arXiv subjects

Christian Kanzow

Publications and source records attributed to Christian Kanzow.

24 records · Page 2Linked to original sources

arXiv2020

New Constraint Qualifications for Optimization Problems in Banach Spaces based on Asymptotic KKT Conditions

Optimization theory in Banach spaces suffers from the lack of available constraint qualifications. Despite the fact that there exist only a very few constraint qualifications, they are, in addition, often violated even in simple applications. This is very much in contrast to finite-dimensional nonlinear programs, where a large number of constraint qualifications is known. Since these constraint qualifications are usually defined using the set of active inequality constraints, it is difficult to extend them to the infinite-dimensional setting. One exception is a recently introduced sequential constraint qualification based on asymptotic KKT conditions. This paper shows that this so-called asymptotic KKT regularity allows suitable extensions to the Banach space setting in order to obtain new constraint qualifications. The relation of these new constraint qualifications to existing ones is discussed in detail. Their usefulness is also shown by several examples as well as an algorithmic application to the class of augmented Lagrangian methods.

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arXiv2018

Quasi-Variational Inequalities in Banach Spaces: Theory and Augmented Lagrangian Methods

This paper deals with quasi-variational inequality problems (QVIs) in a generic Banach space setting. We provide a theoretical framework for the analysis of such problems which is based on two key properties: the pseudomonotonicity (in the sense of Brezis) of the variational operator and a Mosco-type continuity of the feasible set mapping. We show that these assumptions can be used to establish the existence of solutions and their computability via suitable approximation techniques. In addition, we provide a practical and easily verifiable sufficient condition for the Mosco-type continuity property in terms of suitable constraint qualifications. Based on the theoretical framework, we construct an algorithm of augmented Lagrangian type which reduces the QVI to a sequence of standard variational inequalities. A full convergence analysis is provided which includes the existence of solutions of the subproblems as well as the attainment of feasibility and optimality. Applications and numerical results are included to demonstrate the practical viability of the method.

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arXiv2018

Relaxation schemes for mathematical programs with switching constraints

Switching-constrained optimization problems form a difficult class of mathematical programs since their feasible set is almost disconnected while standard constraint qualifications are likely to fail at several feasible points. That is why the application of standard methods from nonlinear programming does not seem to be promising in order to solve such problems. In this paper, we adapt the relaxation method from Kanzow and Schwartz (SIAM J. Optim., 23(2):770-798, 2013) for the numerical treatment of mathematical programs with complementarity constraints to the setting of switching-constrained optimization. It is shown that the proposed method computes M-stationary points under mild assumptions. Furthermore, we comment on other possible relaxation approaches which can be used to tackle mathematical programs with switching constraints. As it turns out, adapted versions of Scholtes' global relaxation scheme as well as the relaxation scheme of Steffensen and Ulbrich only find W-stationary points of switching-constrained optimization problems in general. Some computational experiments visualize the performance of the proposed relaxation method.

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arXiv2018

Augmented Lagrangian Methods for the Solution of Generalized Nash Equilibrium Problems

We propose an augmented Lagrangian-type algorithm for the solution of generalized Nash equilibrium problems (GNEPs). Specifically, we discuss the convergence properties with regard to both feasibility and optimality of limit points. This is done by introducing a secondary GNEP as a new optimality concept. In this context, special consideration is given to the role of suitable constraint qualifications that take into account the particular structure of GNEPs. Furthermore, we consider the behaviour of the method for jointly-convex GNEPs and describe a modification which is tailored towards the computation of variational equilibria. Numerical results are included to illustrate the practical performance of the overall method.

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arXiv2018

An Augmented Lagrangian Method for Optimization Problems in Banach Spaces

We propose a variant of the classical augmented Lagrangian method for constrained optimization problems in Banach spaces. Our theoretical framework does not require any convexity or second-order assumptions and allows the treatment of inequality constraints with infinite-dimensional image space. Moreover, we discuss the convergence properties of our algorithm with regard to feasibility, global optimality, and KKT conditions. Some numerical results are given to illustrate the practical viability of the method.

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arXiv2018

On Error Bounds and Multiplier Methods for Variational Problems in Banach Spaces

This paper deals with a general form of variational problems in Banach spaces which encompasses variational inequalities as well as minimization problems. We prove a characterization of local error bounds for the distance to the (primal-dual) solution set and give a sufficient condition for such an error bound to hold. In the second part of the paper, we consider an algorithm of augmented Lagrangian type for the solution of such variational problems. We give some global convergence properties of the method and then use the error bound theory to provide estimates for the rate of convergence and to deduce boundedness of the sequence of penalty parameters. Finally, numerical results for optimal control, Nash equilibrium problems, and elliptic parameter estimation problems are presented.

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