Generalized Complementarity Problem with Three Classes of Generalized Variational Inequalities Involving ⊕ Operation

In this study, we introduce and study a generalized complementarity problem involving XOR operation and three classes of generalized variational inequalities involving XOR operation. Under certain appropriate conditions, we establish equivalence between them. An iterative algorithm is defined for solving one of the three generalized variational inequalities involving XOR operation. Finally, an existence and convergence result is proved, supported by an example.

Further Application ofH-Differentiability to Generalized Complementarity Problems Based on Generalized Fisher-Burmeister Functions

We study nonsmooth generalized complementarity problems based on the generalized Fisher-Burmeister function and its generalizations, denoted by GCP(f,g) wherefandgareH-differentiable. We describeH-differentials of some GCP functions based on the generalized Fisher-Burmeister function and its generalizations, and their merit functions. Under appropriate conditions on theH-differentials offandg, we show that a local/global minimum of a merit function (or a “stationary point” of a merit function) is coincident with the solution of the given generalized complementarity problem. When specializing GCP(f,g)to the nonlinear complementarity problems, our results not only give new results but also extend/unify various similar results proved forC1, semismooth, and locally Lipschitzian.

A Superlinearly Convergent Method for the Generalized Complementarity Problem over a Polyhedral Cone

Making use of a smoothing NCP-function, we formulate the generalized complementarity problem (GCP) over a polyhedral cone as an equivalent system of equations. Then we present a Newton-type method for the equivalent system to obtain a solution of the GCP. Our method solves only one linear system of equations and performs only one line search at each iteration. Under mild assumptions, we show that our method is both globally and superlinearly convergent. Compared to the previous literatures, our method has stronger convergence results under weaker conditions.