Convex optimization on Banach Spaces
This work addresses optimization problems in infinite-dimensional spaces for researchers in mathematical optimization, but it appears incremental as it extends existing greedy methods to more general settings.
The paper tackles convex optimization in Banach spaces using greedy algorithms based on function evaluations, including approximate evaluations, and provides a priori upper bounds for convergence rates that depend on function smoothness and sparsity of the solution.
Greedy algorithms which use only function evaluations are applied to convex optimization in a general Banach space $X$. Along with algorithms that use exact evaluations, algorithms with approximate evaluations are treated. A priori upper bounds for the convergence rate of the proposed algorithms are given. These bounds depend on the smoothness of the objective function and the sparsity or compressibility (with respect to a given dictionary) of a point in $X$ where the minimum is attained.