  Many optimization problems arising in high-dimensional statistics decompose
naturally into a sum of several terms, where the individual terms are
relatively simple but the composite objective function can only be optimized
with iterative algorithms. In this paper, we are interested in optimization
problems of the form F(Kx) + G(x), where K is a fixed linear transformation,
while F and G are functions that may be nonconvex and/or nondifferentiable. In
particular, if either of the terms are nonconvex, existing alternating
minimization techniques may fail to converge; other types of existing
approaches may instead be unable to handle nondifferentiability. We propose the
MOCCA (mirrored convex/concave) algorithm, a primal/dual optimization approach
that takes local convex approximation to each term at every iteration. Inspired
by optimization problems arising in computed tomography (CT) imaging, this
algorithm can handle a range of nonconvex composite optimization problems, and
offers theoretical guarantees for convergence when the overall problem is
approximately convex (that is, any concavity in one term is balanced out by
convexity in the other term). Empirical results show fast convergence for
several structured signal recovery problems.
