Smoothing Newton Method Research Articles

A Smoothing Method for Solving the NCP Based on a New Smoothing Approximate Function

A new smoothing approximate function of the FischerBurmeister function is given. A modified smoothing Newton method based on the function is proposed for solving a kind of nonlinear complementarity problems. Under suitable conditions, the global convergence of the method is proved. Numerical results show the effectiveness of the method.

A Smoothing Newton Method for Solving Absolute Value Equations

In this paper, we reformulate the system of absolute value equations as afamily of parameterized smooth equations and propose a smoothing Newton method tosolve this class of problems. we prove that the method is globally and locally quadraticallyconvergent under suitable assumptions. The preliminary numerical results demonstratethat the method is effective.

Calibrating low-rank correlation matrix problem: an SCA-based approach

Rank-constrained nearest correlation matrix problems, weighted or not, are reformulated into difference of convex (DC) functions constrained optimization problems. A general sequential convex approximation (SCA) approach for a DC-constrained optimization problem is developed. To overcome difficulties encountered in solving the convex approximation subproblems in the SCA approach, an SCA-based nonsmooth equation approach is proposed to solve the specific rank-constrained problem. In this approach, we use a simple iteration scheme for updating the multiplier variable corresponding to the rank constraint, and an inexact smoothing Newton method for calculating the primal variable and the multiplier variable corresponding to the linear constraint. Numerical experiments are reported and they illustrate the efficiency of our approach.

Nonsingularity in matrix conic optimization induced by spectral norm via a smoothing metric projector

Matrix conic optimization induced by spectral norm (MOSN) has found important applications in many fields. This paper focus on the optimality conditions and perturbation analysis of the MOSN problem. The Karush–Kuhn–Tucker (KKT) conditions of the MOSN problem can be reformulated as a nonsmooth system via the metric projector over the cone. We show in this paper, the nonsingularity of the Clarke’s generalized Jacobian of the smoothing KKT system constructed by a smoothing metric projector, the strong regularity and the strong second-order sufficient condition under constraint nondegeneracy are all equivalent. Moreover, this nonsingularity is used in several globally convergent smoothing Newton methods.

Semi-Smooth Newton Method for Solving 2D Contact Problems with Tresca and Coulomb Friction

The contribution deals with contact problems for two elastic bodies with friction. After the description of the problem we present its discretization based on linear or bilinear finite elements. The semi--smooth Newton method is used to find the solution, from which we derive active sets algorithms. Finally, we arrive at the globally convergent dual implementation of the algorithms in terms of the Langrange multipliers for the Tresca problem. Numerical experiments conclude the paper.

A Spline Smoothing Newton Method for L∞ Distance Regression with Bound Constraints

Orthogonal distance regression is arguably the most common criterion for fitting a model to data with errors in the observations. It is not appropriate to force the distances to be orthogonal, when angular information is available about the measured data points. We consider here a natural generalization of a particular formulation of that problem which involves the replacement of l2 norm by l∞ norm. This criterion may be a more appropriate one in the context of accept/reject decisions for manufacture parts. For l∞ distance regression with bound constraints, we give a smoothing Newton method which uses cubic spline and aggregate function, to smooth max function. The main spline smoothing technique uses a smooth cubic spline instead of max function and only few components in the max function are computed; hence it acts also as an active set technique, so it is more efficient for the problem with large amounts of measured data. Numerical tests in comparison to some other methods show that the new method is very efficient.

A smoothing Newton method for nonlinear complementarity problems

We propose a smoothing Newton method for solving nonlinear complementarity problems based on a new smoothing function. We adopt a variant merit function. Under suitable conditions, we obtain the global convergence and local quadratic convergence of the proposed algorithm. Some numerical results indicate that our algorithm is effective for the given problems.

A smoothing Newton method for the second-order cone complementarity problem

In this paper we introduce a new smoothing function and show that it is coercive under suitable assumptions. Based on this new function, we propose a smoothing Newton method for solving the second-order cone complementarity problem (SOCCP). The proposed algorithm solves only one linear system of equations and performs only one line search at each iteration. It is shown that any accumulation point of the iteration sequence generated by the proposed algorithm is a solution to the SOCCP. Furthermore, we prove that the generated sequence is bounded if the solution set of the SOCCP is nonempty and bounded. Under the assumption of nonsingularity, we establish the local quadratic convergence of the algorithm without the strict complementarity condition. Numerical results indicate that the proposed algorithm is promising.

A Smoothing Method with Appropriate Parameter Control Based on Fischer-Burmeister Function for Second-Order Cone Complementarity Problems

We deal with complementarity problems over second-order cones. The complementarity problem is an important class of problems in the real world and involves many optimization problems. The complementarity problem can be reformulated as a nonsmooth system of equations. Based on the smoothed Fischer-Burmeister function, we construct a smoothing Newton method for solving such a nonsmooth system. The proposed method controls a smoothing parameter appropriately. We show the global and quadratic convergence of the method. Finally, some numerical results are given.

A new one-step smoothing Newton method for second-order cone programming

In this paper, we present a new one-step smoothing Newton method for solving the second-order cone programming (SOCP). Based on a new smoothing function of the well-known Fischer-Burmeister function, the SOCP is approximated by a family of parameterized smooth equations. Our algorithm solves only one system of linear equations and performs only one Armijo-type line search at each iteration. It can start from an arbitrary initial point and does not require the iterative points to be in the sets of strictly feasible solutions. Without requiring strict complementarity at the SOCP solution, the proposed algorithm is shown to be globally and locally quadratically convergent under suitable assumptions. Numerical experiments demonstrate the feasibility and efficiency of our algorithm.

The Jacobian consistency of a smoothed Fischer–Burmeister function associated with second-order cones

This paper deals with the second-order cone complementarity problem (SOCCP), which is an important class of problems containing various optimization problems. The SOCCP can be reformulated as a system of nonsmooth equations. For solving this system of nonsmooth equations, smoothing Newton methods are widely used. The Jacobian consistency property plays an important role for achieving a rapid convergence of the methods. In this paper, we show the Jacobian consistency of a smoothed Fischer–Burmeister function. Moreover, we estimate the distance between the subgradient of the Fischer–Burmeister function and the gradient of its smoothing function.

Convergence of a non-interior smoothing method for variational inequality problems

The variational inequality problem can be reformulated as a system of equations. One can solve the reformulated equations to obtain a solution of the original problem. In this paper, based on a symmetric perturbed min function, we propose a new smoothing function, which has some nice properties. By which we propose a new non-interior smoothing algorithm for solving the variational inequality problem, which is based on both the non-interior continuation method and the smoothing Newton method. The proposed algorithm only needs to solve at most one system of equations at each iteration. In particular, we show that the algorithm is globally linearly and locally quadratically convergent under suitable assumptions. The preliminary numerical results are reported.

Nonsingularity of FB system and constraint nondegeneracy in semidefinite programming

For a KKT point of the linear semidefinite programming (SDP), we show that the nonsingularity of the B-subdifferential of Fischer-Burmeister (FB) nonsmooth system, the nonsingularity of Clarke's Jacobian of this system, and the primal and dual constraint nondegeneracies, are all equivalent. Also, each of these conditions is equivalent to the nonsingularity of Clarke's Jacobian of the smoothed counterpart of FB nonsmooth system, which particularly implies that the FB smoothing Newton method may attain the local quadratic convergence without strict complementarity assumption. We also report numerical results of the FB smoothing method for some benchmark problems.

A smoothing Newton method for mathematical programs governed by second-order cone constrained generalized equations

In this paper, we consider a class of mathematical programs governed by second-order cone constrained parameterized generalized equations. We reformulate the necessary optimality conditions as a system of nonsmooth equations under linear independence constraint qualification and the strict complementarity condition. A set of second order sufficient conditions is proposed, which is proved to be sufficient for the second order growth of the stationary point. The smoothing Newton method in [40] is employed to solve the system of nonsmooth equations whose strongly BD-regularity at a solution point is demonstrated under the second order sufficient conditions. Several illustrative examples are provided and discussed.

Quadratic model updating with gyroscopic structure from partial eigendata

Quadratic eigenvalue model updating problem, which aims to match observed spectral information with some feasibility constraints, arises in many engineering areas. In this paper, we consider a damped gyroscopic model updating problem (GMUP) of constructing five n-by-n real matrices M,C,K,G and N, such that they are closest to the given matrices and the quadratic pencil Q(λ):=λ 2 M+λ(C+G)+K+N possess the measured partial eigendata. In practice, M,C and K, represent the mass, damping and stiffness matrices, are symmetric (with M and K positive definite), G and N, represent the gyroscopic and circulatory matrices, are skew-symmetric. Under mild assumptions, we show that the Lagrangian dual problem of GMUP can be solved by a quadratically convergent inexact smoothing Newton method. Numerical examples are given to show the high efficiency of our method.

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 new class of smoothing functions and a smoothing Newton method for complementarity problems

In this paper, we introduce a new class of smoothing functions, which include some popular smoothing complementarity functions. We show that the new smoothing functions possess a system of favorite properties. The existence and continuity of a smooth path for solving the nonlinear complementarity problem (NCP) with a P 0 function are discussed. The Jacobian consistency of this class of smoothing functions is analyzed. Based on the new smoothing functions, we investigate a smoothing Newton algorithm for the NCP and discuss its global and local superlinear convergence. Some preliminary numerical results are reported.

A smoothing Newton method for generalized Nash equilibrium problems with second-order cone constraints

We consider a type of generalized Nash equilibrium problems withsecond-order cone constraints. The Karush-Kuhn-Tucker system can beformulated as a system of semismooth equations involving metricprojectors. Furthermore, the smoothing Newton method is given to geta Karush-Kuhn-Tucker point of the problem. The nonsingularity ofClarke's generalized Jacobian of the Karush-Kuhn-Tucker system,which is needed in the convergence analysis of smoothing Newtonmethod, is demonstrated under the so-called constraint nondegeneracycondition in generalized Nash equilibrium problems and pseudo-strongsecond order optimality condition. At last, we take someexperiments, in which the smoothing Newton method is applied.Furthermore, we get the normalized equilibria in theconstraint-shared case. The numerical results show that thesmoothing Newton method has a good performance in solving this typeof generalized Nash equilibrium problems.

一类修正的广义拟牛顿法解互补问题

互补问题自被提出至今，人们对它进行了一系列研究，提出了许多有效算法，比较常用的有投影法、内点法、光滑(非光滑)牛顿法等。本文利用Fischer-Burmeister函数将互补问题转化为无约束优化问题，再利用修正的广义拟牛顿算法求解。改进后的算法经数值实验验证有良好的数值效果。 Since the complementarity problem is proposed, people have done series of research, propose a lot of efficient algorithms, more used methods are projection method, interior-point method, smooth (nonsmooth) Newton method, etc. In this paper, complementarity problem is convert into unconstrained optimization by using Fischer-Burmeister function, then unconstrained optimization is solved by modified generalized quasi-Newton algorithm. the improved algorithm has good numerical results verified by numerical experiments.

Smoothing Technique of Nonsmooth Newton Methods for Control-State Constrained Optimal Control Problems

This paper is concerned with the numerical solution of optimal control problems subject to mixed control-state constraints with differential-algebraic equations. The necessary conditions arising from a local minimum principle are transformed into an equivalent nonsmooth equation in appropriate Banach spaces. We then develop a parametric smoothing Newton method for solving this nonsmooth equation. The global and local convergence is proposed under suitable settings. Numerical examples are presented to demonstrate the efficiency of our global smoothing technique.