Browsing by Author "Ulbrich, Michael"

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    A Globally Convergent Primal-Dual Interior-Point Filter Method for Nonconvex Nonlinear Programming
    (2000-04) Ulbrich, Michael; Ulbrich, Stefan; Vicente, Luis N.
    In this paper, the filter technique of Fletcher and Leyffer (1997) is used to globalize the primal-dual interior-point algorithm for nonlinear programming, avoiding the use of merit functions and the updating of penalty parameters. The new algorithm decomposes the primal-dual step obtained from the perturbed first-order necessary conditions into a normal and a tangential step, whose sizes are controlled by a trust-region type parameter. Each entry in the filter is a pair of coordinates: one resulting from feasibility and centrality, and associated with the normal step; the other resulting from optimality (complementarity and duality), and related with the tangential step. Global convergence to first-order critical points is proved for the new primal-dual interior-point filter algorithm.
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    Semismooth Newton Methods for Operator Equations in Function Spaces
    (2000-04) Ulbrich, Michael
    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 alpha-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 alpha-order semismoothness, convergence of q-order 1+alpha is proved. We also establish the semismoothness of composite operators and develop corresponding chain rules. The developed theory is accompanied by illustrating examples and by applications to nonlinear complementarity problems.