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"On the Shoulders of Giants" A brief excursion into the history of mathematical programming

Rainer Tichatschke — 2012

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Similar to many mathematical fields also the topic of mathematical programming has its origin in applied problems. But, in contrast to other branches of mathematics, we don't have to dig too deeply into the past centuries to find their roots. The historical tree of mathematical programming, starting from its conceptual roots to its present shape, is remarkably short, and to quote Isaak Newton, we can say: "We are standing on the shoulders of giants". The goal of...

Interior proximal method for variational inequalities on non-polyhedral sets

Alexander KaplanRainer Tichatschke — 2007

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Interior proximal methods for variational inequalities are, in fact, designed to handle problems on polyhedral convex sets or balls, only. Using a slightly modified concept of Bregman functions, we suggest an interior proximal method for solving variational inequalities (with maximal monotone operators) on convex, in general non-polyhedral sets, including in particular the case in which the set is described by a system of linear as well as strictly convex constraints. The convergence analysis of...

Note on the paper: interior proximal method for variational inequalities on non-polyhedral sets

Alexander KaplanRainer Tichatschke — 2010

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper we clarify that the interior proximal method developed in [6] (vol. 27 of this journal) for solving variational inequalities with monotone operators converges under essentially weaker conditions concerning the functions describing the "feasible" set as well as the operator of the variational inequality.

Prox-regularization and solution of ill-posed elliptic variational inequalities

Alexander KaplanRainer Tichatschke — 1997

Applications of Mathematics

In this paper new methods for solving elliptic variational inequalities with weakly coercive operators are considered. The use of the iterative prox-regularization coupled with a successive discretization of the variational inequality by means of a finite element method ensures well-posedness of the auxiliary problems and strong convergence of their approximate solutions to a solution of the original problem. In particular, regularization on the kernel of the differential operator and regularization...

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