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1.
In this paper, we consider a class of penalized NCP-functions, which includes several existing well-known NCP-functions as
special cases. The merit function induced by this class of NCP-functions is shown to have bounded level sets and provide error
bounds under mild conditions. A derivative free algorithm is also proposed, its global convergence is proved and numerical
performance compared with those based on some existing NCP-functions is reported. 相似文献
2.
A New Class of Semismooth Newton-Type Methods for Nonlinear Complementarity Problems 总被引:2,自引:0,他引:2
We introduce a new, one-parametric class of NCP-functions. This class subsumes the Fischer function and reduces to the minimum function in a limiting case of the parameter. This new class of NCP-functions is used in order to reformulate the nonlinear complementarity problem as a nonsmooth system of equations. We present a detailed investigation of the properties of the equation operator, of the corresponding merit function as well as of a suitable semismooth Newton-type method. Finally, numerical results are presented for this method being applied to a number of test problems. 相似文献
3.
Li-Yong LuWei-Zhe Gu 《Journal of Computational and Applied Mathematics》2011,235(8):2300-2313
Based on the generalized CP-function proposed by Hu et al. [S.L. Hu, Z.H. Huang, J.S. Chen, Properties of a family of generalized NCP-functions and a derivative free algorithm for complementarity problems, J. Comput. Appl. Math. 230 (2009) 69-82], we introduce a smoothing function which is a generalization of several popular smoothing functions. By which we propose a non-interior continuation algorithm for solving the complementarity problem. The proposed algorithm only needs to solve at most one system of linear equations at each iteration. In particular, we show that the algorithm is globally linearly and locally quadratically convergent under suitable assumptions. The preliminary numerical results demonstrate that the algorithm is effective. 相似文献
4.
In this paper, we propose a new family of NCP-functions and the corresponding merit functions, which are the generalization of some popular NCP-functions and the related merit functions. We show that the new NCP-functions and the corresponding merit functions possess a system of favorite properties. Specially, we show that the new NCP-functions are strongly semismooth, Lipschitz continuous, and continuously differentiable; and that the corresponding merit functions have SC1 property (i.e., they are continuously differentiable and their gradients are semismooth) and LC1 property (i.e., they are continuously differentiable and their gradients are Lipschitz continuous) under suitable assumptions. Based on the new NCP-functions and the corresponding merit functions, we investigate a derivative free algorithm for the nonlinear complementarity problem and discuss its global convergence. Some preliminary numerical results are reported. 相似文献
5.
We introduce a new NCP-function in order to reformulate the nonlinear complementarity problem as a nonsmooth system of equations.
This new NCP-function turns out to have stronger theoretical properties than the widely used Fischer-Burmeister function and
other NCP-functions suggested previously. Moreover, numerical experience indicates that a semismooth Newton method based on
this new NCP-function performs considerably better than the corresponding method based on the Fischer-Burmeister function.
Received: March 10, 1997 / Accepted: February 15, 2000?Published online May 12, 2000 相似文献
6.
《Journal of Computational and Applied Mathematics》1999,110(1):181-185
In a recent paper, Chen and Solis investigated the appearance of spurious solutions when first-order ODEs are discretized using Runge–Kutta schemes. They concluded that the reliability of the numerical solutions to a particular ODE could be verified only by constructing several discrete models and comparing their numerical results with the known properties of the exact solutions. We demonstrate that by using nonstandard schemes, all the difficulties found by Chen and Solis can be eliminated, and that qualitatively correct numerical solutions are obtained for all values of the step size. We illustrate these issues by applying nonstandard finite-difference techniques to the logistic, sine, cubic, and Monod equations. 相似文献
7.
8.
Qingchuan Yao 《Numerische Mathematik》1999,81(4):647-677
This paper proposes some modified Halley iterations for finding the zeros of polynomials. We investigate the non-overshoot
properties of the modified Halley iterations and other important properties that play key roles in solving symmetric eigenproblems.
We also extend Halley iteration to systems of polynomial equations in several variables.
Received March 20, 1996 / Revised version received December 5, 1997 相似文献
9.
We consider a regularization method for nonlinear complementarity problems with F being a P0-function which replaces the original problem with a sequence of the regularized complementarity problems. In this paper, this sequence of regularized complementarity problems are solved approximately by applying the generalized Newton method for an equivalent augmented system of equations, constructed by the generalized Fischer–Burmeister (FB) NCP-functions φp with p>1. We test the performance of the regularization semismooth Newton method based on the family of NCP-functions through solving all test problems from MCPLIB. Numerical experiments indicate that the method associated with a smaller p, for example p[1.1,2], usually has better numerical performance, and the generalized FB functions φp with p[1.1,2) can be used as the substitutions for the FB function φ2. 相似文献
10.
Jein-Shan Chen 《Journal of Global Optimization》2006,36(4):565-580
This paper is a follow-up of the work [Chen, J.-S.: J. Optimiz. Theory Appl., Submitted for publication (2004)] where an NCP-function and a descent method were proposed for the nonlinear complementarity problem. An unconstrained reformulation was formulated due to a merit function based on the proposed NCP-function. We continue to explore properties of the merit function in this paper. In particular, we show that the gradient of the merit function is globally Lipschitz continuous which is important from computational aspect. Moreover, we show that the merit function is SC
1 function which means it is continuously differentiable and its gradient is semismooth. On the other hand, we provide an alternative proof, which uses the new properties of the merit function, for the convergence result of the descent method considered in [Chen, J.-S.: J. Optimiz. Theory Appl., Submitted for publication (2004)]. 相似文献
11.
Mehmet Ali Özarslan 《Numerical Functional Analysis & Optimization》2016,37(1):92-105
In this article, we consider modified Bernstein-Kantorovich operators and investigate their approximation properties. We show that the order of approximation to a function by these operators is at least as good as that of ones classically used. We obtain a simultaneous approximation result for our operators. Also, we prove two direct approximation results via the usual second-order modulus of smoothness and the second-order Ditzian-Totik modulus of smoothness, respectively. Finally, some graphical illustrations are provided. 相似文献
12.
13.
We investigate some properties related to the generalized Newton method for the Fischer-Burmeister (FB) function over second-order
cones, which allows us to reformulate the second-order cone complementarity problem (SOCCP) as a semismooth system of equations.
Specifically, we characterize the B-subdifferential of the FB function at a general point and study the condition for every
element of the B-subdifferential at a solution being nonsingular. In addition, for the induced FB merit function, we establish
its coerciveness and provide a weaker condition than Chen and Tseng (Math. Program. 104:293–327, 2005) for each stationary point to be a solution, under suitable Cartesian P-properties of the involved mapping. By this, a damped Gauss-Newton method is proposed, and the global and superlinear convergence
results are obtained. Numerical results are reported for the second-order cone programs from the DIMACS library, which verify
the good theoretical properties of the method.
S. Pan’s work is partially supported by the Doctoral Starting-up Foundation (B13B6050640) of GuangDong Province.
J.-S. Chen is member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. J.-S. Chen’s work is
partially supported by National Science Council of Taiwan. 相似文献
14.
《Journal of Computational and Applied Mathematics》2005,181(1):200-210
The paper gives a definition of the filled function for nonlinear integer programming. This definition is modified from that of the global convexized filled function for continuous global optimization. A filled function with only one parameter which satisfies this definition is presented. We also discuss the properties of the proposed function and give a filled function method to solve the nonlinear integer programming problem. The implementation of the algorithm on several test problems is reported with satisfactory numerical results. 相似文献
15.
D. Goldfarb R. Polyak K. Scheinberg I. Yuzefovich 《Computational Optimization and Applications》1999,14(1):55-74
We present and analyze an interior-exterior augmented Lagrangian method for solving constrained optimization problems with both inequality and equality constraints. This method, the modified barrier—augmented Lagrangian (MBAL) method, is a combination of the modified barrier and the augmented Lagrangian methods. It is based on the MBAL function, which treats inequality constraints with a modified barrier term and equalities with an augmented Lagrangian term. The MBAL method alternatively minimizes the MBAL function in the primal space and updates the Lagrange multipliers. For a large enough fixed barrier-penalty parameter the MBAL method is shown to converge Q-linearly under the standard second-order optimality conditions. Q-superlinear convergence can be achieved by increasing the barrier-penalty parameter after each Lagrange multiplier update. We consider a dual problem that is based on the MBAL function. We prove a basic duality theorem for it and show that it has several important properties that fail to hold for the dual based on the classical Lagrangian. 相似文献
16.
In this paper, our aim is to show some mean value inequalities for the modified Bessel functions of the first and second kind. Our proofs are based on some bounds for the logarithmic derivatives of these functions, which are in fact equivalent to the corresponding Turán-type inequalities for these functions. As an application of the results concerning the modified Bessel function of the second kind, we prove that the cumulative distribution function of the gamma–gamma distribution is log-concave. At the end of this paper, several open problems are posed, which may be of interest for further research. 相似文献
17.
In this paper, we give some conditions for a class of functions related to Bessel functions to be positive definite or strictly positive definite. We present some properties and relationships involving logarithmically completely monotonic functions and strictly positive definite functions. In particular, we are interested with the modified Bessel functions of the second kind. As applications, we prove the logarithmically monotonicity for a class of functions involving the modified Bessel functions of second kind and we established new inequalities for this function. 相似文献
18.
Gap functions for a system of generalized vector quasi-equilibrium problems with set-valued mappings
In this paper, some gap functions for three classes of a system of generalized vector quasi-equilibrium problems with set-valued
mappings (for short, SGVQEP) are investigated by virtue of the nonlinear scalarization function of Chen, Yang and Yu. Three
examples are then provided to demonstrate these gap functions. Also, some gap functions for three classes of generalized finite
dimensional vector equilibrium problems (GFVEP) are derived without using the nonlinear scalarization function method. Furthermore,
a set-valued function is obtained as a gap function for one of (GFVEP) under certain assumptions.
相似文献
19.
In this paper we consider a sum of modified Bessel functions of the first kind of which particular case is used in the study of Kanter’s sharp modified Bessel function bound for concentrations of some sums of independent symmetric random vectors. We present some monotonicity and convexity properties for that sum of modified Bessel functions of the first kind, as well as some Turán type inequalities, lower and upper bounds. Moreover, we point out an error in Kanter’s paper (J Multivariate Anal 6:222–236, 1976). 相似文献
20.
Zhiqing Meng Rui Shen Chuangyin Dang Min Jiang 《Numerical Functional Analysis & Optimization》2013,34(11):1471-1492
Augmented Lagrangian function is one of the most important tools used in solving some constrained optimization problems. In this article, we study an augmented Lagrangian objective penalty function and a modified augmented Lagrangian objective penalty function for inequality constrained optimization problems. First, we prove the dual properties of the augmented Lagrangian objective penalty function, which are at least as good as the traditional Lagrangian function's. Under some conditions, the saddle point of the augmented Lagrangian objective penalty function satisfies the first-order Karush-Kuhn-Tucker condition. This is especially so when the Karush-Kuhn-Tucker condition holds for convex programming of its saddle point existence. Second, we prove the dual properties of the modified augmented Lagrangian objective penalty function. For a global optimal solution, when the exactness of the modified augmented Lagrangian objective penalty function holds, its saddle point exists. The sufficient and necessary stability conditions used to determine whether the modified augmented Lagrangian objective penalty function is exact for a global solution is proved. Based on the modified augmented Lagrangian objective penalty function, an algorithm is developed to find a global solution to an inequality constrained optimization problem, and its global convergence is also proved under some conditions. Furthermore, the sufficient and necessary calmness condition on the exactness of the modified augmented Lagrangian objective penalty function is proved for a local solution. An algorithm is presented in finding a local solution, with its convergence proved under some conditions. 相似文献