Class Of Smooth Functions Research Articles

A smoothing Newton method for NCPs with the P 0-property

In this paper, we first investigate a two-parametric class of smoothing functions which contains the penalized smoothing Fischer–Burmeister function and the penalized smoothing CHKS function as special cases. Then we present a smoothing Newton method for the nonlinear complementarity problem based on the class of smoothing functions. Issues such as line search rule, boundedness of the level set, global and quadratic convergence are studied. In particular, we give a line search rule containing the common used Armijo-type line search rule as a special case. Also without requiring strict complementarity assumption at the P 0-NCP solution or the nonemptyness and boundedness of the solution set, the proposed algorithm is proved to be globally convergent. Preliminary numerical results show the efficiency of the algorithm and provide efficient domains of the two parameters for the complementarity problems.

Improved smoothing Newton methods for symmetric cone complementarity problems

There recently has been much interest in smoothing Newton method for solving nonlinear complementarity problems. We extend such method to symmetric cone complementarity problems (SCCP). In this paper, we first investigate a one-parametric class of smoothing functions in the context of symmetric cones, which contains the Fischer–Burmeister smoothing function and the CHKS smoothing function as special cases. Then we propose a smoothing Newton method for the SCCP based on the one-parametric class of smoothing functions. For the proposed method, besides the classical step length, we provide a new step length and the global convergence is obtained. Finally, preliminary numerical results are reported, which show the effectiveness of the two step lengthes in the algorithm and provide efficient domains of the parameter for the complementarity problems.

Smoothing method for mathematical programs with symmetric cone complementarity constraints

In this article, we consider the mathematical program with symmetric cone complementarity constraints (MPSCCC) in a general form. It contains the mathematical program with second-order-cone complementarity constraints and the mathematical program with complementarity constraints. We present a smoothing method that approximates the primal MPSCCC by means of the Chen–Mangasarian class of smoothing functions. We show that a sequence of stationary points of the approximate programs converges to a C(larke)-stationary point of the primal MPSCCC under suitable conditions.

Nonparametric function estimation subject to monotonicity, convexity and other shape constraints

This paper uses free-knot and fixed-knot regression splines in a Bayesian context to develop methods for the nonparametric estimation of functions subject to shape constraints in models with log-concave likelihood functions. The shape constraints we consider include monotonicity, convexity and functions with a single minimum. A computationally efficient MCMC sampling algorithm is developed that converges faster than previous methods for non-Gaussian models. Simulation results indicate the monotonically constrained function estimates have good small sample properties relative to (i) unconstrained function estimates, and (ii) function estimates obtained from other constrained estimation methods when such methods exist. Also, asymptotic results show the methodology provides consistent estimates for a large class of smooth functions. Two detailed illustrations exemplify the ideas.

Approximating smooth functions using algebraic-trigonometric polynomials

The problem under consideration is that of approximating classes of smooth functions by algebraic-trigonometric polynomials of the form , where is an algebraic polynomial of degree and is a trigonometric polynomial of order . The precise order of approximation by such polynomials in the classes and an upper bound for similar approximations in the class with are found. The proof of these estimates uses mixed series in Legendre polynomials which the author has introduced and investigated previously.Bibliography: 13 titles.

Approximation accuracies of a class of polynomial smoothing functions

In 2007 Xiong et al.proposed a class of polynomial smoothing functions whose approximation accuracy is a problem that has not been solved.This paper applied dichotomy algorithm to solve the problem.To overcome the shortcoming that the root might be missed by dichotomy algorithm the problem of approximation accuracy for smoothing function was firstly expressed by the problem of solving the maximum value of approximation function and the approximation function was divided into 4 segments and the maximum value of the each segment was sought respectively then the maximum value of approximation function was obtained in the whole x-axis.Taking 1st-order and 2nd-order smooth polynomial functions as examples whose approximation accuracies were solved by the dichotomy algorithm.The results show that the dichotomy algorithm is a effective way to solve the approximation accuracy for this class of smoothing functions of support vector machine.

Smoothing newton algorithm for solving generalized complementarity problem

The generalized complementarity problem includes the well-known nonlinear complementarity problem and linear complementarity problem as special cases. In this paper, based on a class of smoothing functions, a smoothing Newton-type algorithm is proposed for solving the generalized complementarity problem. Under suitable assumptions, the proposed algorithm is well-defined and global convergent.

Smoothing Newton algorithm based on a regularized one-parametric class of smoothing functions for generalized complementarity problems over symmetric cones

Based on the KK smoothing function, we introduce a regularizedone-parametric class of smoothing functions and show that it iscoercive under suitable assumptions. By making use of the introducedregularized one-parametric class of smoothing functions, weinvestigate a smoothing Newton algorithm for solving the generalizedcomplementarity problems over symmetric cones (GSCCP), where anonmonotone line search scheme is used. We show that the algorithmis globally and locally superlinearly convergent under suitableassumptions. The theory of Euclidean Jordan algebras is a basic toolin our analysis.

Smoothing Nonlinear Conjugate Gradient Method for Image Restoration Using Nonsmooth Nonconvex Minimization

Image restoration problems are often converted into large-scale, nonsmooth, and nonconvex optimization problems. Most existing minimization methods are not efficient for solving such problems. It is well known that nonlinear conjugate gradient methods are preferred to solve large-scale smooth optimization problems due to their simplicity, low storage, practical computation efficiency, and nice convergence properties. In this paper, we propose a smoothing nonlinear conjugate gradient method where an intelligent scheme is used to update the smoothing parameter at each iteration and guarantees that any accumulation point of a sequence generated by this method is a Clarke stationary point of the nonsmooth and nonconvex optimization problem. Moreover, we present a class of smoothing functions and show their approximation properties. This method is easy to implement without adding any new variables. Three image restoration problems with different pixels and different regularization terms are used in numerical tests. Experimental results and comparison with the continuation method in [M. Nikolova et al., SIAM J. Imaging Sci., 1 (2008), pp. 2–25] show the efficiency of the proposed method.

Asymptotic theory of semiparametric Z-estimators for stochastic processes with applications to ergodic diffusions and time series

This paper generalizes a part of the theory of $Z$-estimation which has been developed mainly in the context of modern empirical processes to the case of stochastic processes, typically, semimartingales. We present a general theorem to derive the asymptotic behavior of the solution to an estimating equation $\theta\leadsto \Psi_n(\theta,\widehat{h}_n)=0$ with an abstract nuisance parameter $h$ when the compensator of $\Psi_n$ is random. As its application, we consider the estimation problem in an ergodic diffusion process model where the drift coefficient contains an unknown, finite-dimensional parameter $\theta$ and the diffusion coefficient is indexed by a nuisance parameter $h$ from an infinite-dimensional space. An example for the nuisance parameter space is a class of smooth functions. We establish the asymptotic normality and efficiency of a $Z$-estimator for the drift coefficient. As another application, we present a similar result also in an ergodic time series model.

Smoothing Newton algorithm for symmetric cone complementarity problems based on a one-parametric class of smoothing functions

In this paper, we introduce a one-parametric class of smoothing functions in the context of symmetric cones which contains two symmetric perturbed smoothing functions as special cases, and show that it is coercive under suitable assumptions. Based on this class of smoothing functions, a smoothing Newton algorithm is extended to solve the complementarity problems over symmetric cones, and it is proved that the proposed algorithm is globally and locally superlinearly convergent under suitable assumptions. The theory of Euclidean Jordan algebras is a basic tool which is extensively used in our analysis. Preliminary numerical results for randomly generated second-order cone programs and several practical second-order cone programs from the DIMACS library are reported.

Interpolating-orthogonal wavelet systems

Based upon Meyer wavelets, new systems of periodic wavelets and wavelets on the whole axis are constructed; these systems are orthogonal and interpolating simultaneously. Estimates of the errors of approximation of different classes of smooth functions by these wavelets are obtained.

Optimization of one class of hyperbolic systems with smooth controls

In this paper we consider the optimal control problem for a system of first-order hyperbolic equations. The right-hand side of this system is determined by a control system of ordinary differential equations. Admissible controls belong to the class of smooth functions. We obtain an optimality condition and propose a general scheme of improvement methods for the mentioned problem.

Smoothing Newton algorithm for linear programming over symmetric cones

By using the theory of Euclidean Jordan algebras, based on a new class of smoothing functions, the Qi-Sun-Zhou’s smoothing Newton algorithm is extended to solve linear programming over symmetric cones (SCLP). The algorithm is globally convergent under suitable assumptions.

Nonlinear widths of classes of smooth functions defined on the unit sphere in ℝ d

Nonlinear widths of classes of smooth functions defined on the unit sphere in ℝ d

A smoothing Newton algorithm based on a one-parametric class of smoothing functions for linear programming over symmetric cones

In this paper, we introduce a one-parametric class of smoothing functions which contains the Fischer–Burmeister smoothing function and the CHKS smoothing function as special cases. Based on this class of smoothing functions, a smoothing Newton algorithm is extended to solve linear programming over symmetric cones. The global and local quadratic convergence results of the algorithm are established under suitable assumptions. The theory of Euclidean Jordan algebras is a basic tool in our analysis.

A Bayesian Approach to Non-Parametric Monotone Function Estimation

SummaryThe paper proposes two Bayesian approaches to non-parametric monotone function estimation. The first approach uses a hierarchical Bayes framework and a characterization of smooth monotone functions given by Ramsay that allows unconstrained estimation. The second approach uses a Bayesian regression spline model of Smith and Kohn with a mixture distribution of constrained normal distributions as the prior for the regression coefficients to ensure the monotonicity of the resulting function estimate. The small sample properties of the two function estimators across a range of functions are provided via simulation and compared with existing methods. Asymptotic results are also given that show that Bayesian methods provide consistent function estimators for a large class of smooth functions. An example is provided involving economic demand functions that illustrates the application of the constrained regression spline estimator in the context of a multiple-regression model where two functions are constrained to be monotone.

Solutions of mKdV in Classes of Functions Unbounded at Infinity

Using P. Lax’s concept of a Lax pair we prove global existence and uniqueness for solutions of the initial value problem for mKdV in classes of smooth functions which can be unbounded at infinity, and in particular, may tend to infinity with respect to the space variable. Moreover, we establish the invariance of the spectrum and the unitary type of the Schrödinger operator under the KdV flow and the invariance of the spectrum and the unitary type of the impedance operator under the mKdV flow for potentials in these classes.

Аппроксимативные свойства смешанных рядов по полиномам Лагерра на классах гладких функций

Аппроксимативные свойства смешанных рядов по полиномам Лагерра на классах гладких функций

Distributed algorithms for reaching consensus on general functions

This paper presents analysis and design results for distributed consensus algorithms in multi-agent networks. We consider continuous consensus functions of the initial state of the network agents. Under mild smoothness assumptions, we obtain necessary and sufficient conditions characterizing any algorithm that asymptotically achieves consensus. This characterization is the building block to obtain various design results for networks with weighted, directed interconnection topologies. We first identify a class of smooth functions for which one can synthesize in a systematic way distributed algorithms that achieve consensus. We apply this result to the family of weighted power mean functions, and characterize the exponential convergence properties of the resulting algorithms. We establish the validity of these results for scenarios with switching interconnection topologies. Finally, we conclude with two discontinuous distributed algorithms that achieve, respectively, max and min consensus in finite time.