Co-NP-completeness of some matrix classification problems

The classes of P-, P ₀-, R ₀-, semimonotone, strictly semimonotone, column sufficient, and nondegenerate matrices play important roles in studying solution properties of equations and complementarity problems and convergence/complexity analysis of methods for solving these problems. It is known that the problem of deciding whether a square matrix with integer/rational entries is a P- (or nondegenerate) matrix is co-NP-complete. We show, through a unified analysis, that analogous decision problems for the other matrix classes are also co-NP-complete. Received: April 1999 / Accepted: March 1, 2000?Published online May 12, 2000     

Smooth methods of multipliers for complementarity problems

This paper describes several methods for solving nonlinear complementarity problems. A general duality framework for pairs of monotone operators is developed and then applied to the monotone complementarity problem, obtaining primal, dual, and primal-dual formulations. We derive Bregman-function-based generalized proximal algorithms for each of these formulations, generating three classes of complementarity algorithms. The primal class is well-known. The dual class is new and constitutes a general collection of methods of multipliers, or augmented Lagrangian methods, for complementarity problems. In a special case, it corresponds to a class of variational inequality algorithms proposed by Gabay. By appropriate choice of Bregman function, the augmented Lagrangian subproblem in these methods can be made continuously differentiable. The primal-dual class of methods is entirely new and combines the best theoretical features of the primal and dual methods. Some preliminary computation shows that this class of algorithms is effective at solving many of the standard complementarity test problems. Received February 21, 1997 / Revised version received December 11, 1998? Published online May 12, 1999     

Feasible descent algorithms for mixed complementarity problems

In this paper we consider a general algorithmic framework for solving nonlinear mixed complementarity problems. The main features of this framework are: (a) it is well-defined for an arbitrary mixed complementarity problem, (b) it generates only feasible iterates, (c) it has a strong global convergence theory, and (d) it is locally fast convergent under standard regularity assumptions. This framework is applied to the PATH solver in order to show viability of the approach. Numerical results for an appropriate modification of the PATH solver indicate that this framework leads to substantial computational improvements. Received April 9, 1998 / Revised version received November 23, 1998?Published online March 16, 1999     

Some Optimization Reformulations of the Extended Linear Complementarity Problem

We consider the extended linear complementarity problem (XLCP) introduced by Mangasarian and Pang [22], of which the horizontal and vertical linear complementarity problems are two special cases. We give some new sufficient conditions for every stationary point of the natural bilinear program associated with XLCP to be a solution of XLCP. We further propose some unconstrained and bound constrained reformulations for XLCP, and study the properties of their stationary points under assumptions similar to those for the bilinear program.     

The Generic Stability and Existence of Essentially Connected Components of Solutions for Nonlinear Complementarity Problems

The aim of this paper is to develop the general generic stability theory for nonlinear complementarity problems in the setting of infinite dimensional Banach spaces. We first show that each nonlinear complementarity problem can be approximated arbitrarily by a nonlinear complementarity problem which is stable in the sense that the small change of the objective function results in the small change of its solution set; and thus we say that almost all complementarity problems are stable from viewpoint of Baire category. Secondly, we show that each nonlinear complementarity problem has, at least, one connected component of its solutions which is stable, though in general its solution set may not have good behaviour (i.e., not stable). Our results show that if a complementarity problem has only one connected solution set, it is then always stable without the assumption that the functions are either Lipschitz or differentiable.     

Smoothing Newton and Quasi-Newton Methods for Mixed Complementarity Problems

The mixed complementarity problem can be reformulated as a nonsmooth equation by using the median operator. In this paper, we first study some useful properties of this reformulation and then derive the Chen-Harker-Kanzow-Smale smoothing function for the mixed complementarity problem. On the basis of this smoothing function, we present a smoothing Newton method for solving the mixed complementarity problem. Under suitable conditions, the method exhibits global and quadratic convergence properties. We also present a smoothing Broyden-like method based on the same smoothing function. Under appropriate conditions, the method converges globally and superlinearly.     

On the finite displacement problem of a hollow sphere under internal and external pressures

Taking advantage of successive approximations the present boundary-value problemis solved.We find the first-order and second-order solutions,and there fore we obtain theformulae in the second approximatior for the displacement,strain,and stress fields.     

A PENALTY TECHNIQUE FOR NONLINEAR COMPLEMENTARITY PROBLEMS

1.IntroductionConsiderthefollowingnonlinearcomplementarityproblemsNCP(F)offindinganxER",suchthatwhereFisamappingfromR"intoitself.ItisanimportantformofthefollowingvariationalinequalityVI(F,X)offindinganxEX,suchthatwhereXCReisaclosedconvexset.WhenX=R7,(1.1)…     

On estimation of the optimal parameter of the modulus-based matrix splitting algorithm for linear complementarity problems on second-order cones

There are many studies on the well-known modulus-based matrix splitting (MMS) algorithm for solving complementarity problems, but very few studies on its optimal parameter, which is of theoretical and practical importance. Therefore and here, by introducing a novel mapping to explicitly cast the implicit fixed point equation and thus obtain the iteration matrix involved, we first present the estimation approach of the optimal parameter of each step of the MMS algorithm for solving linear complementarity problems on the direct product of second-order cones (SOCLCPs). It also works on single second-order cone and the non-negative orthant. On this basis, we further propose an iteration-independent optimal parameter selection strategy for practical usage. Finally, the practicability and effectiveness of the new proposal are verified by comparing with the experimental optimal parameter and the diagonal part of system matrix. In addition, with the optimal parameter, the effectiveness of the MMS algorithm can indeed be greatly improved, even better than the state-of-the-art solvers SCS and SuperSCS that solve the equivalent SOC programming.