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Interior point solutions of variational problems and global inverse function theorems
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.ORCID iD: 0000-0002-2681-8383
2007 (English)In: International Journal of Robust and Nonlinear Control, ISSN 1049-8923, E-ISSN 1099-1239, Vol. 17, no 5-6, 463-481 p.Article in journal (Refereed) Published
Abstract [en]

Variational problems and the solvability of certain nonlinear equations have a long and rich history beginning with calculus and extending through the calculus of variations. In this paper, we are interested in 'well-connected' pairs of such problems which are not necessarily related by critical point considerations. We also study constrained problems of the kind which arise in mathematical programming. We are also interested in interior minimizing points for the variational problem and in the well-posedness (in the sense of Hadamard) of solvability of the related systems of equations. We first prove a general result which implies the existence of interior points and which also leads to the development of certain generalization of the Hadamard-type global inverse function theorem, along the theme that uniqueness quite often implies existence. This result is illustrated by proving the non-existence of shock waves for certain initial data for the vector Burgers' equation. The global inverse function theorem is also illustrated by a derivation of the existence of positive definite solutions of matrix Riccati equations without first analysing the nonlinear matrix Riccati differential equation. The main results on the existence of solutions to geometrically constrained well-connected pairs are then presented and illustrated by a geometric analysis of the existence of interior points for linear programming problems.

Place, publisher, year, edition, pages
2007. Vol. 17, no 5-6, 463-481 p.
Keyword [en]
variational problems, well-posedness of solvability of nonlinear, equations, global inverse function theorems, interior point methods, Burgers' equation, algebraic Riccati equations, constraint
URN: urn:nbn:se:kth:diva-16480DOI: 10.1002/rnc.1138ISI: 000245145700009ScopusID: 2-s2.0-33947432103OAI: diva2:334522
QC 20100525Available from: 2010-08-05 Created: 2010-08-05Bibliographically approved

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