Convex optimization in sums of Banach spaces

Convex Optimization on Banach Spaces

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 compre...

Adaptive Optimization of Convex Functionals in Banach Spaces

This paper is concerned with optimization or minimization problems that are governed by operator equations, such as partial differential or integral equations, and thus are naturally formulated in an infinite dimensional function space V . We first construct a prototype algorithm of steepest descent type in V and prove its convergence. By using a Riesz basis in V we can transform the minimizati...

Descent methods for convex optimization problems in Banach spaces

where f : E→ R is a convex function; see, for example, [1, 2, 8] and the references therein. It is well known that standard iterative methods for solving (1.2), which are designed for finite-dimensional optimization problems, cannot guarantee strong convergence of their iteration sequences to a solution of the initial problem if the cost function does not possess strengthened convexity properti...

Adaptive Convex Optimization in Banach Spaces: a Multilevel Approach

Convex Games in Banach Spaces

We study the regret of an online learner playing a multi-round game in a Banach space B against an adversary that plays a convex function at each round. We characterize the minimax regret when the adversary plays linear functions in terms of the Rademacher type of the dual of B. The cases when the adversary plays bounded and uniformly convex functions respectively are also considered. Our resul...