Convex Optimization on 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...

Some results on functionally convex sets in real Banach spaces

‎We use of two notions functionally convex (briefly‎, ‎F--convex) and functionally closed (briefly‎, ‎F--closed) in functional analysis and obtain more results‎. ‎We show that if $lbrace A_{alpha} rbrace _{alpha in I}$ is a family $F$--convex subsets with non empty intersection of a Banach space $X$‎, ‎then $bigcup_{alphain I}A_{alpha}$ is F--convex‎. ‎Moreover‎, ‎we introduce new definition o...

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

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...