Local Quadratic Convergence Research Articles

A modified Newton iteration for finding nonnegative Z-eigenpairs of a nonnegative tensor

We propose a modified Newton iteration for finding some nonnegative Z-eigenpairs of a nonnegative tensor. When the tensor is irreducible, all nonnegative eigenpairs are known to be positive. We prove local quadratic convergence of the new iteration to any positive eigenpair of a nonnegative tensor, under the usual assumption guaranteeing the local quadratic convergence of the original Newton iteration. A big advantage of the modified Newton iteration is that it seems capable of finding a nonnegative eigenpair starting with any positive unit vector. Special attention is paid to transition probability tensors.

A variant nonmonotone smoothing algorithm with improved numerical results for large-scale LWCPs

This paper considers the linear weighted complementarity problem (denoted by LWCP) introduced by Potra (SIAM J Optim 22:1634–1654, 2012). Based on two weighted smoothing functions, we propose a new nonmonotone smoothing algorithm for solving the LWCP and establish its global and local quadratic convergence without the strict complementarity assumption. Compared to existing nonmonotone smoothing algorithms, the proposed algorithm solves the linear system only approximately which can save the computation work when one solves large-scale LWCPs. Moreover, the nonmonotone line search technique adopted in this paper includes the usual monotone line search and some existing nonmonotone line searches as special cases. Numerical results show that our algorithm is considerably efficient for solving large-scale LWCPs.

A Levenberg–Marquardt method for solving semi-symmetric tensor equations

In this paper, we propose a Levenberg–Marquardt (LM) method for solving tensor equations with semi-symmetric coefficient tensor and prove its global convergence and local quadratic convergence under the local error bound condition, which is weaker than non-singularity. As application, we solve H-eigenvalue of real semi-symmetric tensor by the LM method. At last, some numerical examples are provided to illustrate the efficiency and validity of these methods proposed.

Numerical study of a smoothing algorithm for the complementarity system over the second-order cone

This paper deals with the complementarity system over the second-order cone (denoted by CSSOC) which contains a wide class of problems. We extend a class of regularized Chen–Harker–Kanzow–Smale smoothing functions studied by Huang and Sun (Appl Math Optim 52:237–262, 2005) for the linear complementarity problem to the CSSOC. Based on this class of functions, we propose a smoothing algorithm for solving the CSSOC. Under weak assumptions, we prove that the proposed algorithm has global and local quadratic convergence. The proposed algorithm is different from existing smoothing algorithms for solving the CSSOC because it adopts a new nonmonotone line search rule. In addition, our algorithm solves a new equation reformulation of the CSSOC. Numerical experiments indicate that the proposed algorithm is quite effective.

A smoothing inexact Newton method for variational inequalities with nonlinear constraints

In this paper, we propose a smoothing inexact Newton method for solving variational inequalities with nonlinear constraints. Based on the smoothed Fischer-Burmeister function, the variational inequality problem is reformulated as a system of parameterized smooth equations. The corresponding linear system of each iteration is solved approximately. Under some mild conditions, we establish the global and local quadratic convergence. Some numerical results show that the method is effective.

On local quadratic convergence of inexact simplified Jacobi–Davidson method for interior eigenpairs of Hermitian eigenproblems

For the Hermitian eigenproblems, under proper assumption on an initial approximation to the desired eigenvector, we prove local quadratic convergence of the inexact simplified Jacobi–Davidson method when the involved relaxed correction equation is solved by a standard Krylov subspace iteration, which particularly leads to local cubic convergence when the relaxed correction equation is solved to a prescribed precision proportional to the norm of the current residual. These results are valid for the interior as well as the extreme eigenpairs of the Hermitian eigenproblem and, hence, generalize the results by Bai and Miao (2017) from the extreme eigenpairs to the interior ones.

On local quadratic convergence of inexact simplified Jacobi–Davidson method

For the Hermitian eigenproblems, we prove local quadratic convergence of the inexact simplified Jacobi–Davidson method when the involved relaxed correction equation is solved by a standard Krylov subspace iteration. This method then shows local cubic convergence rate when the relaxed correction equation is solved to a prescribed precision proportional to the norm of the current residual. As a by-product, we obtain local cubic convergence of the simplified Jacobi–Davidson method. These results significantly improve the existing ones that show only local linear convergence for the inexact simplified Jacobi–Davidson method, which lead to local quadratic convergence for the simplified Jacobi–Davidson method when the tolerance of the inexact solve is particularly set to be zero. Numerical experiments confirm these theoretical results.

A nonmonotone smoothing Newton method for circular cone programming

The circular cone programming (CCP) problem is to minimize or maximize a linear function over the intersection of an affine space with the Cartesian product of circular cones. In this paper, we study nondegeneracy and strict complementarity for the CCP, and present a nonmonotone smoothing Newton method for solving the CCP. We reformulate the CCP as a second-order cone programming (SOCP) problem using the algebraic relation between the circular cone and the second-order cone. Then based on a one parametric class of smoothing functions for the SOCP, a smoothing Newton method is developed for the CCP by adopting a new nonmonotone line search scheme. Without restrictions regarding its starting point, our algorithm solves one linear system of equations approximately and performs one line search at each iteration. Under mild assumptions, our algorithm is shown to possess global and local quadratic convergence properties. Some preliminary numerical results illustrate that our nonmonotone smoothing Newton method is promising for solving the CCP.

An improved generalized Newton method for absolute value equations.

In this paper, we suggest and analyze an improved generalized Newton method for solving the NP-hard absolute value equations Ax-|x|=b when the singular values of A exceed 1. We show that the global and local quadratic convergence of the proposed method. Numerical experiments show the efficiency of the method and the high accuracy of calculation.

A new smoothing and regularization Newton method for the symmetric cone complementarity problem

Based on a regularized Chen–Harker–Kanzow–Smale (CHKS) smoothing function, we propose a new smoothing and regularization Newton method for solving the symmetric cone complementarity problem. By using the theory of Euclidean Jordan algebras, we establish the global and local quadratic convergence of the method on certain assumptions. The proposed method uses a nonmonotone line search technique which includes the usual monotone line search as a special case. In addition, our method treats both the smoothing parameter $$\mu $$ and the regularization parameter $$\varepsilon $$ as independent variables. Preliminary numerical results are reported which indicate that the proposed method is effective.

Jacobian consistency of a one-parametric class of smoothing Fischer–Burmeister functions for SOCCP

The Jacobian consistency of smoothing functions plays an important role for achieving the rapid convergence of Newton methods or Newton-like methods with an appropriate parameter control. In this paper, we study the properties, derive the computable formula for the Jacobian matrix and prove the Jacobian consistency of a one-parametric class of smoothing Fischer–Burmeister functions for second-order cone complementarity problems proposed by Tang et al. (Comput Appl Math 33:655–669, 2014). Then we apply its Jacobian consistency to a smoothing Newton method with the appropriate parameter control presented by Chen et al. (Math Comput 67:519–540, 1998), and show the global convergence and local quadratic convergence of the algorithm for solving the SOCCP under rather weak assumptions.

Some new facts about sequential quadratic programming methods employing second derivatives

For the sequential quadratic programming method (SQP), we show that close to a solution satisfying the same assumptions that are required for its local quadratic convergence (namely uniqueness of the Lagrange multipliers and the second-order sufficient optimality condition), the direction given by the SQP subproblem using the Hessian of the Lagrangian is a descent direction for the standard -penalty function. We emphasize that this property is not straightforward at all, because the Hessian of the Lagrangian need not be positive definite under these assumptions or, in fact, under any other reasonable set of assumptions. In particular, this descent property was not known previously, under any assumptions (even including the stronger linear independence constraint qualification, strict complementarity, etc.). We also check the property in question by experiments on nonconvex problems from the Hock–Schittkowski test collection for a model algorithm. While to propose any new and complete SQP algorithm is not our goal here, our experiments confirm that the descent condition, and a model method based on it, work as expected. This indicates that the new theoretical findings that we report might be useful for full/practical SQP implementations which employ second derivatives and linesearch for the -penalty function. In particular, our results imply that in SQP methods where using subproblems without Hessian modifications is an option, this option has a solid theoretical justification at least on late iterations.

Sparse Spike Deconvolution of Seismic Data Using Trust-Region Based SQP Algorithm

A new deconvolution algorithm for retrieving a sparse reflectivity series from noisy seismic traces is proposed. The problem is formulated as a constrained minimization, taking the approximation zero norm of reflectivity as the objective function. The resulting minimization is solved efficiently by the trust-region based sequential quadratic programming (SQP) method, which provides global convergence and local quadratic convergence rates under suitable assumptions. The null space decomposition method and the de-biasing method are employed to reduce computational complexity and further improve the calculation accuracy. Synthetic simulations indicate that the spikes on the reflectivity, both their positions and amplitudes, are recovered effectively by the proposed approach.

A smoothing Broyden-like method with a nonmonotone derivative-free line search for nonlinear complementarity problems

By using the CHKS-function, we propose a smoothing Broyden-like method for general nonlinear complementarity problems (NCPs). The method is based on the smoothing equation for which we consider the smoothing parameter as an independent variable, and makes use of a new nonmonotone derivative-free line search rule. Under suitable assumptions, we show that the iteration sequence generated by the proposed algorithm converges globally and superlinearly. Furthermore, the algorithm has local quadratic convergence under mild assumptions. Some numerical results are reported, which show that the algorithm is quite effective.

A Globally Convergent Trust-Region Method for Large-Scale Symmetric Nonlinear Systems

This study presents a novel adaptive trust-region method for solving symmetric nonlinear systems of equations. The new method uses a derivative-free quasi-Newton formula in place of the exact Jacobian. The global convergence and local quadratic convergence of the new method are established without the nondegeneracy assumption of the exact Jacobian. Using the compact limited memory BFGS, we adapt a version of the new method for solving large-scale problems and develop the dogleg scheme for solving the associated trust-region subproblems. The sufficient decrease condition for the adapted dogleg scheme is established. While the efficiency of the present trust-region approach can be improved by using adaptive radius techniques, utilizing the compact limited memory BFGS adjusts this approach to handle large-scale symmetric nonlinear systems of equations. Preliminary numerical results for both medium- and large-scale problems are reported.

Optimal control of linear systems with interval constraints

For linear systems with interval constraints, a method for computing a time-optimal control is proposed. The method is based on transforming a quasi-optimal control. The properties and features of the quasi-optimal control are examined. A technique is described for dividing the domain of initial conditions into reachable sets over different times and for approximating each set by a family of hyperplanes. An iterative method for computing an optimal control with interval constraints is developed. The convergence of the method is proved, and a sufficient condition for the convergence of the computational process is obtained. The radius of local quadratic convergence is found. Numerical results are presented.

Two globally convergent nonmonotone trust-region methods for unconstrained optimization

This paper addresses some trust-region methods equipped with nonmonotone strategies for solving nonlinear unconstrained optimization problems. More specifically, the importance of using nonmonotone techniques in nonlinear optimization is motivated, then two new nonmonotone terms are proposed, and their combinations into the traditional trust-region framework are studied. The global convergence to first- and second-order stationary points and local superlinear and quadratic convergence rates for both algorithms are established. Numerical experiments on the CUTEst test collection of unconstrained problems and some highly nonlinear test functions are reported, where a comparison among state-of-the-art nonmonotone trust-region methods shows the efficiency of the proposed nonmonotone schemes.

A smoothing Newton algorithm for weighted linear complementarity problem

In this paper, we present a new smoothing Newton method for solving monotone weighted linear complementarity problem (WCP). Our algorithm needs only to solve one linear system of equation and performs one line search per iteration. Any accumulation point of the iteration sequence generated by our algorithm is a solution of WCP. Under suitable conditions, our algorithm has local quadratic convergence rate. Numerical experiments show the feasibility and efficiency of the algorithm.

二阶锥规划的光滑非精确牛顿法

本文给出了一个新的求解二阶锥规划的光滑非精确牛顿法。在每次迭代时，新方法采用非精确牛顿法去求解一个方程组的解，降低了光滑牛顿法的计算量。在较弱条件下，证明了算法具有全局和局部二阶收敛性质。数值试验表明算法是有效的。 A new smoothing inexact Newton method is presented for solving the second-order cone pro-gramming. At each iteration, the method uses an inexact Newton method to solve the system of equations, which saves computation work of smoothing Newton methods. Under weak assumptions, our method is proved to have global and local quadratic convergence. Numerical experiments indicate that the proposed method is quite effective.

An Iterative Minimization Formulation for Saddle Point Search

This paper proposes and analyzes an iterative minimization formulation for searching index-1 saddle points of an energy function. We give a general and rigorous description of eigenvector-following methodology in this iterative scheme by considering an auxiliary optimization problem at each iteration in which the new objective function is locally defined near the current guess. We prove that this scheme has a quadratic local convergence rate in terms of number of iterations, in comparison to the linear rate of the gentlest ascent dynamics [W. E and X. Zhou, Nonlinearity, 24 (2011), pp. 1831--1842] and many other existing methods. We also propose the generalization of the new methodology for saddle points of higher index and for constrained energy functions on the manifold. Preliminary numerical results on the nature of this iterative minimization formulation are presented.