Rank-one Method Research Articles

Variational Density Functional Calculations of Excited States via Direct Optimization

The development of variational density functional theory approaches to excited electronic states is impeded by limitations of the commonly used self-consistent field (SCF) procedure. A method based on a direct optimization approach as well as the maximum overlap method is presented, and the performance is compared with previously proposed SCF strategies. Excited-state solutions correspond to saddle points of the energy as a function of the electronic degrees of freedom. The approach presented here makes use of a preconditioner determined with the help of the maximum overlap method to guide the convergence on a target nth-order saddle point. This method is found to be more robust and converge faster than previously proposed SCF approaches for a set of 89 excited states of molecules. A limited-memory formulation of the symmetric rank-one method for updating the inverse Hessian is found to give the best performance. A conical intersection for the carbon monoxide molecule is calculated without resorting to fractional occupation numbers. Calculations on the excited states of the hydrogen atom and a doubly excited state of the dihydrogen molecule using a self-interaction corrected functional are presented. For these systems, the self-interaction correction is found to improve the accuracy of density functional calculations of excited states.

Scaling strategies for symmetric rank-one method in solving unconstrained optimization problems

Scaling strategies for symmetric rank-one method in solving unconstrained optimization problems

On the performance of a new symmetric rank-one method with restart for solving unconstrained optimization problems

Quasi-Newton (QN) methods are generally held to be the most efficient minimization methods for solving unconstrained optimization problems. Among the QN methods, symmetric rank-one (SR1) is one of the very competitive formulas. In the present paper, we propose a new SR1 method. The new technique attempts to improve the quality of the SR1 Hessian by employing the scaling of the identity in a certain sense. However, since at some iterations these updates might be singular, indefinite or undefined, this paper proposes an updates criterion based on the eigenvalues of the SR1 update to measure this quality. Hence, the new method is employed only to improve the approximation of the SR1 Hessian. It is shown that the numerical results support the theoretical considerations for the usefulness of this criterion and show that the proposed method improves the performance of the SR1 update substantially.

Structured symmetric rank-one method for unconstrained optimization

In this paper, we investigate a symmetric rank-one (SR1) quasi-Newton (QN) formula in which the Hessian of the objective function has some special structure. Instead of approximating the whole Hessian via the SR1 formula, we consider an approach which only approximates part of the Hessian matrix that is not easily acquired. Although the SR1 update possesses desirable features, it is unstable in the sense that, it may not retain positive definiteness and may become undefined. Therefore, we describe some safeguards to overcome these difficulties. Since the structured SR1 method provides a more accurate Hessian approximation, therefore the proposed method reduces significantly the computational efforts needed in solving a problem. The results of a series of experiments on a typical set of standard unconstrained optimization problems are reported, which show that the structured SR1 method exhibits a clear improvement in numerical performance over some existing QN algorithms.

A symmetric rank-one method based on extra updating techniques for unconstrained optimization

In this paper, we present a new symmetric rank-one (SR1) method for the solution of unconstrained optimization problems. The proposed method involves an algorithm in which the usual SR1 Hessian is updated a number of times in a way to be specified in some iterations, to improve the performance of the Hessian approximation. In particular, we discuss how to consider a criterion for indicating at each iteration whether it is necessary to employ extra updates. However it is well known that there are some theoretical difficulties when applying the SR1 update. Even for a current positive definite Hessian approximation, it is possible that the SR1 update may not be defined or the SR1 update may not preserve positive definiteness at some iterations. We then employ a restarting procedure that guarantees that updated matrices will be well-defined while preserving positive definiteness of updates. Numerical results support these theoretical considerations. They show that the implementation of the SR1 method using extra updating techniques improves the performance of the SR1 method substantially for a number of test problems from the literature.

Symmetric rank-one method based on some modified secant conditions for unconstrained optimization

The symmetric rank-one (SR1) method is one of the well-known quasi-Newton methods, and many researchers have studied the SR1 method. On the other hand, to accelerate quasi-Newton methods, some researchers have proposed variants of the secant condition. In this paper, we propose SR1 methods based on some modified secant conditions. We analyze local behaviors of the methods. In order to establish the global convergence of the methods, we apply the trust region method to our methods.

Improved Hessian approximation with modified secant equations for symmetric rank-one method

Symmetric rank-one (SR1) is one of the competitive formulas among the quasi-Newton (QN) methods. In this paper, we propose some modified SR1 updates based on the modified secant equations, which use both gradient and function information. Furthermore, to avoid the loss of positive definiteness and zero denominators of the new SR1 updates, we apply a restart procedure to this update. Three new algorithms are given to improve the Hessian approximation with modified secant equations for the SR1 method. Numerical results show that the proposed algorithms are very encouraging and the advantage of the proposed algorithms over the standard SR1 and BFGS updates is clearly observed.

Convergence of Symmetric Rank-One method based on Modified Quasi-Newton equation

In this paper we investigate on convergence rate of a modified symmetric rank-one (SR1) method for unconstrainedoptimization problems. In general, the modified SR1 method incorporates a modified secant equationinto the standard SR1 method. Also a restart procedure is applied to avoid the loss of positive definiteness and zero denominator.A remarkable feature of the modified SR1 method is that it possesses at most $n+1$-step $q$-superlinearly convergent and$2n$-step quadratic convergent without uniformly independent assumptions of steps.

Novel shooting algorithm for highly efficient analysis of fiber Raman amplifiers

An efficient and stable algorithm based on the simple-shooting method to solve the fiber Raman-amplifier (FRA) equations is proposed and numerically demonstrated. In the algorithm, the conventional Newton-Raphson method is substituted by a more efficient and stable modified Newton method to serve as the correction mechanism in the shooting iterations. By introducing the Broyden's rank-one method into the modified Newton method, the time-consuming calculation of the Jacobian matrix is dramatically relieved, and the efficiency of the proposed algorithm is further remarkably improved. An effective initial-guess-providing technique for the shooting method has also been proposed. Simulation results show that the efficiency of the proposed algorithm is improved more than three times compared with those of the previously reported methods. This algorithm can be used in the design of various kinds of FRAs to predict their gain and noise performance accurately.

Extra updates for the bfgs method*

This paper considers employing extra updates for the BFGS method for unconstrained optimization. The usual BFGS Hessian is updated a number of times, depending on the information of the first order derivatives, to obtain a new Hessian approximation at each iteration. Two approaches are proposed. One of them has the same properties of global and superlinear convergence on convex functions the BFGS method has, and another has the same property of quadratic termination without exact line searches that the symmetric rank-one method has. The new algorithms attempt to combine the best features of certain methods which are intended for either parallel computation or large scale optimization. It is concluded that some new algorithms are competitive with the standard BFGS method