Nonsmooth optimization via quasi-Newton methods

Lewis, Adrian S.; Overton, Michael L.

doi:10.1007/s10107-012-0514-2

Cited by 318 publications

(276 citation statements)

References 39 publications

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“…A method based on the BFGS code HANSO v.2.1 available from http://www.cs.nyu.edu/overton/software/hanso/ and kindly made available to us by Michael Overton [17]. Recall that BFGS is a quasi-Newton method with a particular rank-two correction of the approximation of the Hessian at each iteration.…”

Section: Numerical Experimentsmentioning

confidence: 99%

On computing the distance to stability for matrices using linear dissipative Hamiltonian systems

2017

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In this paper, we consider the problem of computing the nearest stable matrix to an unstable one. We propose new algorithms to solve this problem based on a reformulation using linear dissipative Hamiltonian systems: we show that a matrix A is stable if and only if it can be written as A = (J − R)Q, where J = −J T , R 0 and Q ≻ 0 (that is, R is positive semidefinite and Q is positive definite). This reformulation results in an equivalent optimization problem with a simple convex feasible set. We propose three strategies to solve the problem in variables (J, R, Q): (i) a block coordinate descent method, (ii) a projected gradient descent method, and (iii) a fast gradient method inspired from smooth convex optimization. These methods require O(n 3 ) operations per iteration, where n is the size of A. We show the effectiveness of the fast gradient method compared to the other approaches and to several state-of-the-art algorithms.

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Section: Numerical Experimentsmentioning

confidence: 99%

On computing the distance to stability for matrices using linear dissipative Hamiltonian systems

2017

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“…For minimizing the post-processing objective function (Eq. 12), we used the BFGS-based quasi-Newton method implementation by Lewis and Overton (2012). All the experiments were carried out on an Ubuntu 11.04 Linux PC with a 2.8 GHz Intel Core i7 Quad-Core processor and 8 GB of memory.…”

Section: Methodsmentioning

confidence: 99%

“…For this refinement, we use a BFGS-based quasi-Newton method (Lewis and Overton, 2012) that only requires the value of the objective function and its gradient at each point. Letting X (0) = X SDP , we iteratively minimize the following objective function:…”

Section: Post-processingmentioning

confidence: 99%

Determining Protein Structures from NOESY Distance Constraints by Semidefinite Programming

Alipanahi

Krislock

Ghodsi

et al. 2013

Journal of Computational Biology

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Contemporary practical methods for protein nuclear magnetic resonance (NMR) structure determination use molecular dynamics coupled with a simulated annealing schedule. The objective of these methods is to minimize the error of deviating from the nuclear overhauser effect (NOE) distance constraints. However, the corresponding objective function is highly nonconvex and, consequently, difficult to optimize. Euclidean distance matrix (EDM) methods based on semidefinite programming (SDP) provide a natural framework for these problems. However, the high complexity of SDP solvers and the often noisy distance constraints provide major challenges to this approach. The main contribution of this article is a new SDP formulation for the EDM approach that overcomes these two difficulties. We model the protein as a set of intersecting two-and three-dimensional cliques. Then, we adapt and extend a technique called semidefinite facial reduction to reduce the SDP problem size to approximately one quarter of the size of the original problem. The reduced SDP problem can be solved approximately 100 times faster, and it is also more resistant to numerical problems from erroneous and inexact distance bounds.

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“…The search direction is of type [40]. In the case of non-smooth functions, BFGS typically succeeds in finding a local minimizer, as indicated by Overton et al [41]. However, this requires some attention to the line search conditions.…”

Section: Appendix 1: the L-bfgs-b Algorithmmentioning

confidence: 99%

In Search of the Best Zero Coupon Yield Curve for Nairobi Securities Exchange: Interpolation Methods vs. Parametric Models

Muthoni¹

2015

JMF

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We seek to determine which yield curve construction method produces the best zero coupon yield curve (ZCYC) for Nairobi Securities Exchange (NSE). The ZCYC should be differentiable at all points and at the same time, should produce a continuous and positive forward curve at all knot points. A decreasing discount curve is also expected from the resulting ZCYC, as an indication of monotonicity. For the interpolation method, we will use an improvement of monotone preserving interpolation method on ( ) r t t , while the Nelson and Siegel [1] model is the parametric model of choice. This is because compared to other interpolation methods, the improvement of monotone preserving interpolation method on ( ) r t t produces curves with the desirable trait of differentiability, while the Nelson-Siegel [1] model is shown to produce the best-fit results for Kenyan bond data. We compare the models' performance in terms of accuracy in pricing back the fixed-income securities. For this study, we use bond data from Central Bank of Kenya (CBK). The better of the two methods will be used for the Kenyan securities market and, consequently, the East African Securities markets.

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Nonsmooth optimization via quasi-Newton methods

Cited by 318 publications

References 39 publications

On computing the distance to stability for matrices using linear dissipative Hamiltonian systems

On computing the distance to stability for matrices using linear dissipative Hamiltonian systems

Determining Protein Structures from NOESY Distance Constraints by Semidefinite Programming

In Search of the Best Zero Coupon Yield Curve for Nairobi Securities Exchange: Interpolation Methods vs. Parametric Models

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