Helmut Gfrerer

A posteriori parameter choice for general regularization methods for solving linear ill-posed problems

Abstract For continuous and iterative regularization methods for solving linear ill-posed problems, we propose an a posteriori parameter choice strategy and stopping rule, respectively, that always leads to optimal convergence rates and does not require the knowledge of the smoothness of the exact solution.

First Order and Second Order Characterizations of Metric Subregularity and Calmness of Constraint Set Mappings

A condition ensuring metric subregularity (respectively, calmness) of general multifunctions between Banach spaces is derived. This condition is expressed solely in terms of the given data at the reference point and does not involve any information concerning the solution set of the corresponding inclusion given by the multifunction. In finite dimensions this condition can be expressed in terms of a derivative which appears to be a combination of the coderivative and the contingent derivative. It is further shown that this sufficient condition is in some sense the weakest possible first order condition sufficient for subregularity. We extend this condition under the additional assumption that one part of the multifunction is known to be subregular in advance. We also derive second order conditions for metric subregularity, both sufficient and necessary, for multifunctions associated with constraint systems as they occur in optimization. We show that the main difference between the necessary and sufficient conditions is the replacement of an inequality by a strict inequality, just as in the case of “no gap” second order optimality conditions in optimization.

Hierarchical model for production planning in the case of uncertain demand

The paper deals with hierarchical production planning for a multiperiod model consisting of an aggregate planning level and a detailed planning level in the case of uncertain demand. At the beginning the detailed demand for the actual planning period is known, whereas for future planning periods only upper and lower bounds of detailed demand are known. On the other hand, the aggregate demand on the product group level is supposed to be deterministic at each planning period over the whole planning horizon. For the aggregate planning level, robust production plans (i.e., aggregate production plans which can be disaggregated to a feasible detailed plan for every possible detailed demand) are proposed and a consistent disaggregation is executed for the detailed planning level. Two sets (static and dynamic) of sufficient conditions for robustness of the aggregate plan as well as dynamic disaggregation conditions are derived. These sufficient conditions are a generalization of the results of Lasserre and Merce. The performance of the sufficient conditions are demonstrated by a computational study which shows the potential advantage of the approach.

A globally convergent algorithm based on imbedding and parametric optimization

AbstractThe continuation method, well-established for the solution of nonlinear equations is extended to restricted optimization problems. Only the locally active restrictions are considered along the homotopy path. It is assumed that there are only finitely many critical points, i. e. that there are only finitely many changes of the index set of active restrictions.The globally convergent algorithm which we present proceeds in three stages:1.Within each stability region, the solution is computed by the classical continuation method.2.On the boundary of a stability region, a critical point

On Computation of Generalized Derivatives of the Normal-Cone Mapping and Their Applications

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Complete Characterizations of Tilt Stability in Nonlinear Programming under Weakest Qualification Conditions

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Globally convergent decomposition methods for nonconvex optimization problems

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