A Proximal Minimization Algorithm for Structured Nonconvex and Nonsmooth Problems
Radu Ioan Boţ, Ernö Robert Csetnek, Dang‐Khoa Nguyen · SIAM Journal on Optimization · 2019
We propose a proximal algorithm for minimizing objective functions consisting of three summands: the composition of a nonsmooth function with a linear operator, another nonsmooth function (with each of the nonsmooth summands depending on an independent block variable), and a smooth function which couples the two block variables. The algorithm is a full splitting method, which means that the nonsmooth functions are processed via their proximal operators, the smooth function via gradient steps, and the linear operator via matrix times vector multiplication. We provide sufficient conditions for the boundedness of the generated sequence and prove that any cluster point of the latter is a KKT point of the minimization problem. In the setting of the Kurdyka--Łojasiewicz property, we show global convergence and derive convergence rates for the iterates in terms of the Łojasiewicz exponent.