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Many problems arising in image processing and signal recovery withmulti-regularization can be formulated as minimization of a sum of three convexseparable functions. Typically, the objective function involves a smoothfunction with Lipschitz continuous gradient, a linear composite nonsmoothfunction and a nonsmooth function. In this paper, we propose a primal-dualfixed-point (PDFP) scheme to solve the above class of problems. The proposedalgorithm for three block problems is a fully splitting symmetric scheme, onlyinvolving explicit gradient and linear operators without inner iteration, whenthe nonsmooth functions can be easily solved via their proximity operators,such as $\ell_1$ type regularization. We study the convergence of the proposedalgorithm and illustrate its efficiency through examples on fused LASSO andimage restoration with non-negative constraint and sparse regularization.

A primal-dual fixed-point algorithm for minimization of the sum of three convex separable functions