An inexact coordinate descent method for the weighted \(l_{1}\)-regularized convex optimization problem (Q2862474)

The authors present an inexact coordinate descent (ICD) method with another inexactness description for the subproblem solutions. The weighted \(l_1\)-regularized convex optimization problem NEWLINE\[NEWLINE\text{minimize } F(x): = g(Ax)+(b,x)+\sum^n_{i=1}\tau_i|x_i| \text{ subject to } l \leq x \leq u \tag{1}NEWLINE\]NEWLINE is considered. The authors present an IDC method with inexactness description for the subproblem solutions. The smooth convex problem is extended to that with the \(l_1\)-regularized function. On each iteration step the authors accept an inexact solution of the subproblem instead of the exact solution. The linear convergence rate is proved for the nonsmooth problem. The optimality conditions for problem (1) are derived. The authors also define \(\varepsilon\)-optimality conditions which are related to an inexact solution. A framework of the ICD method and some assumptions for the inexact solutions are presented. The global convergence and linear convergence rate are established. The authors report some numerical experiments for the proposed ICD method and show a comparison with the conjugate gradient method.