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

100%
Tichatschke R.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2012
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tom 32
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nr 1
5-44
EN
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 this paper is to describe briefly the historical growth of mathematical programming from its beginnings to the seventies of the last century and to review its basic ideas for a broad audience. During this process we will demonstrate that optimization is a natural way of thinking which follows some extremal principles.
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Note on the paper: interior proximal method for variational inequalities on non-polyhedral sets

63%
Kaplan A., Tichatschke R.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2010
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tom 30
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nr 1
51-59
EN
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.
3
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Interior proximal method for variational inequalities on non-polyhedral sets

63%
Kaplan A., Tichatschke R.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2007
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tom 27
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nr 1
71-93
EN
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 the method studied admits the use of the 𝝐-enlargement of the operator and an inexact solution of the subproblems.
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