A modified SQP-filter method for nonlinear complementarity problem

The nonlinear complementarity problem can be reformulated as a nonlinear programming. For solving nonlinear programming, sequential quadratic programming (SQP) type method is very effective. Moreover, filter method, for its good numerical results, are extensively studied to handle nonlinear programming problems recently. In this paper, a modified quadratic subproblem is proposed. Based on it, we employ filter technique to tackle nonlinear complementarity problem. This method has no demand on initial point. The restoration phase, which is always used in traditional filter method, is not needed. Global convergence results of the proposed algorithm are established under suitable conditions. Some numerical results are reported in this paper.     

A projection-filter method for solving nonlinear complementarity problems

The Josephy-Newton method attacks nonlinear complementarity problems which consists of solving, possibly inexactly, a sequence of linear complementarity problems. Under appropriate regularity assumptions, this method is known to be locally (superlinearly) convergent. Utilizing the filter method, we presented a new globalization strategy for this Newton method applied to nonlinear complementarity problem without any merit function. The strategy is based on the projection-proximal point and filter methodology. Our linesearch procedure uses the regularized Newton direction to force global convergence by means of a projection step which reduces the distance to the solution of the problem. The resulting algorithm is globally convergent to a solution. Under natural assumptions, locally superlinear rate of convergence was established.     

A Superlinearly Convergent Implicit Smooth SQP Algorithm for Mathematical Programs with Nonlinear Complementarity Constraints

This paper discusses a special class of mathematical programs with nonlinear complementarity constraints, its goal is to present a globally and superlinearly convergent algorithm for the discussed problems. We first reformulate the complementarity constraints as a standard nonlinear equality and inequality constraints by making use of a class of generalized smoothing complementarity functions, then present a new SQP algorithm for the discussed problems. At each iteration, with the help of a pivoting operation, a master search direction is yielded by solving a quadratic program, and a correction search direction for avoiding the Maratos effect is generated by an explicit formula. Under suitable assumptions, without the strict complementarity on the upper-level inequality constraints, the proposed algorithm converges globally to a B-stationary point of the problems, and its convergence rate is superlinear.AMS Subject Classification: 90C, 49MThis work was supported by the National Natural Science Foundation (10261001) and the Guangxi Province Science Foundation (0236001, 0249003) of China.     

Unconstrained minimization approaches to nonlinear complementarity problems

The nonlinear complementarity problem can be reformulated as unconstrained minimization problems by introducing merit functions. Under some assumptions, the solution set of the nonlinear complementarity problem coincides with the set of local minima of the corresponding minimization problem. These results were presented by Mangasarian and Solodov, Yamashita and Fukushima, and Geiger and Kanzow. In this note, we generalize some results of Mangasarian and Solodov, Yamashita and Fukushima, and Geiger and Kanzow to the case where the considered function is only directionally differentiable. Some results are strengthened in the smooth case. For example, it is shown that the strong monotonicity condition can be replaced by the P-uniform property for ensuring a stationary point of the reformulated unconstrained minimization problems to be a solution of the nonlinear complementarity problem. We also present a descent algorithm for solving the nonlinear complementarity problem in the smooth case. Any accumulation point generated by this algorithm is proved to be a solution of the nonlinear complementarity under the monotonicity condition.     

求解特征值互补问题的一类ABS算法

ABS算法是20世纪80年代初,由Abaffy,Broyden和Spedicato完成的用于求解线性方程组的含有三个参量的投影算法,是一类有限次迭代直接法。目前,ABS算法不仅可以求解线性与非线性方程组,还可以求解线性规划和具有线性约束的非线性规划等问题。本文即是利用ABS算法求解特征值互补问题的一种尝试,构造了求解特征值互补问题的ABS算法,证明了求解特征值互补问题的ABS算法的收敛性。数值例子充分验证了求解特征值互补问题的ABS算法的有效性。     

An Interior Point Algorithm for Mixed Complementarity Nonlinear Problems

Nonlinear complementarity and mixed complementarity problems arise in mathematical models describing several applications in Engineering, Economics and different branches of physics. Previously, robust and efficient feasible directions interior point algorithm was presented for nonlinear complementarity problems. In this paper, it is extended to mixed nonlinear complementarity problems. At each iteration, the algorithm finds a feasible direction with respect to the region defined by the inequality conditions, which is also monotonic descent direction for the potential function. Then, an approximate line search along this direction is performed in order to define the next iteration. Global and asymptotic convergence for the algorithm is investigated. The proposed algorithm is tested on several benchmark problems. The results are in good agreement with the asymptotic analysis. Finally, the algorithm is applied to the elastic–plastic torsion problem encountered in the field of Solid Mechanics.     

A Finite Algorithm for Almost Linear Complementarity Problems

We present an algorithm for solving a class of nonlinear complementarity problems called the almost linear complementarity problem (ALCP), which can be used to simulate free boundary problems. The algorithm makes use of a procedure for identifying an active index subset of an ALCP by bounding its solution with an interval vector. It is shown that an acceptable solution of the given ALCP can be obtained by solving at most n systems of equations. Numerical results are reported to illustrate the efficiency of the algorithm for large-scale problems.     

求解非线性互补问题的一类光滑Broyden-like方法

非线性互补问题可以等价地转换为光滑方程组来求解. 基于一种新的非单调线搜索准则, 提出了求解非线性互补问题等价光滑方程组的一类新的非单调光滑 Broyden-like 算法.在适当的假设条件下, 证明了该算法的全局收敛性与局部超线性收敛性. 数值实验表明所提出的算法是有效的.     

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.     

QPCOMP: A quadratic programming based solver for mixed complementarity problems

QPCOMP is an extremely robust algorithm for solving mixed nonlinear complementarity problems that has fast local convergence behavior. Based in part on the NE/SQP method of Pang and Gabriel [14], this algorithm represents a significant advance in robustness at no cost in efficiency. In particular, the algorithm is shown to solve any solvable Lipschitz continuous, continuously differentiable, pseudo-monotone mixed nonlinear complementarity problem. QPCOMP also extends the NE/SQP method for the nonlinear complementarity problem to the more general mixed nonlinear complementarity problem. Computational results are provided, which demonstrate the effectiveness of the algorithm. This material is based on research supported by National Science Foundation Grant CCR-9157632, Department of Energy Grant DE-FG03-94ER61915, and the Air Force Office of Scientific Research Grant F49620-94-1-0036.     

A semismooth equation approach to the solution of nonlinear complementarity problems

In this paper we present a new algorithm for the solution of nonlinear complementarity problems. The algorithm is based on a semismooth equation reformulation of the complementarity problem. We exploit the recent extension of Newton's method to semismooth systems of equations and the fact that the natural merit function associated to the equation reformulation is continuously differentiable to develop an algorithm whose global and quadratic convergence properties can be established under very mild assumptions. Other interesting features of the new algorithm are an extreme simplicity along with a low computational burden per iteration. We include numerical tests which show the viability of the approach.     

Generalized mildly nonlinear complementarity problems for fuzzy mappings

In this paper, we introduce and study a new class of generalized mildly nonlinear complementarity problems for fuzzy mappings. We use the change of variables technique to establish the equivalence between the generalized mildly nonlinear complementarity problems and the Wiener-Hopf equations. This equivalence is used to suggest and analyze a number of iterative algorithm for solving the generalized mildly nonlinear complementarity problems.     

A nonsmooth inexact Newton method for the solution of large-scale nonlinear complementarity problems

A new algorithm for the solation of large-scale nonlinear complementarity problems is introduced. The algorithm is based on a nonsmooth equation reformulation of the complementarity problem and on an inexact Levenberg-Marquardt-type algorithm for its solution. Under mild assumptions, and requiring only the approximate solution of a linear system at each iteration, the algorithm is shown to be both globally and superlinearly convergent, even on degenerate problems. Numerical results for problems with up to 10 000 variables are presented. Partially supported by Agenzia Spaziale Italiana, Roma, Italy.     

Interior-point methods for nonlinear complementarity problems

We present a potential reduction interior-point algorithm for monotone nonlinear complementarity problems. At each iteration, one has to compute an approximate solution of a nonlinear system such that a certain accuracy requirement is satisfied. For problems satisfying a scaled Lipschitz condition, this requirement is satisfied by the approximate solution obtained by applying one Newton step to that nonlinear system. We discuss the global and local convergence rates of the algorithm, convergence toward a maximal complementarity solution, a criterion for switching from the interior-point algorithm to a pure Newton method, and the complexity of the resulting hybrid algorithm.This research was supported in part by NSF Grant DDM-89-22636.The authors would like to thank Rongqin Sheng and three anonymous referees for their comments leading to a better presentation of the results.     

Implicit complementarity problems on isotone projection cones

Isac and Németh [G. Isac and A. B. Németh, Projection methods, isotone projection cones and the complementarity problem, J. Math. Anal. Appl. 153 (1990), pp. 258–275] proved that solving a coincidence point equation (fixed point problem) in turn solves the corresponding implicit complementarity problem (nonlinear complementarity problem) and they exploited the isotonicity of the metric projection onto isotone projection cones to solve implicit complementarity problems (nonlinear complementarity problems) defined by these cones. In this article an iterative algorithm is studied in connection with an implicit complementarity problem. It is proved that if the sequence generated through the defined algorithm is convergent, then its limit is a solution of the coincidence point equation and thus solves the implicit complementarity problem. Sufficient conditions are given for this sequence to be convergent for implicit complementarity problems defined by isotone projection cones, extending the results of Németh [S.Z. Németh, Iterative methods for nonlinear complementarity problems on isotone projection cones, J. Math. Anal. Appl. 350 (2009), pp. 340–370]. Some existing concepts from the latter paper are extended to solve the problem of finding nonzero solutions of the implicit complementarity problem.     

Finding solutions of implicit complementarity problems by isotonicity of the metric projection

Isac and Németh [G. Isac and A. B. Németh, Projection method, isotone projection cones and the complementarity problem, J. Math. Anal. App., 153, 258-275(1990)] proved that solving a coincidence point equation (fixed point problem) in turn solves the corresponding implicit complementarity problem (nonlinear complementarity problem) and they exploited the isotonicity of the metric projection onto isotone projection cones to solve implicit complementarity problems (nonlinear complementarity problems) defined by these cones. In this paper, the notion of *-isotone projection cones is employed and an iterative algorithm is presented in connection with an implicit complementarity problem on *-isotone projection cones. It is proved that if the sequence generated through the defined algorithm is convergent, then its limit is a solution of the coincidence point equation and thus solves the implicit complementarity problem. Sufficient conditions are given for this sequence to be convergent for implicit complementarity problems defined by *-isotone projection cones. The question of finding nonzero solutions of these problems is also studied.     

非线性互补问题的组合同伦算法

针对拟P_*-映射和P(τ,α,β)-映射所对应的非线性互补问题,本文对其解的存在性及有效求解算法进行了研究.文中利用组合同伦方法给出了这两类非线性互补问题存在有界解的构造性证明,并利用预估校正方法对同伦路径进行跟踪,得到了互补问题的解.通过数值算例验证了该算法的有效性.     

Local path-following property of inexact interior methods in nonlinear programming

We study the local behavior of a primal-dual inexact interior point methods for solving nonlinear systems arising from the solution of nonlinear optimization problems or more generally from nonlinear complementarity problems. The algorithm is based on the Newton method applied to a sequence of perturbed systems that follows by perturbation of the complementarity equations of the original system. In case of an exact solution of the Newton system, it has been shown that the sequence of iterates is asymptotically tangent to the central path (Armand and Benoist in Math. Program. 115:199?C222, 2008). The purpose of the present paper is to extend this result to an inexact solution of the Newton system. We give quite general conditions on the different parameters of the algorithm, so that this asymptotic property is satisfied. Some numerical tests are reported to illustrate our theoretical results.     

An interior point potential reduction method for constrained equations

We study the problem of solving a constrained system of nonlinear equations by a combination of the classical damped Newton method for (unconstrained) smooth equations and the recent interior point potential reduction methods for linear programs, linear and nonlinear complementarity problems. In general, constrained equations provide a unified formulation for many mathematical programming problems, including complementarity problems of various kinds and the Karush-Kuhn-Tucker systems of variational inequalities and nonlinear programs. Combining ideas from the damped Newton and interior point methods, we present an iterative algorithm for solving a constrained system of equations and investigate its convergence properties. Specialization of the algorithm and its convergence analysis to complementarity problems of various kinds and the Karush-Kuhn-Tucker systems of variational inequalities are discussed in detail. We also report the computational results of the implementation of the algorithm for solving several classes of convex programs. The work of this author was based on research supported by the National Science Foundation under grants DDM-9104078 and CCR-9213739 and the Office of Naval Research under grant N00014-93-1-0228. The work of this author was based on research supported by the National Science Foundation under grant DMI-9496178 and the Office of Naval Research under grants N00014-93-1-0234 and N00014-94-1-0340.     

求解非线性P_0互补问题的非单调磨光算法

提出了一种新的磨光函数,在分析它与已有磨光函数不同特性的基础上,研究了将它用于求解非线性P_0互补问题时,其磨光路径的存在性和连续性,进而设计了求解一类非线性P_0互补问题的非单调磨光算法.在适当的假设条件下,证明了该算法的全局收敛性和局部超线性收敛性.数值算例验证了算法的有效性.