On construction of new NCP functions

We report a new method to construct complementarity functions for the nonlinear complementarity problem (NCP). Basic properties related to growth behavior, convexity and semismoothness of the newly discovered NCP functions are proved. We also present some variants, generalizations and other transformations of these NCP functions. Finally, we propose some interesting research directions that can be explored in the NCP research.     

Properties of a family of merit functions and a merit function method for the NCP

A family of merit functions are proposed, which are the generalization of several existing merit functions. A number of favorable properties of the proposed merit functions are established. By using these properties, a merit function method for solving nonlinear complementarity problem is investigated, and the global convergence of the proposed algorithm is proved under some standard assumptions. Some preliminary numerical results are given.     

Properties of Restricted NCP Functions for Nonlinear Complementarity Problems

In this paper, we study restricted NCP functions which may be used to reformulate the nonlinear complementarity problem as a constrained minimization problem. In particular, we consider three classes of restricted NCP functions, two of them introduced by Solodov and the other proposed in this paper. We give conditions under which a minimization problem based on a restricted NCP function enjoys favorable properties, such as equivalence between a stationary point of the minimization problem and the nonlinear complementarity problem, strict complementarity at a solution of the minimization problem, and boundedness of the level sets of the objective function. We examine these properties for three restricted NCP functions and show that the merit function based on the restricted NCP function proposed in this paper enjoys favorable properties compared with those based on the other restricted NCP functions.     

A one-parametric class of smoothing functions and an improved regularization Newton method for the NCP

In this paper, we introduce a one-parametric class of smoothing functions, which enjoys some favourable properties and includes two famous smoothing functions as special cases. Based on this class of smoothing functions, we propose a regularization Newton method for solving the non-linear complementarity problem. The main feature of the proposed method is that it uses a perturbed Newton equation to obtain the direction. This not only allows our method to have global and local quadratic convergences without strict complementarity conditions, but also makes the regularization parameter converge to zero globally Q-linearly. In addition, we use a new non-monotone line search scheme to obtain the step size. Some numerical results are reported which confirm the good theoretical properties of the proposed method.     

A family of NCP functions and a descent method for the nonlinear complementarity problem

In last decades, there has been much effort on the solution and the analysis of the nonlinear complementarity problem (NCP) by reformulating NCP as an unconstrained minimization involving an NCP function. In this paper, we propose a family of new NCP functions, which include the Fischer-Burmeister function as a special case, based on a p-norm with p being any fixed real number in the interval (1,+∞), and show several favorable properties of the proposed functions. In addition, we also propose a descent algorithm that is indeed derivative-free for solving the unconstrained minimization based on the merit functions from the proposed NCP functions. Numerical results for the test problems from MCPLIB indicate that the descent algorithm has better performance when the parameter p decreases in (1,+∞). This implies that the merit functions associated with p∈(1,2), for example p=1.5, are more effective in numerical computations than the Fischer-Burmeister merit function, which exactly corresponds to p=2. J.-S. Chen is a 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.     

A construction of bent functions from plateaued functions

In this presentation, a technique for constructing bent functions from plateaued functions is introduced and analyzed. This generalizes earlier techniques for constructing bent from near-bent functions. Using this construction, we obtain a big variety of inequivalent bent functions, some weakly regular and some non-weakly regular. Classes of bent functions having some additional properties that enable the construction of strongly regular graphs are formed, and explicit expressions for bent functions with maximal degree are presented.     

A construction of convex functions

We describe a construction of convex functions on infinite-dimensional spaces and apply this construction to give an illustration to a theorem of Borwein-Fabian from [1]. Namely, we give a simple explicit example of a continuous convex function on l _p , p ≥ 1, which is everywhere compactly differentiable, but not Fréchet differentiable at zero.     

The Jackson interpolation operator and construction of functions

Summary We investigate the approximation behavior of the so-called Jackson interpolation operator. We will also study the K-functional deduced from this operator. As a result we obtain the asymptotic expression of this K-functional.     

非线性互补问题解的存在性检验

在数学规划研究领域中这是一个受到广泛关注的问题,并提出多种求解的迭代算法,但是几乎没有算法可以在计算数值解的同时自动给出解的误差界,Alefeld、Chen和Potra在[1],[2]中讨论了求解线性与非线性互补问题的可靠性算法,该算法是将非线性互补问题归结为：等价的非线性方程（经常被称为Pang方程[7]）     

Joint characteristic functions construction via copulas

When modelling dependent risks it is important to be able to generate joint distributions with given marginals. One of the ways which may be useful in connection with using the Fast Fourier Transform is to construct joint characteristic functions from marginal characteristic functions. In this paper a class of n-dimensional continuous copulas is presented which in turn lead to a simple construction of joint characteristic functions with given marginal characteristic functions. Bounds on various measures of correlation are also given.     

Inverse optimal control and construction of control Lyapunov functions

In this paper, the construction of CLFs for nonlinear systems and a new inverse optimal control law are presented. The construction of a CLF for an affine nonlinear system is reduced to the construction of a CLF for a simpler system, and a new L _g V type control law with respect to a CLF is provided. This control law is a generalization of Sontag’s formula and contains a design parameter. Tuning this parameter gives many suboptimal solutions for the optimization problem. Also, the gain margin and sector margin of the control law are calculated. Examples are provided to illustrate the main theoretical results of the paper. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 61, Optimal Control, 2008.     

A construction of weakly and non-weakly regular bent functions

In this article a technique for constructing p-ary bent functions from near-bent functions is presented. This technique is then used to obtain both weakly regular and non-weakly regular bent functions. In particular we present the first known infinite class of non-weakly regular bent functions.     

Quasi-interpolatory refinable functions and construction of biorthogonal wavelet systems

We present a new family of compactly supported and symmetric biorthogonal wavelet systems. Each refinement mask in this family has tension parameter ω. When ω = 0, it becomes the minimal length biorthogonal Coifman wavelet system (Wei et al., IEEE Trans Image Proc 7:1000–1013, 1998). Choosing ω away from zero, we can get better smoothness of the refinable functions at the expense of slightly larger support. Though the construction of the new biorthogonal wavelet systems, in fact, starts from a new class of quasi-interpolatory subdivision schemes, we find that the refinement masks accidently coincide with the ones by Cohen et al. (Comm Pure Appl Math 45:485–560, 1992, §6.C) (or Daubechies 1992, §8.3.5), which are designed for the purpose of generating biorthogonal wavelets close to orthonormal cases. However, the corresponding mathematical analysis is yet to be provided. In this study, we highlight the connection between the quasi-interpolatory subdivision schemes and the masks by Cohen, Daubechies and Feauveau, and then we study the fundamental properties of the new biorthogonal wavelet systems such as regularity, stability, linear independence and accuracy.     

Properties of differences of meromorphic functions

Let f be a transcendental meromorphic function. We propose a number of results concerning zeros and fixed points of the difference g(z) = f(z + c) − f(z) and the divided difference g(z)/f(z).     

Properties of algebras of binary functions

Translated from Matematicheskie Zametki, Vol. 55, No. 1, pp. 130–140, January, 1994.     

Properties of functions with monotone graphs

A metric space (X,d) is monotone if there is a linear order < on X and a constant c>0 such that d(x,y)≦cd(x,z) for all x<y<z∈X. Properties of continuous functions with monotone graph (considered as a planar set) are investigated. It is shown, for example, that such a function can be almost nowhere differentiable, but must be differentiable at a dense set, and that the Hausdorff dimension of the graph of such a function is 1.     

带NCP函数的信赖域滤子方法

滤子方法最初是由Fletcher和Leyffer在2002年提出的.这种方法的原理是:在一个试探步,如果相应的目标函数值或约束违反度函数值下降,那么该试探步就会被接受.利用Fischer-Burmeister NCP函数来修正滤子中的约束违反度函数,同时证明了这个新的滤子方法具有全局收敛性.     

A Gilmore-Gomory construction of integer programming value functions

In this paper, we analyze how sequentially introducing decision variables into an integer program (IP) affects the value function and its level sets. We use a Gilmore-Gomory approach to find parametrized IP value functions over a restricted set of variables. We introduce the notion of maximal connected subsets of level sets - volumes in which changes to the constraint right-hand side have no effect on the value function - and relate these structures to IP value functions and optimal solutions.