关于空间l∞上的算子理想

讨论算子代数Ｂ（ｌ∞）中各算子理想之间的关系，说明严格奇异算子理想Ｓ（ｌ∞）与弱紧算子理想、也与非本性算子理想重合，成为Ｂ（Ｌ∞）中最大的、非平凡的闭算子理想；它们不与紧算子理想重合。     

Newton’s Method for M-Tensor Equations

Journal of Optimization Theory and Applications - We are concerned with the tensor equations whose coefficient tensors are M-tensors. We first propose a Newton method for solving the equation with...     

A Smoothing Method for Zero–One Constrained Extremum Problems

In this paper, the zero–one constrained extremum problem is reformulated as an equivalent smooth mathematical program with complementarity constraints (MPCC), and then as a smooth ordinary nonlinear programming problem with the help of the Fischer–Burmeister function. The augmented Lagrangian method is adopted to solve the resulting problem, during which the non-smoothness may be introduced as a consequence of the possible inequality constraints. This paper incorporates the aggregate constraint method to construct a uniform smooth approximation to the original constraint set, with approximation controlled by only one parameter. Convergence results are established, showing that under reasonable conditions the limit point of the sequence of stationary points generated by the algorithm is a strongly stationary point of the original problem and satisfies the second order necessary conditions of the original problem. Unlike other penalty type methods for MPCC, the proposed algorithm can guarantee that the limit point of the sequence is feasible to the original problem.     

A Newton Method for Minimizing One-Order Lipschitz Functions

In this paper, we consider the following unconstrained optimization problem (P) min{f(x)|x∈R~n}, where f(x) is a one order Lipschitz function on R~n, i.e., g(x)—the gradient of f(x)—is Lipschitzian. We will represent a kind of Newton method for solving the problem (P).     

The Josephy–Newton Method for Semismooth Generalized Equations and Semismooth SQP for Optimization

While generalized equations with differentiable single-valued base mappings and the associated Josephy–Newton method have been studied extensively, the setting with semismooth base mapping had not been previously considered (apart from the two special cases of usual nonlinear equations and of Karush–Kuhn–Tucker optimality systems). We introduce for the general semismooth case appropriate notions of solution regularity and prove local convergence of the corresponding Josephy–Newton method. As an application, we immediately recover the known primal-dual local convergence properties of semismooth sequential quadratic programming algorithm (SQP), but also obtain some new results that complete the analysis of the SQP primal rate of convergence, including its quasi-Newton variant.     

The Newton Method for Operators with Hölder Continuous First Derivative

We analyze the convergence of the Newton method when the first Fréchet derivative of the operator involved is Hölder continuous. We calculate also the R-order of convergence and provide some a priori error bounds. Based on this study, we give some results on the existence and uniqueness of the solution for a nonlinear Hammerstein integral equation of the second kind.     

Kantorovich-Like Convergence Theorems for Newton’s Method Using Restricted Convergence Domains

The convergence set for Newton’s method is small in general using Lipschitz-type conditions. A center-Lipschitz-type condition is used to determine a subset of the convergence set containing the Newton iterates. The rest of the Lipschitz parameters and functions are then defined based on this subset instead of the usual convergence set. This way the resulting parameters and functions are more accurate than in earlier works leading to weaker sufficient semi-local convergence criteria. The novelty of the paper lies in the observation that the new Lipschitz-type functions are special cases of the ones given in earlier works. Therefore, no additional computational effort is required to obtain the new results. The results are applied to solve Hammerstein nonlinear integral equations of Chandrasekhar type in cases not covered by earlier works.     

Convergence of Newton’s Method for Sections on Riemannian Manifolds

The present paper is concerned with the convergence problems of Newton’s method and the uniqueness problems of singular points for sections on Riemannian manifolds. Suppose that the covariant derivative of the sections satisfies the generalized Lipschitz condition. The convergence balls of Newton’s method and the uniqueness balls of singular points are estimated. Some applications to special cases, which include the Kantorovich’s condition and the γ-condition, as well as the Smale’s γ-theory for sections on Riemannian manifolds, are given. In particular, the estimates here are completely independent of the sectional curvature of the underlying Riemannian manifold and improve significantly the corresponding ones due to Dedieu, Priouret and Malajovich (IMA J. Numer. Anal. 23:395–419, 2003), as well as the ones in Li and Wang (Sci. China Ser. A. 48(11):1465–1478, 2005).     

New Complexity Analysis of the Primal—Dual Newton Method for Linear Optimization

We deal with the primal–dual Newton method for linear optimization (LO). Nowadays, this method is the working horse in all efficient interior point algorithms for LO, and its analysis is the basic element in all polynomiality proofs of such algorithms. At present there is still a gap between the practical behavior of the algorithms and the theoretical performance results, in favor of the practical behavior. This is especially true for so-called large-update methods. We present some new analysis tools, based on a proximity measure introduced by Jansen et al., in 1994, that may help to close this gap. This proximity measure has not been used in the analysis of large-update methods before. The new analysis does not improve the known complexity results but provides a unified way for the analysis of both large-update and small-update methods.     

Convergence Properties of the Regularized Newton Method for the Unconstrained Nonconvex Optimization

The regularized Newton method (RNM) is one of the efficient solution methods for the unconstrained convex optimization. It is well-known that the RNM has good convergence properties as compared to the steepest descent method and the pure Newton’s method. For example, Li, Fukushima, Qi and Yamashita showed that the RNM has a quadratic rate of convergence under the local error bound condition. Recently, Polyak showed that the global complexity bound of the RNM, which is the first iteration k such that ‖∇ f(x _k )‖≤ε, is O(ε ⁻⁴), where f is the objective function and ε is a given positive constant. In this paper, we consider a RNM extended to the unconstrained “nonconvex” optimization. We show that the extended RNM (E-RNM) has the following properties. (a) The E-RNM has a global convergence property under appropriate conditions. (b) The global complexity bound of the E-RNM is O(ε ⁻²) if ∇ ² f is Lipschitz continuous on a certain compact set. (c) The E-RNM has a superlinear rate of convergence under the local error bound condition.     

Adaptive Strategy for the Damping Parameters in an Iteratively Regularized Gauss–Newton Method

In this paper, a convergence analysis of an adaptive choice of the sequence of damping parameters in the iteratively regularized Gauss–Newton method for solving nonlinear ill-posed operator equations is presented. The selection criterion is motivated from the damping parameter choice criteria, which are used for the efficient solution of nonlinear least-square problems. The performance of this selection criterion is tested for the solution of nonlinear ill-posed model problems.     

Primal–Dual–Infeasible Newton Approach for the Analytic Center Deep-Cutting Plane Method

The convergence and complexity of a primal–dual column generation and cutting plane algorithm from approximate analytic centers for solving convex feasibility problems defined by a deep cut separation oracle is studied. The primal–dual–infeasible Newton method is used to generate a primal–dual updating direction. The number of recentering steps is O(1) for cuts as deep as half way to the deepest cut, where the deepest cut is tangent to the primal–dual variant of Dikin's ellipsoid.     

Newton’s Method for Computing the Nearest Correlation Matrix with a Simple Upper Bound

The standard nearest correlation matrix can be efficiently computed by exploiting a recent development of Newton’s method (Qi and Sun in SIAM J. Matrix Anal. Appl. 28:360–385, 2006). Two key mathematical properties, that ensure the efficiency of the method, are the strong semismoothness of the projection operator onto the positive semidefinite cone and constraint nondegeneracy at every feasible point. In the case where a simple upper bound is enforced in the nearest correlation matrix in order to improve its condition number, it is shown, among other things, that constraint nondegeneracy does not always hold, meaning Newton’s method may lose its quadratic convergence. Despite this, the numerical results show that Newton’s method is still extremely efficient even for large scale problems. Through regularization, the developed method is applied to semidefinite programming problems with simple bounds.     

Mesh-Independence of the Lagrange–Newton Method for Nonlinear Optimal Control Problems and their Discretizations

In a recent paper we proved a mesh-independence principle for Newton's method applied to stable and consistent discretizations of generalized equations. In this paper we introduce a new consistency condition which is easier to check in applications. Using this new condition we show that the mesh-independence principle holds for the Lagrange–Newton method applied to nonlinear optimal control problems with mixed control-state constraints and their discretizations by Euler's method or Ritz type methods.     

Newton’s Method for Minimizing a Convex Twice Differentiable Function on a Preconvex Set

The problem of minimizing a convex twice differentiable function on the set-theoretic difference between a convex set and the union of several convex sets is considered. A generalization of Newton’s method for solving problems with convex constraints is proposed. The convergence of the algorithm is analyzed.     

On the Convergence of Newton’s Method for Polynomial Equations and Applications in Radiative Transfer

 Newton’s method is used to approximate a locally unique zero of a polynomial operator F of degree in Banach space. So far, convergence conditions have been found for Newton’s method based on the Newton-Kantorovich hypothesis that uses Lipschitz-type conditions and information only on the first Fréchet-derivative of F. Here we provide a new semilocal convergence theorem for Newton’s method that uses information on all Fréchet-derivatives of F except the first. This way, we obtain sufficient convergence conditions different from the Newton-Kantorovich hypothesis. Our results are extended to include the case when F is a nonlinear operator whose kth Fréchet-derivative satisfies a H?lder continuity condition. An example is provided to show that our conditions hold where all previous ones fail. Moreover, some applications of our results to the solution of polynomial systems and differential equations are suggested. Furthermore, our results apply to solve a nonlinear integral equation appearing in radiative transfer in connection with the problem of determination of the angular distribution of the radiant-flux emerging from a plane radiation field. Received 9 December 1997 in revised form 30 March 1998