The non-interior continuation methods for solving theP0 function nonlinear complementarity problem

In this paper, we propose a new smooth function that possesses a property not satisfied by the existing smooth functions. Based on this smooth function, we discuss the existence and continuity of the smoothing path for solving theP ₀ function nonlinear complementarity problem ( NCP). Using the characteristics of the new smooth function, we investigate the boundedness of the iteration sequence generated by the non-interior continuation methods for solving theP ₀ function NCP under the assumption that the solution set of the NCP is nonempty and bounded. We show that the assumption that the solution set of the NCP is nonempty and bounded is weaker than those required by a few existing continuation methods for solving the NCP     

The non-interior continuation methods for solving theP 0 function nonlinear complementarity problem

In this paper, we propose a new smooth function that possesses a property not satisfied by the existing smooth functions. Based on this smooth function, we discuss the existence and continuity of the smoothing path for solving theP ₀ function nonlinear complementarity problem ( NCP). Using the characteristics of the new smooth function, we investigate the boundedness of the iteration sequence generated by the non-interior continuation methods for solving theP ₀ function NCP under the assumption that the solution set of the NCP is nonempty and bounded. We show that the assumption that the solution set of the NCP is nonempty and bounded is weaker than those required by a few existing continuation methods for solving the NCP     

On the convergence of iterative methods for symmetric linear complementarity problems

We prove convergence of the whole sequence generated by any of a large class of iterative algorithms for the symmetric linear complementarity problem (LCP), under the only hypothesis that a quadratic form associated with the LCP is bounded below on the nonnegative orthant. This hypothesis holds when the matrix is strictly copositive, and also when the matrix is copositive plus and the LCP is feasible. The proof is based upon the linear convergence rate of the sequence of functional values of the quadratic form. As a by-product, we obtain a decomposition result for copositive plus matrices. Finally, we prove that the distance from the generated sequence to the solution set (and the sequence itself, if its limit is a locally unique solution) have a linear rate of R-convergence.Research for this work was partially supported by CNPq grant No. 301280/86.     

Predictor-Corrector Smoothing Newton Method,Based on a New Smoothing Function,for Solving the Nonlinear Complementarity Problem with a P 0 Function

By smoothing a perturbed minimum function, we propose in this paper a new smoothing function. The existence and continuity of a smooth path for solving the nonlinear complementarity problem (NCP) with a P ₀ function are discussed. We investigate the boundedness of the iteration sequence generated by noninterior continuation/smoothing methods under the assumption that the solution set of the NCP is nonempty and bounded. Based on the new smoothing function, we present a predictor-corrector smoothing Newton algorithm for solving the NCP with a P ₀ function, which is shown to be globally linearly and locally superlinearly convergent under suitable assumptions. Some preliminary computational results are reported.     

On the resolution of monotone complementarity problems

A reformulation of the nonlinear complementarity problem (NCP) as an unconstrained minimization problem is considered. It is shown that any stationary point of the unconstrained objective function is a solution of NCP if the mapping F involved in NCP is continuously differentiable and monotone, and that the level sets are bounded if F is continuous and strongly monotone. A descent algorithm is described which uses only function values of F. Some numerical results are given.     

Smoothing Newton method for NCP with the identification of degenerate indices

We present a new smoothing Newton method for nonlinear complementarity problems (NCP(F)) by using an NCP function to reformulate the problem to its equivalent form. Compared with most current smoothing methods, our method contains an estimating technique based on the active-set strategy. This technique focuses on the identification of the degenerate set for a solution x^∗ of the NCP(F). The proposed method has the global convergence, each accumulation point is a solution of the problem. The introduction of the active-set strategy effectively reduces the scale of the problem. Under some regularity assumption, the degenerate set can be identified correctly near the solution and local superlinear convergence is obtained as well.     

Fixed Point Methods for the Complementarity Problem

This paper is concerned with iterative procedures for the monotone complementarity problem. Our iterative methods consist of finding fixed points of appropriate continuous maps. In the case of the linear complementarity problem, it is shown that the problem is solvable if and only if the sequence of iterates is bounded in which case summability methods are used to find a solution of the problem. This procedure is then used to find a solution of the nonlinear complementarity problem satisfying certain regularity conditions for which the problem has a nonempty bounded solution set.     

Strong convergence properties of a modified nonmonotone smoothing algorithm for the SCCP

The smoothing algorithms have been successfully applied to solve the symmetric cone complementarity problem (denoted by SCCP), which in general have the global and local superlinear/quadratic convergence if the solution set of the SCCP is nonempty and bounded. Huang, Hu and Han [Science in China Series A: Mathematics, 52: 833–848, 2009] presented a nonmonotone smoothing algorithm for solving the SCCP, whose global convergence is established by just requiring that the solution set of the SCCP is nonempty. In this paper, we propose a new nonmonotone smoothing algorithm for solving the SCCP by modifying the version of Huang-Hu-Han’s algorithm. We prove that the modified nonmonotone smoothing algorithm not only is globally convergent but also has local superlinear/quadratical convergence if the solution set of the SCCP is nonempty. This convergence result is stronger than those obtained by most smoothing-type algorithms. Finally, some numerical results are reported.     

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.     

Smoothing Trust Region Methods for Nonlinear Complementarity Problems with P 0-Functions

By using the Fischer–Burmeister function to reformulate the nonlinear complementarity problem (NCP) as a system of semismooth equations and using Kanzow’s smooth approximation function to construct the smooth operator, we propose a smoothing trust region algorithm for solving the NCP with P ₀ functions. We prove that every accumulation point of the sequence generated by the algorithm is a solution of the NCP. Under a nonsingularity condition, local Q-superlinear/Q-quadratic convergence of the algorithm is established without the strict complementarity condition. This work was partially supported by the Research Grant Council of Hong Kong and the National Natural Science Foundation of China (Grant 10171030).     

Quadratic Convergence of a Primal-Dual Interior Point Method for Degenerate Nonlinear Optimization Problems

Recently studies of numerical methods for degenerate nonlinear optimization problems have been attracted much attention. Several authors have discussed convergence properties without the linear independence constraint qualification and/or the strict complementarity condition. In this paper, we are concerned with quadratic convergence property of a primal-dual interior point method, in which Newton’s method is applied to the barrier KKT conditions. We assume that the second order sufficient condition and the linear independence of gradients of equality constraints hold at the solution, and that there exists a solution that satisfies the strict complementarity condition, and that multiplier iterates generated by our method for inequality constraints are uniformly bounded, which relaxes the linear independence constraint qualification. Uniform boundedness of multiplier iterates is satisfied if the Mangasarian-Fromovitz constraint qualification is assumed, for example. By using the stability theorem by Hager and Gowda (1999), and Wright (2001), the distance from the current point to the solution set is related to the residual of the KKT conditions.By controlling a barrier parameter and adopting a suitable line search procedure, we prove the quadratic convergence of the proposed algorithm.     

A new hybrid method for nonlinear complementarity problems

In this paper, a new hybrid method is proposed for solving nonlinear complementarity problems (NCP) with P ₀ function. In the new method, we combine a smoothing nonmonotone trust region method based on a conic model and line search techniques. We reformulate the NCP as a system of semismooth equations using the Fischer-Burmeister function. Using Kanzow’s smooth approximation function to construct the smooth operator, we propose a smoothing nonmonotone trust region algorithm of a conic model for solving the NCP with P ₀ functions. This is different from the classical trust region methods, in that when a trial step is not accepted, the method does not resolve the trust region subproblem but generates an iterative point whose steplength is defined by a line search. We prove that every accumulation point of the sequence generated by the algorithm is a solution of the NCP. Under a nonsingularity condition, the superlinear convergence of the algorithm is established without a strict complementarity condition.     

A Smoothing Newton Method for General Nonlinear Complementarity Problems

Smoothing Newton methods for nonlinear complementarity problems NCP(F) often require F to be at least a P ₀-function in order to guarantee that the underlying Newton equation is solvable. Based on a special equation reformulation of NCP(F), we propose a new smoothing Newton method for general nonlinear complementarity problems. The introduction of Kanzow and Pieper's gradient step makes our algorithm to be globally convergent. Under certain conditions, our method achieves fast local convergence rate. Extensive numerical results are also reported for all complementarity problems in MCPLIB and GAMSLIB libraries with all available starting points.     

Equivalence of the Generalized Vertical Block Linear Complementarity Problems and Linear Complementarity Problems

In this paper, generalization of a vertical block linear complementarity problem associated with two different types of matrices, one of which is a square matrix and the other is a vertical block matrix, is proposed. The necessary and sufficient conditions for the existence of the solution of the generalized vertical block linear complementarity problem is derived and the relationship between the solution set of the generalized vertical block linear complementarity problem and the linear complementarity problem is established. It is proved that the generalized vertical block linear complementarity problem has the P-property if and only if the vertical block linear complementarity problem has the P-property.     

A TRULY GLOBALLY CONVERGENT FEASIBLE NEWTON-TYPE METHOD FOR MIXED COMPLEMENTARITY PROBLEMS

Typical solution methods for solving mixed complementarity problems either generatefeasible iterates but have to solve relatively complicated subproblems such as quadraticprograms or linear complementarity problems,or(those methods)have relatively simplesubproblems such as system of linear equations but possibly generate infeasible iterates.In this paper,we propose a new Newton-type method for solving monotone mixed com-plementarity problems,which ensures to generate feasible iterates,and only has to solve asystem of well-conditioned linear equations with reduced dimension per iteration.Withoutany regularity assumption,we prove that the whole sequence of iterates converges to a so-lution of the problem(truly globally convergent).Furthermore,under suitable conditions,the local superlinear rate of convergence is also established.     

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.     

On Minimizing and Stationary Sequences of a New Class of Merit Functions for Nonlinear Complementarity Problems

Motivated by the work of Fukushima and Pang (Ref. 1), we study the equivalent relationship between minimizing and stationary sequences of a new class of merit functions for nonlinear complementarity problems (NCP). These merit functions generalize that obtained via the squared Fischer–Burmeister NCP function, which was used in Ref. 1. We show that a stationary sequence {x^k} /Reⁿ is a minimizing sequence under the condition that the function value sequence {F(x ^k)} is bounded above or the Jacobian matrix sequence {F(x ^k)} is bounded, where F is the function involved in NCP. The latter condition is also assumed by Fukushima and Pang. The converse is true under the assumption of {F(x ^k)} bounded. As an example shows, even for a bounded function F, the boundedness of the sequence {F(x ^k)} is necessary for a minimizing sequence to be a stationary sequence.     

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     

Convergence Analysis of an Infeasible Interior Point Algorithm Based on a Regularized Central Path for Linear Complementarity Problems

Most existing interior-point methods for a linear complementarity problem (LCP) require the existence of a strictly feasible point to guarantee that the iterates are bounded. Based on a regularized central path, we present an infeasible interior-point algorithm for LCPs without requiring the strict feasibility condition. The iterates generated by the algorithm are bounded when the problem is a P _* LCP and has a solution. Moreover, when the problem is a monotone LCP and has a solution, we prove that the convergence rate is globally linear and it achieves `-feasibility and `-complementarity in at most O(n ² ln(1/`)) iterations with a properly chosen starting point.     

A General Approach to Convergence Properties of Some Methods for Nonsmooth Convex Optimization

Based on the notion of the ε -subgradient, we present a unified technique to establish convergence properties of several methods for nonsmooth convex minimization problems. Starting from the technical results, we obtain the global convergence of: (i) the variable metric proximal methods presented by Bonnans, Gilbert, Lemaréchal, and Sagastizábal, (ii) some algorithms proposed by Correa and Lemaréchal, and (iii) the proximal point algorithm given by Rockafellar. In particular, we prove that the Rockafellar—Todd phenomenon does not occur for each of the above mentioned methods. Moreover, we explore the convergence rate of {||x _k || } and {f(x _k ) } when {x _k } is unbounded and {f(x _k ) } is bounded for the non\-smooth minimization methods (i), (ii), and (iii). Accepted 15 October 1996