Fischer Burmeister Smoothing Function Research Articles

An open-source unconstrained stress updating algorithm for the modified Cam-clay model

This paper presents an unconstrained stress updating algorithm for a critical state plastic model of clay soil, where the loading/unloading estimations and the consideration of the stress behaviour transition from elasticity to plasticity can be bypassed by using the Fischer–Burmeister smoothing function instead of the Kuhn–Tucker complementarity conditions. A smoothing tangent operator consistent with the unconstrained stress updating strategy is derived from preserving the quadratic convergence speed for the global solution. Specifically, the relationship and difference between the consistent and continuum tangent operators are analysed from the perspectives of algebra and geometry. In addition, the nonlinear constitutive equations obtained by the backward Euler integration scheme are solved by the double dogleg trust region method (improved by non-monotonic technology), where a larger strain increment than that of the newton method is allowed for the stress updating. Then, the modified Cam-clay model for soil is implemented in ABAQUS/Standard by the proposed algorithm. The results of numerical examples show that the algorithm has significant advantages in terms of computation efficiency and robustness in contrast to the ABAQUS/Standard default integration algorithm, especially for the condition of large load increments and cyclic loadings. More than twice the computational efficiency of the ABAQUS/Standard default integration algorithm can be observed in the representative numerical examples. The source code of the proposed algorithm is freely available at https://github.com/zhouxin615/NMTR_Method.

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

Based on the generalized Fischer–Burmeister smoothing function, we propose a regularization Newton method for solving the nonlinear complementarity problem. The proposed method views the regularization parameter as an independent variable. Moreover, it solves a perturbed Newton equation to obtain the search direction and adopts a simple nonmonotone line search scheme to improve the numerical results. Under suitable assumptions, we prove that our method has global and local quadratic convergence and the regularization parameter converges to zero globally Q-linearly. Numerical results shows that there does exist new smoothing function which is better than the Fischer–Burmeister smoothing function.

Nonmonotone smoothing inexact Newton method for the nonlinear complementarity problem

Smoothing Newton methods have been successfully applied to solve the nonlinear complementarity problem (NCP). In this paper, we first study some properties of the generalized Fischer–Burmeister smoothing function. Based on this function, we then design a smoothing inexact Newton method for the NCP. At each iteration, a system of linear equations is solved only inexactly. Moreover, our method uses a nonmonotone line search technique which is much simpler than existing nonmonotone line searches used in smoothing Newton methods. Under suitable assumptions, we prove that the proposed method is globally and locally superlinearly convergent. Some numerical results are also reported.

A partially smoothing Jacobian method for nonlinear complementarity problems with [formula omitted] function

In this paper, we propose a partially smoothing function for solving the nonlinear complementarity problems (NCP). Some properties of such a smoothing approach are analyzed and are employed to develop a well defined and efficient Jacobian Newton algorithm to find the solution of NCP. Under the condition that the level set of a merit function is bounded, global convergence and super-linear convergence are established for the developed algorithm. Compared with the similar theoretical results available in the literature, the assumption of nonsingularity is removed in virtue of P0 property and the smoothing approach. Numerical experiments show that the proposed smoothing method outperforms the existent ones, particularly in comparison with the state-of-art methods derived from the classical Fischer–Burmeister smoothing function and the aggregation function.

An inexact smoothing method for SOCCPs based on a one-parametric class of smoothing function

In this paper, we introduce a one-parametric class of smoothing functions which contains the Fischer–Burmeister smoothing function as special case. Based on this class of smoothing functions, an inexact smoothing method for solving second-order cone complementarity problems (SOCCPs) is proposed. In each iteration the corresponding linear system is solved only approximately. Under mild assumptions, it is proved that the proposed method has global convergence and local superlinear convergence properties. Preliminary numerical results indicate that the method is effective for large-scale problems.

A one-parametric class of smoothing functions for second-order cone programming

We present a new one-parametric class of smoothing functions for second-order cone programming (denoted by SOCP). This class is fairly general and includes the well-known Fischer–Burmeister smoothing function and the Chi–Liu smoothing function (J Comput Appl Math 223:114–123, 2009) as special cases. Based on these functions, a smoothing-type method is proposed for solving SOCP. The proposed algorithm solves only one linear system of equations and performs only one line search at each iteration. This algorithm can start from an arbitrary point and it is locally superlinearly convergent under a mild assumption. Some numerical results are reported.

A non-monotone inexact regularized smoothing Newton method for solving nonlinear complementarity problems

In the paper [S.P. Rui and C.X. Xu, A smoothing inexact Newton method for nonlinear complementarity problems, J. Comput. Appl. Math. 233 (2010), pp. 2332–2338], the authors proposed an inexact smoothing Newton method for nonlinear complementarity problems (NCP) with the assumption that F is a uniform P function. In this paper, we present a non-monotone inexact regularized smoothing Newton method for solving the NCP which is based on Fischer–Burmeister smoothing function. We show that the proposed algorithm is globally convergent and has a locally superlinear convergence rate under the weaker condition that F is a P 0 function and the solution of NCP is non-empty and bounded. Numerical results are also reported for the test problems, which show the effectiveness of the proposed algorithm.

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.

A new class of smoothing complementarity functions over symmetric cones

A popular approach to solving the complementarity problem is to reformulate it as an equivalent system of smooth equations via a smoothing complementarity function. In this paper, first we propose a new class of smoothing complementarity functions, which contains the natural residual smoothing function and the Fischer–Burmeister smoothing function for symmetric cone complementarity problems. Then we give some unified formulae of the Fréchet derivatives associated with Jordan product. Finally, the derivative of the new proposed class of smoothing complementarity functions is deduced over symmetric cones.

A Predictor-Corrector Smoothing Method for Second Order Cone Programming

Based on the Fischer-Burmeister smoothing function,a predictor-corrector smoothing Newton method is presented for solving the second-order cone programming(SOCP). With this method,the SOCP is reconstructed as a nonlinear system of equations and then the Newton's method is used to the perturbation of this system of equations.It is shown that the method is globally and locally quadratically convergent under suitable assumptions.Numerical results show the effectiveness of the method.

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.