Generalized Fischer-Burmeister Function Research Articles

A new difference of anisotropic and isotropic total variation regularization method for image restoration.

Total variation (TV) regularizer has diffusely emerged in image processing. In this paper, we propose a new nonconvex total variation regularization method based on the generalized Fischer-Burmeister function for image restoration. Since our model is nonconvex and nonsmooth, the specific difference of convex algorithms (DCA) are presented, in which the subproblem can be minimized by the alternating direction method of multipliers (ADMM). The algorithms have a low computational complexity in each iteration. Experiment results including image denoising and magnetic resonance imaging demonstrate that the proposed models produce more preferable results compared with state-of-the-art methods.

A neural network based on the generalized FB function for nonlinear convex programs with second-order cone constraints

This paper proposes a neural network approach to efficiently solve nonlinear convex programs with the second-order cone constraints. The neural network model is designed by the generalized Fischer–Burmeister function associated with second-order cone. We study the existence and convergence of the trajectory for the considered neural network. Moreover, we also show stability properties for the considered neural network, including the Lyapunov stability, the asymptotic stability and the exponential stability. Illustrative examples give a further demonstration for the effectiveness of the proposed neural network. Numerical performance based on the parameter being perturbed and numerical comparison with other neural network models are also provided. In overall, our model performs better than two comparative methods.

A descent algorithm for generalized complementarity problems based on generalized Fischer-Burmeister functions

We study an unconstrained minimization approach to the generalized complementarity problem GCP(f, g) based on the generalized Fischer-Burmeister function and its generalizations when the underlying functions are $$C^1$$ . Also, we show how, under appropriate regularity conditions, minimizing the merit function corresponding to f and g leads to a solution of the generalized complementarity problem. Moreover, we propose a descent algorithm for GCP(f, g) and show a result on the global convergence of a descent algorithm for solving generalized complementarity problem. Finally, we present some preliminary numerical results. Our results further give a unified/generalization treatment of such results for the nonlinear complementarity problem based on generalized Fischer-Burmeister function and its generalizations.

Symmetrization of generalized natural residual function for NCP

In contrast to the generalized Fischer–Burmeister function that is a natural extension of the popular Fischer–Burmeister function NCP-function, the generalized natural residual NCP-function based on discrete extension, recently proposed by Chen, Ko, and Wu, does not possess symmetric graph. In this paper we symmetrize the generalized natural residual NCP-function, and construct not only new NCP-functions and merit functions for the nonlinear complementarity problem, but also provide parallel functions to the generalized Fischer–Burmeister function.

Geometric views of the generalized Fischer–Burmeister function and its induced merit function

In this paper, we study geometric properties of surfaces of the generalized Fischer–Burmeister function and its induced merit function. Then, a visualization is proposed to explain how the convergent behaviors are influenced by two descent directions in merit function approach. Based on the geometric properties and visualization, we have more intuitive ideas about how the convergent behavior is affected by changing parameter. Furthermore, geometric view indicates how to improve the algorithm to achieve our goal by setting proper value of the parameter in merit function approach.

Proximal point algorithm for nonlinear complementarity problem based on the generalized Fischer-Burmeister merit function

This paper is devoted to the study of the proximal point algorithm for solving monotone and nonmonotone nonlinear complementarity problems. The proximal point algorithm is to generate a sequence by solving subproblems that are regularizations of the original problem. After given an appropriate criterion for approximate solutions of subproblems by adopting a merit function, the proximal point algorithm is verified to have global and superlinear convergence properties. For the purpose of solving the subproblems efficiently, we introduce a generalized Newton method and show that only one Newton step is eventually needed to obtain a desired approximate solution that approximately satisfies the appropriate criterion under mild conditions. The motivations of this paper are twofold. One is analyzing the proximal point algorithm based on the generalized Fischer-Burmeister function which includes the Fischer-Burmeister function as special case, another one is trying to see if there are relativistic change on numerical performance when we adjust the parameter in the generalized Fischer-Burmeister.

A continuation approach for solving binary quadratic program based on a class of NCP-functions

In the paper, we consider a continuation approach for the binary quadratic program (BQP) based on a class of NCP-functions. More specifically, we recast the BQP as an equivalent minimization and then seeks its global minimizer via a global continuation method. Such approach had been considered in [11] which is based on the Fischer–Burmeister function. We investigate this continuation approach again by using a more general function, called the generalized Fischer–Burmeister function. However, the theoretical background for such extension can not be easily carried over. Indeed, it needs some subtle analysis.

A Smoothing Trust Region Method for Ncps Based on the Smoothing Generalized Fischer-Burmeister Function

Based on a reformulation of the complementarity problem as a system of nonsmooth equations by using the generalized Fischer-Burmeister function, a smoothing trust region algorithm with line search is proposed for solving general (not necessarily monotone) nonlinear complementarity problems. Global convergence and, under a nonsingularity assumption, local Q-superlinear/Q-quadratic convergence of the algorithm are established. In particular, it is proved that a unit step size is always accepted after a finite number of iterations. Numerical results also confirm the good theoretical properties of our approach.

Regularization semismooth Newton method for p 0-NCPs with non-monotone line search

Based on the generalized Fischer-Burmeister function, Chen et al in 2008 put forward a regularization semismooth Newton method for solving the nonlinear complementarity problem with a P0-function. In this paper, we investigate the above algorithm with the monotone line search replaced by a non-monotone line search. It is shown that the non-monotone algorithm is well-defined, and is globally and locally superlinearly convergent under standard assumptions.

A smoothing Newton method based on the generalized Fischer–Burmeister function for MCPs

We present a smooth approximation for the generalized Fischer–Burmeister function where the 2-norm in the FB function is relaxed to a general p -norm ( p > 1 ), and establish some favorable properties for it — for example, the Jacobian consistency. With the smoothing function, we transform the mixed complementarity problem (MCP) into solving a sequence of smooth system of equations, and then trace a smooth path generated by the smoothing algorithm proposed by Chen (2000) [28] to the solution set. In particular, we investigate the influence of p on the numerical performance of the algorithm by solving all MCPLIP test problems, and conclude that the smoothing algorithm with p ∈ ( 1 , 2 ] has better numerical performance than the one with p > 2 .

A neural network based on the generalized Fischer–Burmeister function for nonlinear complementarity problems

In this paper, we consider a neural network model for solving the nonlinear complementarity problem (NCP). The neural network is derived from an equivalent unconstrained minimization reformulation of the NCP, which is based on the generalized Fischer–Burmeister function ϕ p ( a , b ) = ‖ ( a , b ) ‖ p - ( a + b ) . We establish the existence and the convergence of the trajectory of the neural network, and study its Lyapunov stability, asymptotic stability as well as exponential stability. It was found that a larger p leads to a better convergence rate of the trajectory. Numerical simulations verify the obtained theoretical results.

A regularization semismooth Newton method based on the generalized Fischer–Burmeister function for [formula omitted]-NCPs

We consider a regularization method for nonlinear complementarity problems with F being a P 0 -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 φ p 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 φ p with p ∈ [ 1.1 , 2 ) can be used as the substitutions for the FB function φ 2 .

ON SOME NCP-FUNCTIONS BASED ON THE GENERALIZED FISCHER–BURMEISTER FUNCTION

In this paper, we study several NCP-functions for the nonlinear complementarity problem (NCP) which are indeed based on the generalized Fischer–Burmeister function, ϕp(a, b) = ||(a, b)||p - (a + b). It is well known that the NCP can be reformulated as an equivalent unconstrained minimization by means of merit functions involving NCP-functions. Thus, we aim to investigate some important properties of these NCP-functions that will be used in solving and analyzing the reformulation of the NCP.