## MOCCA: Mirrored Convex/Concave Optimization for Nonconvex Composite Functions

*Rina Foygel Barber, Emil Y. Sidky*; 17(144):1−51, 2016.

### Abstract

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 a 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.

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