Accelerating energy minimization process in charged polymer-biomolecular systems: An enhanced nonlinear conjugate gradient method.

<span lang="EN-US">Nonlinear conjugate gradient (CG) methods are extensively used as an important technique for addressing large-scale unconstrained optimization problems which are arise in many aspects of science, engineering, and economics. That is due to their simplicity, convergence properties, and low memory requirements. To generate a new approximation solution in each iteration, the CG methods usually implement under the strong Wolfe line search. For good performance, many studies have been carried out to modify well-known CG methods. In this paper, we did some modifications on one of CG method called RMIL+ in order to obtain a new CG method possesses the sufficient descent property and the global convergence under strong Wolfe line search. The numerical results demonstrate that the suggested method outperforms other CG methods.</span>

For fast Fourier transform (FFT)-based computational micromechanics, solvers need to be fast, memory-efficient, and independent of tedious parameter calibration. In this work, we investigate the benefits of nonlinear conjugate gradient (CG) methods in the context of FFT-based computational micromechanics. Traditionally, nonlinear CG methods require dedicated line-search procedures to be efficient, rendering them not competitive in the FFT-based context. We contribute to nonlinear CG methods devoid of line searches by exploiting similarities between nonlinear CG methods and accelerated gradient methods. More precisely, by letting the step-size go to zero, we exhibit the Fletcher–Reeves nonlinear CG as a dynamical system with state-dependent nonlinear damping. We show how to implement nonlinear CG methods for FFT-based computational micromechanics, and demonstrate by numerical experiments that the Fletcher–Reeves nonlinear CG represents a competitive, memory-efficient and parameter-choice free solution method for linear and nonlinear homogenization problems, which, in addition, decreases the residual monotonically.

The gradient-based optimization methods are preferable for the large-scale three-dimensional (3D) magnetotelluric (MT) inverse problem. Compared with the popular nonlinear conjugate gradient (NLCG) method, however, the limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) method is less adopted. This paper aims to implement a L-BFGS-based inversion algorithm for the 3D MT problem. And we develop our code on top of the ModEM package, which is highly extensible and popular among the MT community. To accelerate the convergence speed, the preconditioning technique by the affine linear transformation of the original model parameters is used. Two modifications of the conventional L-BFGS algorithm are also made to get a comparable convergence rate with the NLCG method. The impacts of the preconditioner parameters, the regularization parameters, the starting model, etc., on the inversion are evaluated by synthetic examples for both L-BFGS and NLCG methods. And the real MT Kayabe dataset is also inverted by the inversion algorithms. The synthetic tests show that through our L-BFGS inversion algorithm the similar resistivity models can be obtained with that from the NLCG method. For the real data inversion, the L-BFGS method performs more efficiently and reasonable results could be obtained by less iterations of the inversion process than the NLCG method. Thus, we suggest the common usage of the L-BFGS method for the 3D MT inverse problem.

Convergence Conditions for Ascent Methods

An algorithm is presented for the optimization of molecular geometries and general nonquadratic functions using the nonlinear conjugate gradient method with a restricted step and restart procedure. The algorithm only requires the evaluation of the energy function and its gradient, therefor less memory storage is needed than for other conjugate gradient algorithms. Some numerical results are also presented and the efficiency and behaviour of the algorithm is compared with the standard conjugate gradient method. We also present comparisons of both conjugate gradient and variable metric methods with and without the trust region technique. One of the main conclusions of the present work is that a trust region always improves the converge of any optimization method. A sketch of the algorithm is also given.

Conjugate gradient method is verified to be efficient for nonlinear optimization problems of large-dimension data. In this paper, a penalized linear and nonlinear combined conjugate gradient method for the reconstruction of fluorescence molecular tomography (FMT) is presented. The algorithm combines the linear conjugate gradient method and the nonlinear conjugate gradient method together based on a restart strategy, in order to take advantage of the two kinds of conjugate gradient methods and compensate for the disadvantages. A quadratic penalty method is adopted to gain a nonnegative constraint and reduce the illposedness of the problem. Simulation studies show that the presented algorithm is accurate, stable, and fast. It has a better performance than the conventional conjugate gradient-based reconstruction algorithms. It offers an effective approach to reconstruct fluorochrome information for FMT.

In this chapter, we investigate recently proposed nonlinear conjugate gradient (NCG) methods for shape optimization problems. We briefly introduce the methods as well as the corresponding theoretical background and investigate their performance numerically. The obtained results confirm that the NCG methods are efficient and attractive solution algorithms for shape optimization problems.

This thesis is concerned with the formulation of a continuum theory of packaging of DNA in bacterial viruses based on a director-field representation of the encapsidated DNA. The point values of the director field give the local direction and density of the DNA. The continuity of the DNA strand requires that the director field be divergence-free and tangent to the capsid wall. The energy of the DNA is defined as a functional of the director field which accounts for bending, torsion, and for electrostatic interactions through a density-dependent interaction energy. The operative principle which determines the encapsidated DNA conformation is assumed to be energy minimization. The director-field theory is used for the direct formulation and study of two low-energy DNA conformations: the inverse spool and torsionless toroidal solenoids. Analysis of the inverse spool configuration yields predictions of the interaxial spacing and the dependence of the packing force on the packed genome fraction which are found to be in agreement with experiments. Further analysis shows that torsionless toroidal solenoids can achieve lower energy than the inverse spool configuration. Also, the theory is adapted to a framework of numerical optimization, wherein all fields are discretized on a computational lattice, and energy minimizing configurations are sought via simulated annealing and the nonlinear conjugate gradient method. It is shown that the inverse spool conformation is stable in all regions of the virus capsid except in a central core, where the DNA tends to buckle out of the spooling plane.

The nonlinear conjugate gradient method (CGM) is a very effective way in solving large-scale optimal problems. In this paper, a modification to the Dai–Yuan (DY) nonlinear CGM is discussed, and then a sufficient descent CGM for unconstrained optimization is proposed. Unlike the DY CGM, at each iteration, the presented CGM always generates a sufficient descent direction depending on no line search. Under usual assumptions, the modified DY CGM with the Wolfe line search is proved to possess global convergence. Moreover, the idea is further extended to the Fletcher–Reeves (FR) CGM. Finally, a large amount of numerical experiments are executed and reported, which show that the proposed methods are effective.

In this decade, nonlinear conjugate gradient methods have been focused on as effective numerical methods for solving large-scale unconstrained optimization problems. Especially, nonlinear conjugate gradient methods with the sufficient descent property have been studied by many researchers. In this paper, we review sufficient descent nonlinear conjugate gradient methods.

It is very important to generate a descent search direction independent of line searches in showing the global convergence of conjugate gradient methods. The method of Hager and Zhang (2005) satisfies the sufficient descent condition. In this paper, we treat two subjects. We first consider a unified formula of parameters which establishes the sufficient descent condition and follows the modification technique of Hager and Zhang. In order to show the global convergence of the conjugate gradient method with the unified formula of parameters, we define some property (say Property A). We prove the global convergence of the method with Property A. Next, we apply the unified formula to a scaled conjugate gradient method and show its global convergence property. Finally numerical results are given.

Parameters estimation for a new anomalous thermal diffusion model in layered media

Nonlinear conjugate gradient (NCG) methods can generate search directions using only first-order information and a few dot products, making them attractive algorithms for solving large-scale optimization problems. However, even the most modern NCG methods can require large numbers of iterations and, therefore, many function evaluations to converge to a solution. This poses a challenge for simulation-constrained problems where the function evaluation entails expensive partial or ordinary differential equation solutions. Preconditioning can accelerate convergence and help compute a solution in fewer function evaluations. However, general-purpose preconditioners for nonlinear problems are challenging to construct. In this paper, we review a selection of classical and modern NCG methods, introduce their preconditioned variants, and propose a preconditioner based on the diagonalization of the BFGS formula. As with the NCG methods, this preconditioner utilizes only first-order information and requires only a small number of dot products. Our numerical experiments using CUTEst problems indicate that the proposed preconditioner successfully reduces the number of function evaluations at negligible additional cost for its update and application.

We propose an improved estimation method for the Box-Cox transformation (BCT) cure rate model parameters. Specifically, we propose a generic maximum likelihood estimation algorithm through a non-linear conjugate gradient (NCG) method with an efficient line search technique. We then apply the proposed NCG algorithm to BCT cure model. Through a detailed simulation study, we compare the model fitting results of the NCG algorithm with those obtained by the existing expectation maximization (EM) algorithm. First, we show that our proposed NCG algorithm allows simultaneous maximization of all model parameters unlike the EM algorithm when the likelihood surface is flat with respect to the BCT index parameter. Then, we show that the NCG algorithm results in smaller bias and noticeably smaller root mean square error of the estimates of the model parameters that are associated with the cure rate. This results in more accurate and precise inference on the cure rate. In addition, we show that when the sample size is large the NCG algorithm, which only needs the computation of the gradient and not the Hessian, takes less CPU time to produce the estimates. These advantages of the NCG algorithm allows us to conclude that the NCG method should be the preferred estimation method over the already existing EM algorithm in the context of BCT cure model. Finally, we apply the NCG algorithm to analyze a well-known melanoma data and show that it results in a better fit when compared to the EM algorithm.

در تحلیل قابلیت اعتماد سازه‌ها محاسبه‌ی صحیح شاخص قابلیت اعتماد بسیار حائز اهمیت است. به‌علت ماهیت غیرخطی و هم‌چنین داشتن چندین نقطه کمینه‌ی موضعی در برخی از توابع شرایط حدی، روش‌های محاسباتی قابلیت اعتماد، ممکن است برآورد مناسبی از احتمال خرابی ارائه ندهند و به سمت این نقاط کمینه‌ی موضعی همگرا گردند. در این مقاله، ابتدا فرمولاسیون گرادیان مزدوج غیرخطی ارائه‌شده توسط (FR) Fletcher & Reeves و Dai & Yuan (DY) برای محاسبه‌ی شاخص قابلیت اعتماد بیان و سپس کارایی و همگرایی روش‌های گرادیان مزدوج غیرخطی برای مثال‌های مختلفی بررسی شده است. از آن‌جایی که روش‌های ارائه‌شده‌ی گرادیان مزدوج FR و DY در برخی از این مثال‌ها همگرا نشده‌اند لذا طی یک الگوریتم ترکیبی بهینه‌سازی، یک روش جدید اصلاح‌شده گرادیان مزدوج غیر خطی به نحوی که بتوان به‌طور مناسب همگرایی مسائل قابلیت اعتماد را تضمین نمود، ارائه‌شده است. نتایج به‌دست‌آمده نشان می‌دهد که روش‌های جدید گرادیان مزدوج غیرخطی اصلاح‌شده، نسبت به روش‌های قدیمی با سرعت و تعداد تکرار کم‌تری همگرا شده و از کارایی و دقت کافی جهت محاسبه احتمال خرابی سازه‌ها، برخوردار هستند.