Abstract

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.