On the rate of superlinear convergence of a class of variable metric methods

Ritter, Klaus

doi:10.1007/bf01396414

Cited by 16 publications

(7 citation statements)

References 18 publications

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“…x k 2 ): (Powell, 1983) gives a slightly better result and performs numerical tests on small problems to measure the rate observed in practice. Faster rates of convergence can be established (Schuller, 1974), (Ritter, 1980), under the assumption that the search directions are uniformly linearly independent, but this does not often occur in practice. Several interesting results assuming asymptotically exact line searches are given by Baptist and Stoer (1977) and Stoer (1977).…”

Section: Conjugate Gradient Methodsmentioning

confidence: 99%

Theory of algorithms for unconstrained optimization

Nocedal¹

1992

Acta Numerica

250

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A few months ago, while preparing a lecture to an audience that included engineers and numerical analysts, I asked myself the question: from the point of view of a user of nonlinear optimization routines, how interesting and practical is the body of theoretical analysis developed in this field? To make the question a bit more precise, I decided to select the best optimization methods known to date – those methods that deserve to be in a subroutine library – and for each method ask: what do we know about the behaviour of this method, as implemented in practice? To make my task more tractable, I decided to consider only algorithms for unconstrained optimization.

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Section: Conjugate Gradient Methodsmentioning

confidence: 99%

Theory of algorithms for unconstrained optimization

Nocedal¹

1992

Acta Numerica

250

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“…Thus, in the following simulations, we replace the Step 1 above with a constant step length. In practice, the proposed CG method often leads to linear or superlinear convergence speed [17,[28][29][30].…”

Section: Iteration Kmentioning

confidence: 99%

Pixel-based OPC optimization based on conjugate gradients

Ma¹,

Arce²

2011

Opt. Express

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Optical proximity correction (OPC) methods are resolution enhancement techniques (RET) used extensively in the semiconductor industry to improve the resolution and pattern fidelity of optical lithography. In pixel-based OPC (PBOPC), the mask is divided into small pixels, each of which is modified during the optimization process. Two critical issues in PBOPC are the required computational complexity of the optimization process, and the manufacturability of the optimized mask. Most current OPC optimization methods apply the steepest descent (SD) algorithm to improve image fidelity augmented by regularization penalties to reduce the complexity of the mask. Although simple to implement, the SD algorithm converges slowly. The existing regularization penalties, however, fall short in meeting the mask rule check (MRC) requirements often used in semiconductor manufacturing. This paper focuses on developing OPC optimization algorithms based on the conjugate gradient (CG) method which exhibits much faster convergence than the SD algorithm. The imaging formation process is represented by the Fourier series expansion model which approximates the partially coherent system as a sum of coherent systems. In order to obtain more desirable manufacturability properties of the mask pattern, a MRC penalty is proposed to enlarge the linear size of the sub-resolution assistant features (SRAFs), as well as the distances between the SRAFs and the main body of the mask. Finally, a projection method is developed to further reduce the complexity of the optimized mask pattern.

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“…This result was later on extended by Stachurski [13] to all meth-ods of Broyden's fl-class which are not "too far away" from the BFGS-method in a certain sense, excluding methods "too close" to the DFP-algorithm. However, Griewank and Toint [7] For exact line searches and uniformly convex functions f Powell [8] proved the global convergence of the DFP method, in particular the Q-superlinear convergence (1.6)b) of the x k. There are several local convergence results for exact line searches and the class of Broyden methods, the latest being an estimate of Ritter [12] Ilxk+.…”

Section: Introductionmentioning

confidence: 96%