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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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Ill-posed quadratic optimization in Banach spaces

Faker Ben Belgacem1

1Université de Technologie de Compiègne, BP 20529, 60205 Compiegne Cedex, France, LAMSIN, Ecole Nationale d'Ingénieurs de Tunis, B.P. 37, 1002 Le Belvédère, Tunisia. E-mail: ,

Citation Information: Journal of Inverse and Ill-posed Problems. Volume 18, Issue 3, Pages 263–279, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip.2010.010, July 2010

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Quadratic optimization in Banach spaces is explored in the context of ill-posedness. We elucidate the conditions under which the cost function is bounded from below. Issues related to the ill-posedness in Hilbert scales have been comprehensively studied in a constructive way in Inverse Probl. 24 (2008), 055002. Things are entirely different in Banach spaces where the spectral theory does not work and calculating fractional powers of the quadratic operator we are involved in does not make sense anymore. Specific functional analysis tools such as the interpolation of Banach spaces are required. A by-product of the results we state is the possibility to handle numerically the quadratic optimization problem as if it were a least-squares problem, for a class of data that is described accurately. Needless to recall the important impact of such results on the regularization procedures, necessary for a safe computational treatment of the problem.

Keywords.: Ill-posed quadratic optimization; interpolation of Banach spaces; optimal control and inverse problems

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