System Of Nonsmooth Equations Research Articles

A Smoothing Newton Method with Fischer-Burmeister Function for Second-Order Cone Complementarity Problems

The second-order cone complementarity problem (SOCCP) is an important class of problems containing a lot of optimization problems. The SOCCP can be transformed into a system of nonsmooth equations. To solve this nonsmooth system, smoothing techniques are often used. Fukushima, Luo and Tseng (SIAM J. Optim. 12:436–460, 2001) studied concrete theories and properties of smoothing functions for the SOCCP. Recently, a practical computational method using the smoothed natural residual function to solve the SOCCP was given by Chen, Sun and Sun (Comput. Optim. Appl. 25:39–56, 2003). In the present paper, we propose an algorithm to solve the SOCCP by using the smoothed Fischer-Burmeister function. Some preliminary numerical results are given.

A globally and superlinearly convergent quasi-Newton method for general box constrained variational inequalities without smoothing approximation

A new quasi-Newton algorithm for the solution of general box constrained variational inequality problem (GVI(l, u, F, f)) is proposed in this paper. It is based on a reformulation of the variational inequality problem as a nonsmooth system of equations by using the median operator. Without smoothing approximation, the proposed quasi-Newton algorithm is directly applied to solve this class of nonsmooth equations. Under appropriate assumptions, it is proved that the algorithmic sequence globally and superlinearly converges to a solution of the equation reformulation and also of GVI(l, u, F, f). Numerical results show that our new algorithm works quite well.

A smoothing Newton-type method for solving the L 2 spectral estimation problem with lower and upper bounds

This paper discusses the L 2 spectral estimation problem with lower and upper bounds. To the best of our knowledge, it is unknown if the existing methods for this problem have superlinear convergence property or not. In this paper we propose a nonsmooth equation reformulation for this problem. Then we present a smoothing Newton-type method for solving the resulting system of nonsmooth equations. Global and local superlinear convergence of the proposed method are proved under some mild conditions. Numerical tests show that this method is promising.

A predictor-corrector smoothing Newton method for solving the mixed complementarity problem with a P0-function

The mixed complementarity problem (denoted by MCP(F)) can be reformulated as the solution of a nonsmooth system of equations. In the paper, based on a perturbed mid function, we contract a new smoo...

A predictor-corrector smoothing Newton method for solving the mixed complementarity problem with a P 0-function

The mixed complementarity problem (denoted by MCP(F)) can be reformulated as the solution of a nonsmooth system of equations. In the paper, based on a perturbed mid function, we contract a new smoothing function. The existence and continuity of a smooth path for solving the mixed complementarity problem with a P 0 function are discussed. Then we presented a predictor-corrector smoothing Newton algorithm to solve the MCP with a P 0-function. The global convergence of the proposed algorithm is verified under mild conditions. And by using the smooth and semismooth technique, the local superlinear convergence of the method is proved under some suitable assumptions.

On the Local Convergence of Semismooth Newton Methods for Linear and Nonlinear Second-Order Cone Programs Without Strict Complementarity

The optimality conditions of a nonlinear second-order cone program can be reformulated as a nonsmooth system of equations using a projection mapping. This allows the application of nonsmooth Newton methods for the solution of the nonlinear second-order cone program. Conditions for the local quadratic convergence of these nonsmooth Newton methods are investigated. Related conditions are also given for the special case of a linear second-order cone program. An interesting and important feature of these conditions is that they do not require strict complementarity of the solution. Some numerical results are included in order to illustrate the theoretical considerations.

A verified inexact implicit Runge–Kutta method for nonsmooth ODEs

Structural pounding and oscillations have been extensively investigated by using ordinary differential equations (ODEs). In many applications, force functions are defined by piecewise continuously differentiable functions and the ODEs are nonsmooth. Implicit Runge–Kutta (IRK) methods for solving the nonsmooth ODEs are numerically stable, but involve systems of nonsmooth equations that cannot be solved exactly in practice. In this paper, we propose a verified inexact IRK method for nonsmooth ODEs which gives a global error bound for the inexact solution. We use the slanting Newton method to solve the systems of nonsmooth equations, and interval method to compute the set of matrices of slopes for the enclosure of solution of the systems. Numerical experiments show that the algorithm is efficient for verification of solution of systems of nonsmooth equations in the inexact IRK method. We report numerical results of nonsmooth ODEs arising from simulation of the collapse of the Tacoma Narrows suspension bridge, steel to steel impact experiment, and pounding between two adjacent structures in 27 ground motion records for 12 different earthquakes.

Convergence analysis of nonsmooth equations for the general nonlinear complementarity problem

This paper deals with a general nonlinear complementarity problem, where the underlying functions are assumed to be continuous. Based on a nonlinear complementarity function, it is transformed into a system of nonsmooth equations. Then, two kinds of approximate Newton methods for the nonsmooth equations are developed and their convergence are proved. Finally, numerical tests are also listed.

Newton method of solving Karush-Kuhn-Tucker systems for a constrained Minimax Problem

Using Karush-Kuhn-Tucker optimality condition, we establish an equivalent system of nonsmooth equations for a constrained minimax problem by Fischer-Burmeister NCP function. Then, Newton method is applied to solving this system of nonsmooth equations. To perform the Newton method, the B-differential and its nonsingularity of the corresponding function in the nonsmooth equations are investigated.

A quadratically convergent algorithm for the generalized linear complementarity problem

In this paper, the generalized linear complementarity problem (GLCP) is reformulated as a system of nonsmooth equations via the Fischer function. Based on this reformulation, the famous Levenberg-Marquardt(LM) algorithm is employed for obtaining its solution. Theoretical results that relate the stationary points of the merit function to the solution of the GLCP are presented. We show that the L-M algorithm is both globally and Quadratically convergent without nondegenerate solution. Moreover, a method to calculate a generalized Jacobian is also given. Mathematics Subject Classifications: 65H10, 90C33, 90C30

A Nonsmooth L-M Method for Solving the Generalized Nonlinear Complementarity Problem over a Polyhedral Cone

In this paper the generalized nonlinear complementarity problem (GNCP) defined on a polyhedral cone is reformulated as a system of nonsmooth equations. Based on this reformulation, the famous Levenberg-Marquardt (L-M) algorithm is employed to obtain its solution. Theoretical results that relate the stationary points of the merit function to the solution of the GNCP are presented. Under mild assumptions, we show that the L-M algorithm is both globally and superlinearly convergent. Moreover, a method to calculate a generalized Jacobian is given and numerical experimental results are presented.

An MPCC approach for stochastic Stackelberg–Nash–Cournot equilibrium

In this article, we investigate a Stochastic Stackelberg–Nash–Cournot Equilibrium problem by reformulating it as a Mathematical Program with Complementarity Constraints (MPCC). The complementarity constraints are further reformulated as a system of nonsmooth equations. We characterize the followers’ Nash–Cournot equilibria by studying the implicit solution of a system of equations. We outline numerical methods for the solution of a stochastic Stackelberg–Nash–Cournot Equilibrium problem with finite distribution of market demand scenarios and propose a discretization approach based on implicit numerical integration to deal with stochastic Stackelberg–Nash–Cournot Equilibrium problem with continuous distribution of demand scenarios. Finally, we discuss the two-leader Stochastic Stackelberg–Nash–Cournot Equilibrium problem.

Quadratic Convergence of a Nonsmooth Newton-Type Method for Semidefinite Programs Without Strict Complementarity

We consider a Newton-type method for the solution of semidefinite programs. This Newton-type method is based on a semismooth reformulation of the semidefinite program as a nonsmooth system of equations. We establish local quadratic convergence of this method under a linear independence assumption and a weak nondegeneracy condition that implies uniqueness of the optimal solution but does not imply strict complementarity.

Non-interior smoothing algorithm for frictional contact problems

A new algorithm for solving the three-dimensional elastic contact problem with friction is presented. The algorithm is a non-interior smoothing algorithm based on an NCP-function. The parametric variational principle and parametric quadratic programming method were applied to the analysis of three-dimensional frictional contact problem. The solution of the contact problem was finally reduced to a linear complementarity problem, which was reformulated as a system of nonsmooth equations via an NCP-function. A smoothing approximation to the nonsmooth equations was given by the aggregate function. A Newton method was used to solve the resulting smoothing nonlinear equations. The algorithm presented is easy to understand and implement. The reliability and efficiency of this algorithm are demonstrated both by the numerical experiments of LCP in mathematical way and the examples of contact problems in mechanics.

A Homotopy-Based Algorithm for Mixed Complementarity Problems

This paper develops an algorithm for solving mixed complementarity problems that is based upon probability-one homotopy methods. After the complementarity problem is reformulated as a system of nonsmooth equations, a homotopy method is used to solve a sequence of smooth approximations to this system of equations. The global convergence properties of this approach are qualitatively different from those of other recent methods, which rely upon decrease of a merit function. This enables the algorithm to reliably solve certain classes of problems that prove troublesome for other methods. To improve efficiency, the homotopy algorithm is embedded in a generalized Newton method.

Improved smoothing-type methods for the solution of linear programs

We consider a smoothing-type method for the solution of linear programs. Its main idea is to reformulate the primal-dual optimality conditions as a nonlinear and nonsmooth system of equations, and to apply a Newton-type method to a smooth approximation of this nonsmooth system. The method presented here is a predictor-corrector method, and is closely related to some methods recently proposed by Burke and Xu on the one hand, and by the authors on the other hand. However, here we state stronger global and/or local convergence properties. Moreover, we present quite promising numerical results for the whole netlib test problem collection.

PREDICTION OF TRANSMISSION ERROR IN SPUR GEARS AS A CONSEQUENCE OF WEAR1*

The aim of this article is to investigate how the so-called transmission error in a gear develops in a wear process. A finite element procedure for calculating wear between two elastic bodies in contact is developed that includes constitutive laws, representing the behavior of the interacting surfaces. When treating a gear problem, large rotations must be included. This is done by analyzing a sequence of finite-element models along with an interpolation procedure. The constitutive laws of interacting surfaces are Signorini's contact law, Coulomb's law of friction, and Archard's law of wear. By formulating both Signorini's contact law and Coulomb's law of friction by means of projection equations, the complete formulation of a time increment of a quasistatic wear-friction evolution problem becomes a system of nonsmooth equations. Then, a modified Newton method is applied to solve this system. The transmission error can be found from the infinitesimal displacement field superposed on the rigid body configuration. Results from the simulations show that wear will increase the peak-to-peak value of the transmission error. This is most significant in a gear when some kind of modification of the gear flank is applied to minimize the transmission error in an initial state. *Communicated by N. Olhoff.

Newton Methods for Solving Two Classes of Nonsmooth Equations

The paper is devoted to two systems of nonsmooth equations. One is the system of equations of max-type functions and the other is the system of equations of smooth compositions of max-type functions. The Newton and approximate Newton methods for these two systems are proposed. The Q-superlinear convergence of the Newton methods and the Q-linear convergence of the approximate Newton methods are established. The present methods can be more easily implemented than the previous ones, since they do not require an element of Clarke generalized Jacobian, of B-differential, or of b-differential, at each iteration point.

A new non-smooth model for three dimensional frictional contact problems

This paper presents a new non-smooth model for three dimensional contact problems with Coulomb friction. The problem is formulated exactly as a system of non-smooth equations without employing any external variables or approximation. As compared with the existing models, the present model does not utilize the slip angle as a variable. Therefore, transformation of variables is not required and the formulation is simpler. For solving a three dimensional contact problem, the nodus is to determine the slip direction at the contact nodes because the relative slipping of the contact may occur in any direction on the contact interface. The proposed model solves this problem in a simple manner by formulating it as an equivalent non-smooth equation. Based on the theory of non-smooth analysis, a generalized derivative is introduced to solve the non-smooth equations. Thus, the non-smooth damped Newton method can be implemented directly. The proposed method has been tested using a number of numerical examples.

A penalized Fischer-Burmeister NCP-function

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.