  The Frank-Wolfe algorithm, a very first optimization method and also known as
the conditional gradient method, was introduced by Frank and Wolfe in 1956. Due
to its simple linear subproblems, the Frank-Wolfe algorithm has recently been
received much attention for solving large-scale structured optimization
problems arising from many applied areas such as signal processing and machine
learning. In this paper we will discuss in detail the convergence analysis of
the Frank-Wolfe algorithm in Banach spaces. Two ways of the selections of the
stepsizes are discussed: the line minimization search method and the open loop
rule. In both cases, we prove the convergence of the Frank-Wolfe algorithm in
the case where the objective function $f$ has uniformly continuous (on bounded
sets) Fr\'echet derivative $f'$. We introduce the notion of the curvature
constant of order $\sigma\in (1,2]$ and obtain the rate
$O(\frac{1}{k^{\sigma-1}})$ of convergence of the Frank-Wolfe algorithm. In
particular, this rate reduces to $O(\frac{1}{k^{\nu}})$ if $f'$ is
$\nu$-H\"older continuous for $\nu\in (0,1]$, and to $O(\frac{1}{k})$ if $f'$
is Lipschitz continuous. A generalized Frank-Wolfe algorithm is also introduced
to address the problem of minimizing a composite objective function.
Convergence of iterates of both Frank-Wolfe and generalized Frank-Wolfe
algorithms are investigated.
