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We propose an interior penalty method to solve a nonlinear obstacle problem arising from the discretization of an infinite-dimensional optimization problem. An interior penalty equation is proposed to approximate the mixed nonlinear complementarity problem representing the Karush-Kuhn-Tucker conditions of the obstacle problem. We prove that the penalty equation is uniquely solvable and present a convergence analysis for the solution of the penalty equation when the problem is strictly convex. We also propose a Newton’s algorithm for solving the penalty equation. Numerical experiments are performed to demonstrate the convergence and usefulness of the method when it is used for the two non-trivial test problems.

In Andreani et al. (Numer. Algorithms 57:457–485, 2011), an interior point method for the horizontal nonlinear complementarity problem was introduced. This method was based on inexact Newton directions and safeguarding projected gradient iterations. Global convergence, in the sense that every cluster point is stationary, was proved in Andreani et al. (Numer. Algorithms 57:457–485, 2011). In Andreani et al. (Eur. J. Oper. Res. 249:41–54, 2016), local fast convergence was proved for the underdetermined problem in the case that the Newtonian directions are computed exactly. In the present paper, it will be proved that the method introduced in Andreani et al. (Numer. Algorithms 57:457–485, 2011) enjoys fast (linear, superlinear, or quadratic) convergence in the case of truly inexact Newton computations. Some numerical experiments will illustrate the accuracy of the convergence theory.

In this paper, we propose a new generalized penalized Fischer---Burmeister merit function, and show that the function possesses a system of favorite properties. Moreover, for the merit function, we establish the boundedness of level set under a weaker condition. We also propose a derivative-free algorithm for nonlinear complementarity problems with a nonmonotone line search. More specifically, we show that the proposed algorithm is globally convergent and has a locally linear convergence rate. Numerical comparisons are also made with the merit function used by Chen (J Comput Appl Math 232:455---471, 2009), which confirm the superior behaviour of the new merit function.

We prove a Kantorovich-type theorem on the existence and uniqueness of the solution of a generalized equation of the form $f(u)+g(u)\owns 0$ where f is a Frechet-differentiable function and g is a maximal monotone operator defined on a Hilbert space. The depth and scope of this theorem is such that when we specialize it to nonlinear operator equations, variational inequalities and nonlinear complementarity problems we obtain novel results for these problems as well. Our approach to the solution of a generalized equation is iterative, and the solution is obtained as the limit of the solutions of partially linearized generalized Newton subproblems of the type $Az+g(z)\owns b$ where A is a linear operator.

A new version of finite difference approximation of the generalized Jacobian for a finite max function is constructed. Numerical results are reported for the generalized Newton methods using this approximation.