A predictor-corrector smoothing Newton method for symmetric cone complementarity problems

In this paper, we present a predictor-corrector smoothing Newton method for solving nonlinear symmetric cone complementarity problems (SCCP) based on the symmetrically perturbed smoothing function. Under a mild assumption, the solution set of the problem concerned is just nonempty, we show that the proposed algorithm is globally and locally quadratic convergent. Also, the algorithm finds a maximally complementary solution to the SCCP. Numerical results for second order cone complementarity problems (SOCCP), a special case of SCCP, show that the proposed algorithm is effective.     

A descent method for a reformulation of the second-order cone complementarity problem

Analogous to the nonlinear complementarity problem and the semi-definite complementarity problem, a popular approach to solving the second-order cone complementarity problem (SOCCP) is to reformulate it as an unconstrained minimization of a certain merit function over Rⁿ${\mathbb{R}}^n$. In this paper, we present a descent method for solving the unconstrained minimization reformulation of the SOCCP which is based on the Fischer–Burmeister merit function (FBMF) associated with second-order cone [J.-S. Chen, P. Tseng, An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Programming 104 (2005) 293–327], and prove its global convergence. Particularly, we compare the numerical performance of the method for the symmetric affine SOCCP generated randomly with the FBMF approach [J.-S. Chen, P. Tseng, An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Programming 104 (2005) 293–327]. The comparison results indicate that, if a scaling strategy is imposed on the test problem, the descent method proposed is comparable with the merit function approach in the CPU time for solving test problems although the former may require more function evaluations.     

A smoothing-type algorithm for the second-order cone complementarity problem with a new nonmonotone line search

The second-order cone complementarity problem (denoted by SOCCP) can be effectively solved by smoothing-type algorithms, which in general are designed based on some monotone line search. In this paper, based on a new smoothing function of the Fischer–Burmeister function, we propose a smoothing-type algorithm for solving the SOCCP. The proposed algorithm uses a new nonmonotone line search scheme, which contains the usual monotone line search as a special case. Under suitable assumptions, we show that the proposed algorithm is globally and locally quadratically convergent. Some numerical results are reported which indicate the effectiveness of the proposed algorithm.     

A smoothing Newton method for a type of inverse semi-definite quadratic programming problem

We consider an inverse problem arising from the semi-definite quadratic programming (SDQP) problem. We represent this problem as a cone-constrained minimization problem and its dual (denoted ISDQD) is a semismoothly differentiable (SC¹${\text{SC}}^1$) convex programming problem with fewer variables than the original one. The Karush–Kuhn–Tucker conditions of the dual problem (ISDQD) can be formulated as a system of semismooth equations which involves the projection onto the cone of positive semi-definite matrices. A smoothing Newton method is given for getting a Karush–Kuhn–Tucker point of ISDQD. The proposed method needs to compute the directional derivative of the smoothing projector at the corresponding point and to solve one linear system per iteration. The quadratic convergence of the smoothing Newton method is proved under a suitable condition. Numerical experiments are reported to show that the smoothing Newton method is very effective for solving this type of inverse quadratic programming problems.     

A new one‐step smoothing newton method for the second‐order cone complementarity problem

In this paper, we present a new one‐step smoothing Newton method for solving the second‐order cone complementarity problem (SOCCP). Based on a new smoothing function, the SOCCP is approximated by a family of parameterized smooth equations. At each iteration, the proposed algorithm only need to solve one system of linear equations and perform only one Armijo‐type line search. The algorithm is proved to be convergent globally and superlinearly without requiring strict complementarity at the SOCCP solution. Moreover, the algorithm has locally quadratic convergence under mild conditions. Numerical experiments demonstrate the feasibility and efficiency of the new algorithm. Copyright © 2010 John Wiley & Sons, Ltd.     

Generalized Newton’s method based on graphical derivatives

This paper concerns developing a numerical method of the Newton type to solve systems of nonlinear equations described by nonsmooth continuous functions. We propose and justify a new generalized Newton algorithm based on graphical derivatives, which have never been used to derive a Newton-type method for solving nonsmooth equations. Based on advanced techniques of variational analysis and generalized differentiation, we establish the well-posedness of the algorithm, its local superlinear convergence, and its global convergence of the Kantorovich type. Our convergence results hold with no semismoothness and Lipschitzian assumptions, which is illustrated by examples. The algorithm and main results obtained in the paper are compared with well-recognized semismooth and B-differentiable versions of Newton’s method for nonsmooth Lipschitzian equations.     

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 nonmonotone smoothing-type algorithm for solving a system of equalities and inequalities

In this paper, we investigate a smoothing-type algorithm with a nonmonotone line search for solving a system of equalities and inequalities. We prove that the nonmonotone algorithm is globally and locally superlinearly convergent under suitable assumptions. The preliminary numerical results are reported.     

Finite termination of a Newton-type algorithm for a class of affine variational inequality problems

Many optimization problems can be reformulated as a system of equations. One may use the generalized Newton method or the smoothing Newton method to solve the reformulated equations so that a solution of the original problem can be found. Such methods have been powerful tools to solve many optimization problems in the literature. In this paper, we propose a Newton-type algorithm for solving a class of monotone affine variational inequality problems (AVIPs for short). In the proposed algorithm, the techniques based on both the generalized Newton method and the smoothing Newton method are used. In particular, we show that the algorithm can find an exact solution of the AVIP in a finite number of iterations under an assumption that the solution set of the AVIP is nonempty. Preliminary numerical results are reported.     

A perturbed version of an inexact generalized Newton method for solving nonsmooth equations

In this paper, we present the combination of the inexact Newton method and the generalized Newton method for solving nonsmooth equations F(x)?=?0, characterizing the local convergence in terms of the perturbations and residuals. We assume that both iteration matrices taken from the B-differential and vectors F(x ^(k)) are perturbed at each step. Some results are motivated by the approach of C?tina? regarding to smooth equations. We study the conditions, which determine admissible magnitude of perturbations to preserve the convergence of method. Finally, the utility of these results is considered based on some variant of the perturbed inexact generalized Newton method for solving some general optimization problems.     

Applications of the Differential Calculus to Nonlinear Elliptic Operators with Discontinuous Coefficients

The paper concerns Dirichlet’s problem for second order quasilinear non-divergence form elliptic equations with discontinuous coefficients. We start with suitable structure, growth, and regularity conditions ensuring solvability of the problem under consideration. Fixing then a solution u ₀ such that the linearized at u ₀ problem is non-degenerate, we apply the Implicit Function Theorem. As a result we get that for all small perturbations of the coefficients there exists exactly one solution u ≈ u ₀ which depends smoothly (in W ^2,p with p larger than the space dimension) on the data. For that, no structure and growth conditions are needed and the perturbations of the coefficients can be general L ^∞-functions of the space variable x. Moreover, we show that the Newton Iteration Procedure can be applied in order to obtain a sequence of approximate (in W ^2,p ) solutions for u ₀.     

A generalized Newton method for absolute value equations associated with second order cones

In this paper, we introduce the absolute value equations associated with second order cones (SOCAVE in short), which is a generalization of the absolute value equations discussed recently in the literature. It is proved that the SOCAVE is equivalent to a class of second order cone linear complementarity problems (SOCLCP in short). In particular, we propose a generalized Newton method for solving the SOCAVE and show that the proposed method is globally linearly and locally quadratically convergent under suitable assumptions. We also report some preliminary numerical results of the proposed method for solving the SOCAVE and the SOCLCP, which show the efficiency of the proposed method.     

The penalized Fischer-Burmeister SOC complementarity function

In this paper, we study the properties of the penalized Fischer-Burmeister (FB) second-order cone (SOC) complementarity function. We show that the function possesses similar desirable properties of the FB SOC complementarity function for local convergence; for example, with the function the second-order cone complementarity problem (SOCCP) can be reformulated as a (strongly) semismooth system of equations, and the corresponding nonsmooth Newton method has local quadratic convergence without strict complementarity of solutions. In addition, the penalized FB merit function has bounded level sets under a rather weak condition which can be satisfied by strictly feasible monotone SOCCPs or SOCCPs with the Cartesian R ₀₁-property, although it is not continuously differentiable. Numerical results are included to illustrate the theoretical considerations.     

A semismooth Newton method for SOCCPs based on a one-parametric class of SOC complementarity functions

In this paper, we present a detailed investigation for the properties of a one-parametric class of SOC complementarity functions, which include the globally Lipschitz continuity, strong semismoothness, and the characterization of their B-subdifferential. Moreover, for the merit functions induced by them for the second-order cone complementarity problem (SOCCP), we provide a condition for each stationary point to be a solution of the SOCCP and establish the boundedness of their level sets, by exploiting Cartesian P-properties. We also propose a semismooth Newton type method based on the reformulation of the nonsmooth system of equations involving the class of SOC complementarity functions. The global and superlinear convergence results are obtained, and among others, the superlinear convergence is established under strict complementarity. Preliminary numerical results are reported for DIMACS second-order cone programs, which confirm the favorable theoretical properties of the method.     

Smoothing Newton algorithm for the second-order cone programming with a nonmonotone line search

The smoothing-type algorithms, which are in general designed based on some monotone line search, have been successfully applied to solve the second-order cone programming (denoted by SOCP). In this paper, we propose a nonmonotone smoothing Newton algorithm for solving the SOCP. Under suitable assumptions, we show that the proposed algorithm is globally and locally quadratically convergent. To compare with the existing smoothing-type algorithms for the SOCP, our algorithm has the following special properties: (i) it is based on a new smoothing function of the vector-valued natural residual function; (ii) it uses a nonmonotone line search scheme which contains the usual monotone line search as a special case. Preliminary numerical results demonstrate that the smoothing-type algorithm using the nonmonotone line search is promising for solving the SOCP.     

An efficient fourth order weighted-Newton method for systems of nonlinear equations

In this paper, we develop a fourth order method for solving the systems of nonlinear equations. The algorithm is composed of two weighted-Newton steps and requires the information of one function and two first Fréchet derivatives. Therefore, for a system of n equations, per iteration it uses n?+?2n ² evaluations. Computational efficiency is compared with Newton’s method and some other recently published methods. Numerical tests are performed, which confirm the theoretical results. From the comparison with known methods it is observed that present method shows good stability and robustness.     

A homogeneous smoothing-type algorithm for symmetric cone linear programs

In this paper, we investigate a smoothing-type algorithm for solving the symmetric cone linear program （（SCLP） for short） by making use of an augmented system of its optimality conditions. The algorithm only needs to solve one system of linear equations and to perform one line search at each iteration. It is proved that the algorithm is globally convergent without assuming any prior knowledge of feasibility/infeasibility of the problem. In particular, the algorithm may correctly detect solvability of （SCLP）. Furthermore, if （SCLP） has a solution, then the algorithm will generate a solution of （SCLP）, and if the problem is strongly infeasible, the algorithm will correctly detect infeasibility of （SCLP）.     

A Regularization Newton Method for Solving Nonlinear Complementarity Problems

In this paper we construct a regularization Newton method for solving the nonlinear complementarity problem (NCP(F )) and analyze its convergence properties under the assumption that F is a P ₀ -function. We prove that every accumulation point of the sequence of iterates is a solution of NCP(F ) and that the sequence of iterates is bounded if the solution set of NCP(F ) is nonempty and bounded. Moreover, if F is a monotone and Lipschitz continuous function, we prove that the sequence of iterates is bounded if and only if the solution set of NCP(F ) is nonempty by setting , where is a parameter. If NCP(F) has a locally unique solution and satisfies a nonsingularity condition, then the convergence rate is superlinear (quadratic) without strict complementarity conditions. At each step, we only solve a linear system of equations. Numerical results are provided and further applications to other problems are discussed. Accepted 25 March 1998     

Homotopy method for solving ball-constrained variational inequalities

In this paper, we present a smoothing homotopy method for solving ball-constrained variational inequalities by utilizing a similar Chen-Harker-Kanzow-Smale function to smooth Robinson’s normal equation. Without any monotonicity condition on the defining map F, for the starting point chosen almost everywhere in Rⁿ, the existence and convergence of the homotopy pathway are proven. Numerical experiments illustrate that the method is feasible and effective.     

A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities

In this paper we take a new look at smoothing Newton methods for solving the nonlinear complementarity problem (NCP) and the box constrained variational inequalities (BVI). Instead of using an infinite sequence of smoothing approximation functions, we use a single smoothing approximation function and Robinson’s normal equation to reformulate NCP and BVI as an equivalent nonsmooth equation H(u,x)=0, where H:ℜ ²ⁿ →ℜ ²ⁿ , u∈ℜ ⁿ is a parameter variable and x∈ℜ ⁿ is the original variable. The central idea of our smoothing Newton methods is that we construct a sequence {z ^k =(u ^k ,x ^k )} such that the mapping H(·) is continuously differentiable at each z ^k and may be non-differentiable at the limiting point of {z ^k }. We prove that three most often used Gabriel-Moré smoothing functions can generate strongly semismooth functions, which play a fundamental role in establishing superlinear and quadratic convergence of our new smoothing Newton methods. We do not require any function value of F or its derivative value outside the feasible region while at each step we only solve a linear system of equations and if we choose a certain smoothing function only a reduced form needs to be solved. Preliminary numerical results show that the proposed methods for particularly chosen smoothing functions are very promising. Received June 23, 1997 / Revised version received July 29, 1999?Published online December 15, 1999