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In this note, we propose a FISTA-type first order algorithm, VAR-FISTA, tosolve a composite optimization problem. A distinctive feature of VAR-FISTA isits ability to exploit the convexity of the function in the problem, resultingin an improved iteration complexity when the function is convex compared towhen it is nonconvex. The iteration complexity result for the convex andnonconvex case obtained in the note are compatible to the best known in theliterature so far.

A FISTA-Type First Order Algorithm on Composite Optimization Problems that is Adaptable to the Convex Situation