Solving systems of monotone inclusions via primal-dual splitting techniques (Q395636)

As the authors announce in the abstract, they propose a primal-dual splitting algorithm for solving the following system of coupled inclusions in Hilbert spaces: NEWLINE\[NEWLINE0\in L_i^*(A_i\square C_i)(L_i\bar x_i)+B_i(\bar x_1,\dots,\bar x_m), \;i=1,\dots,m,NEWLINE\]NEWLINE together with its dual system NEWLINE\[NEWLINE0\in L_i^*\bar v_i+B_i(x_1,\dots,x_m), \;i=1,\dots,m,NEWLINE\]NEWLINE where \(A_i\square C_i=(A_i^{-1}+C_i^{-1})^{-1}\), \(A_i:G_i\to 2^{G_i}\) is maximal monotone, \(C_i:G_i\to 2^{G_i}\) is monotone with a Lipschitz continuous inverse, \(L_i:H_i\to G_i\) is linear continuous, and \(B_i\) is a Lipschitz continuous map satisfying NEWLINE\[NEWLINE\sum_{i=1}^m \langle x_i-y_i|B_i(x_1,\dots,x_m)-B_i(y_1,\dots,y_m)\rangle \geq 0.NEWLINE\]NEWLINE The main result of the paper is applied to convex minimization problems. Two numerical experiments are given. The first one concerns the problem of average consensus on colored networks, the second one is a problem of classifying images via support vector machines.