Fischer-Burmeister NCP Function Research Articles

Elastoplastic models and oscillators solved by a Lie-group differential algebraic equations method

In this paper, the viewpoint of non-linear complementarity problem (NCP) is adopted to derive a system of index-one differential algebraic equations (DAEs) for elastoplastic models, by recasting the complementary trio to an algebraic equation through the Fischer–Burmeister NCP-function. Then, we develop a novel algorithm based on the Lie-group GL(n,R) to iteratively solve the resultant DAEs at each time marching step. The Lie-group differential algebraic equations (LGDAE) method is convergent very fast, rendering efficient numerical schemes which can long-term preserve the yield-surface for plasticity models, without resorting on two-phase equations and on-off switching criteria. Several examples, including two non-linear elastoplastic oscillators whose restoring forces are modeled by elastoplastic constitutive equations, are used to assess the performance of the presently developed index-one formulation of elastoplasticity and test the efficiency and accuracy of LGDAE.

Generalized finite difference method for solving two-dimensional non-linear obstacle problems

In this study, the obstacle problems, also known as the non-linear free boundary problems, are analyzed by the generalized finite difference method (GFDM) and the fictitious time integration method (FTIM). The GFDM, one of the newly-developed domain-type meshless methods, is adopted in this study for spatial discretization. Using GFDM can avoid the tasks of mesh generation and numerical integration and also retain the high accuracy of numerical results. The obstacle problem is extremely difficult to be solved by any numerical scheme, since two different types of governing equations are imposed on the computational domain and the interfaces between these two regions are unknown. The obstacle problem will be mathematically formulated as the non-linear complementarity problems (NCPs) and then a system of non-linear algebraic equations (NAEs) will be formed by using the GFDM and the Fischer–Burmeister NCP-function. Then, the FTIM, a simple and powerful solver for NAEs, is used solve the system of NAEs. The FTIM is free from calculating the inverse of Jacobian matrix. Three numerical examples are provided to validate the simplicity and accuracy of the proposed meshless numerical scheme for dealing with two-dimensional obstacle problems.

The numerical study of a regularized smoothing Newton method for solving P0-NCP based on the generalized smoothing Fischer–Burmeister function

The nonlinear complementarity problems (denoted by NCPs) usually are reformulated as the solution of a nonsmooth system of equations. In this paper, we will present a regularized smoothing Newton method for solving nonlinear complementarity problems with P0-function (P0-NCPs) based on the generalized smoothing Fischer–Burmeister NCP-function ϕp(μ,a,b) with p>1, where μ is smoothing parameter. Without requiring strict complementarity assumption at the P0-NCPs solution, the proposed algorithm is proved to be globally and superlinearly convergent under suitable assumptions. Furthermore, the algorithm is locally quadratic convergent under mild conditions. Numerical experiments indicate that the proposed method is quite effective. In addition, in this paper, the regularization parameter ε in our algorithm is viewed as an independent variable, hence, our algorithm seems to be simpler and more easily implemented compared to many existing literatures.

A Smoothing Inexact Newton Method for Generalized Nonlinear Complementarity Problem

Based on the smoothing function of penalized Fischer‐Burmeister NCP‐function, we propose a new smoothing inexact Newton algorithm with non‐monotone line search for solving the generalized nonlinear complementarity problem. We view the smoothing parameter as an independent variable. Under suitable conditions, we show that any accumulation point of the generated sequence is a solution of the generalized nonlinear complementarity problem. We also establish the local superlinear (quadratic) convergence of the proposed algorithm under the BD‐regular assumption. Preliminary numerical experiments indicate the feasibility and efficiency of the proposed algorithm.

The semismooth and smoothing Newton methods for solving Pareto eigenvalue problem

In this paper, we study the numerical behavior of the semismooth and smoothing Newton methods for solving Pareto eigenvalue problem of the form x ⩾ 0 , Ax - λ Bx ⩾ 0 , 〈 x , Ax - λ Bx 〉 = 0 , where ( A, B) is a pair of possibly asymmetric matrices of order n. Such an eigenvalue problem arises in mechanics and in other areas of applied mathematics. By using the (smoothing) Fischer-Burmeister NCP function and the normalization condition e T x = 1, the Pareto eigenvalue problem can be converted into a equivalent semismooth (or smoothing) system of equations, where e = (1, … , 1) T . Then a semismooth (or smoothing) Newton algorithm is designed to solve such a semismooth (or smoothing) system of equations. Some numerical results are reported in the paper, which indicates that the proposed algorithms are very effective.

Computation of generalized differentials in nonlinear complementarity problems

Let f and g be continuously differentiable functions on R n . The nonlinear complementarity problem NCP(f,g), 0?f(x)?g(x)?0, arises in many applications including discrete Hamilton-Jacobi-Bellman equations and nonsmooth Dirichlet problems. A popular method to find a solution of the NCP(f,g) is the generalized Newton method which solves an equivalent system of nonsmooth equations F(x)=0 derived by an NCP function. In this paper, we present a sufficient and necessary condition for F to be Frechet differentiable, when F is defined by the "min" NCP function, the Fischer-Burmeister NCP function or the penalized Fischer-Burmeister NCP function. Moreover, we give an explicit formula of an element in the Clarke generalized Jacobian of F defined by the "min" NCP function, and the B-differential of F defined by other two NCP functions. The explicit formulas for generalized differentials of F lead to sharper global error bounds for the NCP(f,g).

Solving models with inequalities using standard econometric software

Simultaneous econometric models may contain pairs of complementary inequalities. It is discussed how to reformulate such models and solve them with econometric software which can handle only equalities. Two approaches are applied: the normal map representation and the Fischer–Burmeister NCP function. The latter seems to work best. The software programs TSP, SAS/ETS and EViews are tested. The test model describes two markets for electricity, each with fluctuating demand and an endogenous production capacity; the capacity of the trade link between the regions is also endogenous.

A global continuation algorithm for solving binary quadratic programming problems

In this paper, we propose a new continuous approach for the unconstrained binary quadratic programming (BQP) problems based on the Fischer-Burmeister NCP function. Unlike existing relaxation methods, the approach reformulates a BQP problem as an equivalent continuous optimization problem, and then seeks its global minimizer via a global continuation algorithm which is developed by a sequence of unconstrained minimization for a global smoothing function. This smoothing function is shown to be strictly convex in the whole domain or in a subset of its domain if the involved barrier or penalty parameter is set to be sufficiently large, and consequently a global optimal solution can be expected. Numerical results are reported for 0-1 quadratic programming problems from the OR-Library, and the optimal values generated are made comparisons with those given by the well-known SBB and BARON solvers. The comparison results indicate that the continuous approach is extremely promising by the quality of the optimal values generated and the computational work involved, if the initial barrier parameter is chosen appropriately.

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 nonlinear Lagrangian based on Fischer-Burmeister NCP function

This paper proposes a nonlinear Lagrangian Based on Fischer-Burmeister NCP function for solving nonlinear programming problems with inequality constraints. The convergence theorem shows that the sequence of points generated by this nonlinear Lagrange algorithm is locally convergent when the penalty parameter is less than a threshold under a set of suitable conditions on problem functions, and the error bound of solution, depending on the penalty parameter, is also established. Moreover, the paper develops the dual approach associated with the proposed nonlinear Lagrangian, in which the related duality theorem is demonstrated. Furthermore, it is shown that the condition number of the nonlinear Lagrangian Hessian at the optimal solution is proportional to the controlling penalty parameter. Numerical results for solving several nonlinear programming problems are reported, showing that the new nonlinear Lagrangian is superior over other known nonlinear Lagrangians for solving some nonlinear programming problems.