ИСТИНА

We discuss the mixed complementarity problem (MCP), that is, the variational inequality on a generalized box. Based on the identification of indices active at a solution of MCP, we propose a class of Newton-type methods for which local superlinear convergence holds under extremely mild assumptions. In particular, the error bound condition needed for the identification procedure and the nondegeneracy condition needed for the convergence of the resulting Newton method are individually and collectively strictly weaker than the property of semistability of a solution. Thus the superlinear local convergence conditions of the presented method are weaker than conditions required for the semismooth (generalized) Newton methods applied to MCP reformulations through complementarity functions. Moreover, they are also weaker than convergence conditions of the linearization (Josephy--Newton) method. The relations of different regularity conditions are studied. For the special case of optimality systems with primal-dual structure, we further consider the question of superlinear convergence of primal variables. We illustrate our theoretical results with numerical experiments on some specially constructed MCPs whose solutions do not satisfy the usual regularity assumptions. We also discuss a globalization scheme for a class of active-set Newton methods for solving MCP. We describe the basic hybrid globalization scheme and its convergence properties. Numerical experiments on some test problems are presented, including results on the MCPLIB collection.