Exact Line Research Articles

A new hyhbrid coefficient of conjugate gradient method

<p><span>Hybridization is one of the popular approaches in modifying the conjugate gradient method. In this paper, a new hybrid conjugate gradient is suggested and analyzed in which the parameter <!--[if gte mso 9]><xml>
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 </xml><![endif]--> while using exact line search. The proposed method is shown to possess both sufficient descent and global convergence properties. Numerical performances show that the proposed method is promising and has overpowered other hybrid conjugate gradient methods in its number of iterations and central processing unit per time. </span></p>

Malignant Ossifying Fibromyxoid Tumor of the Tongue: A Rare Case

AbstractOssifying fibromyxoid tumors (OFMTs) are rare musculoskeletal tumors of uncertain malignant potential. Most OFMTs arise from the subcutaneous tissue, skeletal muscles of extremities, trunk, or head-and-neck region. OFMTs are usually benign, but atypical and malignant OFMT have been described with recurrence and distant metastasis. Till date, the etiopathogenesis and exact line of differentiation of OFMT is still doubtful. We are presenting an unusual case of malignant OFMT of the tongue. It is a rare case of malignant OFMT affecting an unusual site.

Analyzing random permutations for cyclic coordinate descent

We consider coordinate descent methods on convex quadratic problems, in which exact line searches are performed at each iteration. (This algorithm is identical to Gauss-Seidel on the equivalent symmetric positive definite linear system.) We describe a class of convex quadratic problems for which the random-permutations version of cyclic coordinate descent (RPCD) outperforms the standard cyclic coordinate descent (CCD) approach, yielding convergence behavior similar to the fully-random variant (RCD). A convergence analysis is developed to explain the empirical observations.

Adaptive iterative Hessian sketch via A-optimal subsampling

Iterative Hessian sketch (IHS) is an effective sketching method for modeling large-scale data. It was originally proposed by Pilanci and Wainwright (J Mach Learn Res 17(1):1842–1879, 2016) based on randomized sketching matrices. However, it is computationally intensive due to the iterative sketch process. In this paper, we analyze the IHS algorithm under the unconstrained least squares problem setting and then propose a deterministic approach for improving IHS via A-optimal subsampling. Our contributions are threefold: (1) a good initial estimator based on the A-optimal design is suggested; (2) a novel ridged preconditioner is developed for repeated sketching; and (3) an exact line search method is proposed for determining the optimal step length adaptively. Extensive experimental results demonstrate that our proposed A-optimal IHS algorithm outperforms the existing accelerated IHS methods.

Primal–dual accelerated gradient methods with small-dimensional relaxation oracle

In this paper, a new variant of accelerated gradient descent is proposed. The proposed method does not require any information about the objective function, uses exact line search for the practical accelerations of convergence, converges according to the well-known lower bounds for both convex and non-convex objective functions, possesses primal–dual properties and can be applied in the non-euclidian set-up. As far as we know this is the first such method possessing all of the above properties at the same time. We also present a universal version of the method which is applicable to non-smooth problems. We demonstrate how in practice one can efficiently use the combination of line-search and primal-duality by considering a convex optimization problem with a simple structure (for example, linearly constrained).

Shape fluctuations and optical transition of He2* excimer tracers in superfluid He4

Metastable ${\mathrm{He}}_{2}^{*}$ excimer molecules have been utilized as tracer particles of the normal component in superfluid $^{4}\mathrm{He}$ (He II) which can be imaged via laser-induced fluorescence. These excimer molecules form tiny bubbles in He II and can bind to quantized vortices at sufficiently low temperatures, thereby allowing for direct visualization of vortex dynamics in an inviscid superfluid. However, the ${a}^{3}{\mathrm{\ensuremath{\Sigma}}}_{u}^{+}\ensuremath{\rightarrow}{c}^{3}{\mathrm{\ensuremath{\Sigma}}}_{g}^{+}$ optical absorption line, which is responsible for the fluorescence imaging of the ${\mathrm{He}}_{2}^{*}$ molecules, is controlled by fluctuations on the bubble shape, and its exact line profile is not known at low temperatures. In this paper, we present a bubble model for evaluating the surface fluctuation eigenmodes of the excimers in He II. The line profile of the ${a}^{3}{\mathrm{\ensuremath{\Sigma}}}_{u}^{+}\ensuremath{\rightarrow}{c}^{3}{\mathrm{\ensuremath{\Sigma}}}_{g}^{+}$ transition is calculated at different temperatures by considering both the zero-point and thermal fluctuations on the bubble shape. We show that as the temperature drops from 2 K to 20 mK, the peak absorption strength is enhanced by a factor of about five, accompanying a blueshift of the peak location by about 2 nm. A double-peak line profile due to the rotational levels of the molecular core can be resolved. This bubble model also allows us to evaluate the stiffness of the ${\mathrm{He}}_{2}^{*}$ bubbles and hence their diffusion constant in He II due to scattering off thermal phonons. Our results will aid the design of future experiments on imaging quantized vortices in He II using ${\mathrm{He}}_{2}^{*}$ tracers.

Visualizing different crystalline states during the infrared imaging of calcium phosphates

Methods utilizing relatively simple mathematical operations during physical analyses to enable the visualization of otherwise invisible correlations and effects are of particular appeal to researchers and students in pedagogical settings. At the same time, discerning the amorphous phase from its crystalline counterpart in materials is challenging with the use of vibrational spectroscopy and is nowhere as straightforward as in phase composition analytical methods such as X-ray diffraction. A method is demonstrated for the use of first- and second-order differentiation of Fourier transform infrared spectra of calcium phosphates to distinguish their amorphous states from the crystalline ones based on the exact line positioning rather than on comparatively vaguer band broadening and splitting effects. The study utilizes a kinetic approach, focusing on the comparison of spectral features of amorphous precursors annealed in air at different temperatures or aged for different periods of time in an aqueous solution until transforming to one or a mixture of crystalline phases, including hydroxyapatite and α- and β-tricalcium phosphate. One of the findings challenges the concept of the nucleation lag time preceding the crystallization from amorphous precursors as a “dead” period and derives a finite degree of constructive changes occurring at the atomic scale in its course. The differential method for highlighting spectral differences depending on the sample crystallinity allows for monitoring in situ the process of conversion of the amorphous calcium phosphate phase to its crystalline analogue(s). One such method can be of practical significance for synthetic solid state chemists testing for the chemical stability and/or concentration of the reactive amorphous phase in these materials, but also for biologists measuring the maturity of bone and medical researchers evaluating its phase composition and, thus, the state of metabolic and mechanical stability.

Improved utility harmonic impedance measurement based on robust independent component analysis and bootstrap check

In this study, a new method for utility harmonic impedance measurement based on robust independent component analysis (RICA) and bootstrap check is proposed. The RICA approach which adopts exact line search to seek the optimal step-size in iteration has better performance for short data records. At the same time, a bootstrap check technique to remove the singular solutions of this sort of independent component analyses methods is presented as well. This verification technique is performed through judging the dispersion degree of the estimates of the bootstrap samples which is quantified by the coefficient of variation. Based on the results of computer simulation and filed test, it is proved that the combination of the RICA approach with the bootstrap check technique can reduce the adverse impact of the background harmonic fluctuation and the singular solutions on impedance estimation effectively, and satisfactory measurement results can be acquired.

A COMPARATIVE STUDY OF SOME MODIFICATIONS OF CG METHODS UNDER EXACT LINE SEARCH

Conjugate Gradient (CG) method is a technique used in solving nonlinear unconstrained optimization problems. In this paper, we analysed the performance of two modifications and compared the results with the classical conjugate gradient methods of. These proposed methods possesse global convergence properties for general functions using exact line search. Numerical experiments show that the two modifications are more efficient for the test problems compared to classical CG coefficients.

An exact line search scheme to accelerate the EM algorithm: Application to Gaussian mixture models identification

This paper tackles the slowness issue of the well-known expectation-maximization (EM) algorithm in the context of Gaussian mixture models. To cope with this slowness problem, an Exact Line Search scheme is proposed. It is based on exact computation of the step size required to jump, for a given search direction, towards the final solution. Computing this exact step size is easily done by only rooting a second-order polynomial computed from the initial log-likelihood maximization problem. Numerical results using both simulated and real dataset showed the efficiency of the proposed exact line search scheme when applied to the conventional EM algorithm as well as the anti-annealing based acceleration techniques based on either the EM or the expectation conjugate gradient algorithm.

Conjugate Gradient Methods for Computing Weighted Analytic Center for Linear Matrix Inequalities Using Exact and Quadratic Interpolation Line Searches

We study the problem of computing the weighted analytic center for linear matrix inequality constraints. In this paper, we apply conjugate gradient (CG) methods to find the weighted analytic center. CG methods have low memory requirements and strong local and global convergence properties. The methods considered are the classical methods by Hestenes-Stiefel (HS), Fletcher and Reeves (FR), Polak and Ribiere (PR) and a relatively new method by Rivaie, Abashar, Mustafa and Ismail (RAMI). We compare performance of each method on random test problems by observing the number of iterations and time required by the method to find the weighted analytic center for each test problem. We use Newton’s method exact line search and Quadratic Interpolation inexact line search. Our numerical results show that PR is the best method, followed by HS, then RAMI, and then FR. However, PR and HS performed about the same with exact line search. The results also indicate that both line searches work well, but exact line search handles weights better than the inexact line search when some weight is relatively much larger than the other weights. We also find from our results that with Quadratic interpolation line search, FR is more susceptible to jamming phenomenon than both PR and HS.

Achievable Secrecy Rate Region for Buffer-Aided Multiuser MISO Systems

In this paper, we consider a buffer-aided multiuser multiple-input single-output (MISO) network consisting of one multi-antenna access point (AP) and multiple single-antenna users, in which the AP is provisioned with data buffers for temporarily storing the data for each user either from the upper layer applications or the message by the AP. The data for one specific user must be kept confidential from all other unintended users. For such a system, we aim at maximizing the long-term average achievable secrecy rate region by carefully designing the flow control, the information signal and artificial noise beamforming, as well as the user selection. To address this issue, we first transform the time average optimization problem into a real-time one by using the Lyapunov optimization framework. Then it is proposed to decompose the optimization problem into several sub-problems by using the optimization decomposition technique. Although the information signal and artificial noise beamforming sub-problem is non-convex, we show that it can be decomposed into a two-stage optimization problem to effectively solve it by using the exact line search and DC (difference of two convex functions) algorithms. Moreover, we extend the average secrecy rate region maximization problem to the worst-case scenario, in which all unintended users are colluding in eavesdropping. Our analysis discloses that, there exists an inherent tradeoff between the average achievable secrecy rate region and the average queueing length. It is shown that a better average secrecy rate region can be realized by fully taking advantage of the buffer-aided transmission potentials in the MISO network, if a certain queueing delay is tolerable.

New Parameter of CG-Method with Exact Line Search for Unconstraint Optimization

In this paper, a new CG method has been introduced to solve nonlinear equations systems. This method achieved the conditions of descent and global convergence, using the exact line search. The numerical results were good compared to other methods in terms of the number of iterations and the number of functions evaluation.

New Conjugate Gradient Method Addressing Large Scale Unconstrained Optimization Problem

An iterative conjugate gradient (CG) method is prominently known for dealing with unconstrained optimization problem. A new CG method which is modified by Wei Yao Liu (WYL) method is tested by standard test functions. Moreover, the step size is calculated using exact line search. Theoretical proofs on convergence analysis are shown. As a result, this new CG is comparable to the other methods in finding the optimal points by measuring the total iterations required as well as the computing time. Numerical results showed the execution between three CG methods in details.

Descent Condition for a Scalar Parameter of Spectral HS (SpHS) Conjugate Gradient Method

Spectral conjugate gradient method has been used in most cases as an alternative to the conjugate gradient (CG) method in order to solve nonlinear unconstrained problems. In this paper, we introduced a spectral parameter of HS conjugate gradient method resultant from the classical CG search direction and used some of the standard test functions with numerous variables to prove its sufficient descent and global convergence properties, the numerical outcome is verified by exact line search procedures.

A Hybrid of Quasi-Newton Method with CG Method for Unconstrained Optimization

The quasi-Newton is a well-known method for solving small to medium-scale unconstrained optimization problems due to its simplicity and convergence. This leads to many modifications to improve its performance, and one of them is by hybridizing it with another optimization method. In this study, the quasi-Newton method is combined with the ARM method, which is a type of conjugate gradient method. The resulting hybrid algorithm is globally convergent under exact line search.

On the extension of the Hager–Zhang conjugate gradient method for vector optimization

The extension of the Hager–Zhang (HZ) nonlinear conjugate gradient method for vector optimization is discussed in the present research. In the scalar minimization case, this method generates descent directions whenever, for example, the line search satisfies the standard Wolfe conditions. We first show that, in general, the direct extension of the HZ method for vector optimization does not yield descent (in the vector sense) even when an exact line search is employed. By using a sufficiently accurate line search, we then propose a self-adjusting HZ method which possesses the descent property. The proposed HZ method with suitable parameters reduces to the classical one in the scalar minimization case. Global convergence of the new scheme is proved without regular restarts and any convex assumption. Finally, numerical experiments illustrating the practical behavior of the approach are presented, and comparisons with the Hestenes–Stiefel conjugate gradient and the steepest descent methods are discussed.

An Alternative Modified Conjugate Gradient Coefficient for Solving Nonlinear System of Equations

In mathematical term, the method of solving models and finding the best alternatives is known as optimization. Conjugate gradient (CG) method is an evolution of computational method in solving optimization problems. In this article, an alternative modified conjugate gradient coefficient for solving large-scale nonlinear system of equations is presented. The method is an improved version of the Rivaie et el conjugate gradient method for unconstrained optimization problems. The new CG is tested on a set of test functions under exact line search. The approach is easy to implement due to its derivative-free nature and has been proven to be effective in solving real-life application. Under some mild assumptions, the global convergence of the proposed method is established. The new CG coefficient also retains the sufficient descent condition. The performance of the new method is compared to the well-known previous PRP CG methods based on number of iterations and CPU time. Numerical results using some benchmark problems show that the proposed method is promising and has the best efficiency amongst all the methods tested.

Analysis of the gradient method with an Armijo–Wolfe line search on a class of non-smooth convex functions

ABSTRACTIt has long been known that the gradient (steepest descent) method may fail on non-smooth problems, but the examples that have appeared in the literature are either devised specifically to defeat a gradient or subgradient method with an exact line search or are unstable with respect to perturbation of the initial point. We give an analysis of the gradient method with steplengths satisfying the Armijo and Wolfe inexact line search conditions on the non-smooth convex function . We show that if a is sufficiently large, satisfying a condition that depends only on the Armijo parameter, then, when the method is initiated at any point with , the iterates converge to a point with , although f is unbounded below. We also give conditions under which the iterates , using a specific Armijo–Wolfe bracketing line search. Our experimental results demonstrate that our analysis is reasonably tight.

The convergence properties of RMIL+ conjugate gradient method under the strong Wolfe line search

In Dai (2016), based on the global convergence of RMIL conjugate gradient method, Dai has modified it and called the modified version RMIL+ which has good numerical results and globally convergent under the exact line search. In this paper, we established the sufficient descent property and the global convergence of RMIL+ via strong Wolfe line search method. Moreover, numerical results based on well-known optimization problems show that the modified method is competitive when compare with other conjugate gradient methods.