On the Asymptotic Directions of the S-Dimensional Optimum Gradient Method

Abstract: The

Cited by 27 publications

References 0 publications

The cyclic Barzilai-–Borwein method for unconstrained optimization

The cyclic Barzilai-–Borwein method for unconstrained optimization

Estimation of Spectral Bounds in Gradient Algorithms

A new fast‐lock, low‐jitter, and all‐digital frequency synthesizer for DVB‐T receivers

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