On perturbed steepest descent methods with inexact line search for bilevel convex optimization

We use a general framework for solving convex constrained optimization problems introduced in an earlier work to obtain algorithms for problems with a constraint set defined as the set of minimizers of a given function. Also, the algorithms allow the objective function to be decomposed as a sum of other convex functions that can be treated separately. We prove that the general algorithm converges to the optimum of the objective function over the set of minima of a convex Lipschitz-differentiable function chosen previously. When using orthogonal projections onto the convex constraints, we retrieve a Cimmino-like algorithm that converges to the optimum over the set of weighted least squares solutions. Furthermore, we show an important application of our approach to compressed sensing and inverse problems. (AU)