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Biorthogonal greedy algorithms in convex optimization
The study of greedy approximation in the context of convex optimization is
becoming a promising research direction as greedy algorithms are actively being
employed to construct sparse minimizers for convex functions with respect to
given sets of elements. In this paper we propose a unified way of analyzing a
certain kind of greedy-type algorithms for the minimization of convex functions
on Banach spaces. Specifically, we define the class of Weak Biorthogonal Greedy
Algorithms for convex optimization that contains a wide range of greedy
algorithms. We analyze the introduced class of algorithms and establish the
properties of convergence, rate of convergence, and numerical stability, which
is understood in the sense that the steps of the algorithm are allowed to be
performed not precisely but with controlled computational inaccuracies. We show
that the following well-known algorithms for convex optimization --- the Weak
Chebyshev Greedy Algorithm (co) and the Weak Greedy Algorithm with Free
Relaxation (co) --- belong to this class, and introduce a new algorithm --- the
Rescaled Weak Relaxed Greedy Algorithm (co). Presented numerical experiments
demonstrate the practical performance of the aforementioned greedy algorithms
in the setting of convex minimization as compared to optimization with
regularization, which is the conventional approach of constructing sparse
minimizers