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A continuous dynamical splitting method for solving ‘strongly+weakly’ convex programming problems

ABSTRACT The Douglas–Rachford splitting method is a classical and powerful method for minimizing the sum of two convex functions. In this paper, we introduce two dynamical systems based on this method for solving the minimum value problem of the sum of a strongly convex function and a weakly convex function. Under mild conditions, it is shown that the proposed dynamical systems are globally convergent to fixed point sets of the corresponding Douglas–Rachford operators, respectively, and are globally asymptotically stable if the corresponding fixed point sets are singleton. Furthermore, the globally exponential convergence of the proposed dynamical systems is established under some regularity conditions. A numerical example is reported to illustrate the effectiveness of the dynamical splitting method.