We develop a semismoothness concept for nonsmooth superposition operators in function spaces. The considered class of operators includes NCP-function-based reformulations of infinite-dimensional nonlinear complementarity problems, and thus covers a very comprehensive class of applications. Our results generalize semismoothness and ff-order semismoothness from finite-dimensional spaces to a Banach space setting. Hereby, a new generalized differential is used that can be seen as an extension of Qi's finite-dimensional C-subdifferential to our infinite-dimensional framework. We apply these semismoothness results to develop a Newton-like method for nonsmooth operator equations and prove its local q-superlinear convergence to regular solutions. If the underlying operator is ff-order semismoothness, convergence of q-order 1 + ff is proved. We also establish the semismoothness of composite operators and develop corresponding chain rules. The developed theory is accompanied by illustrating e...

We introduce a new algorithm for the solution of the mixed complementarity problem (MCP) which has stronger properties than most existing methods. In fact, typical solution methods for the MCP either generate feasible iterates but have to solve relatively complicated subproblems (like quadratic programs or linear complementarity problems), or they have relatively simple subproblems (like linear systems of equations) but generate not necessarily feasible iterates. The method to be presented here combines the nice features of these two classes of methods: It has to solve only one linear system of equations (of reduced dimension) at each iteration, and it generates feasible (more precisely: strictly feasible) iterates. The new method has some nice global and local convergence properties. Some preliminary numerical results will also be given.

We propose and analyze a class of penalty-function-free nonmonotone trust-region methods for nonlinear equality constrained optimization problems. The algorithmic framework yields global convergence without using a merit function and allows nonmonotonicity independently for both, the constraint violation and the value of the Lagrangian function. Similar to the Byrd--Omojokun class of algorithms, each step is composed of a quasinormal and a tangential step. Both steps are required to satisfy a decrease condition for their respective trust-region subproblems. The proposed mechanism for accepting steps combines nonmonotone decrease conditions on the constraint violation and/or the Lagrangian function, which leads to a flexibility and acceptance behavior comparable to filter-based methods. We establish the global convergence of the method. Furthermore, transition to quadratic local convergence is proved. Numerical tests are presented that confirm the robustness and efficiency of the approach.

We propose and analyze a semismooth Newton-type method for the solution of a pointwise constrained optimal control problem governed by the time-dependent incompressible Navier-Stokes equations. The method is based on a reformulation of the optimality system as an equivalent nonsmooth operator equation. We analyze the flow control problem and establish q-superlinear convergence of the method. In the numerical implementation, adjoint techniques are combined with a truncated conjugate gradient method. Numerical results are presented that support our theoretical results and confirm the viability of the approach.

. We consider the problem of finding a solution of a nonlinear system of equations subject to some box constraints. To this end, we introduce a new active set-type Newton method with global and local fast convergence properties. The method generates feasible iterates and has to solve only one linear system of equations at each iteration. Due to our active set strategy, this linear system is of reduced dimension. Key Words. Nonlinear equations, box constraints, Newton's method, active set strategy, projected gradient, global convergence, quadratic convergence. 1 The research of this author was partially supported by the DFG (Deutsche Forschungsgemeinschaft). 1 Introduction The problem we address in this paper is to find a solution of the constrained nonlinear system F (x) = 0; x 2 [l; u]; (1) where F : [l; u] ! IR n is a given function which is assumed to be continuously differentiable in an open set containing the box [l; u], and where l = (l 1 ; : : : ; l n ) T ; u = (u 1 ; ...

: A popular approach to solve the Karush-Kuhn-Tucker (KKT) system, mainly arising from the variational inequality problem and the constrained optimization problem, is to reformulate it as a constrained minimization problem with simple bounds. In this paper, we propose a trust region method for solving the reformulation problem with the trust region subproblem being solved by the truncated conjugate gradient (CG) method, which is cost eective. Other advantages of the proposed method over the existing ones include that a good approximation solution to the trust region subproblem can be found by the truncated CG method and is judged in a simple way; and that the working matrix in each iteration is H, instead of the condensed H T H, where H is a matrix element of the generalized Jacobian of the function used in the reformulation. As a matter of fact, the matrix used is of reduced dimension. Extra attention is taken to ensure the success of the truncated CG method as well as the feasibil...

Many applications in mathematical modeling and optimal control lead to problems that are posed in function spaces and contain pointwise complementarity conditions. In this paper, a projected Newton method for nonlinear complementarity problems in the innite dimensional function space L p is proposed and analyzed. Hereby, an NCP-function is used to reformulate the problem as a nonsmooth operator equation. The method stays feasible with respect to prescribed bound-constraints. The convergence analysis is based on semismoothness results for superposition operators in function spaces. The proposed algorithm is shown to converge locally q-superlinearly to a regular solution. As an important tool for applications, we establish a sufficient condition for regularity. The application of the algorithm to the distributed bound-constrained control of an elliptic partial dierential equation is discussed in detail. Numerical results confirm the efficiency of the method.

We consider the problem of finding a solution of a nonlinear system of equations subject to some box constraints. To this end, we introduce a new active set-type Newton method with global and local fast convergence properties. The method generates feasible iterates and has to solve only one linear system of equations at each iteration. Due to our active set strategy, this linear system is of reduced dimension.