A Modified Primal-Dual Algorithm for Structured Convex Optimization with a Lipschitzian Term

This paper focuses on solving structured convex optimization problems that consist of a smooth term with a Lipschitzian gradient, and two nonsmooth terms. One of the nonsmooth functions is composed with a linear operator, and both nonsmooth functions are proximal-friendly. To solve these problems, we propose a modified primal-dual algorithm, denoted by MPD3O, that combines ideas from the primal-dual splitting algorithm PD3O and the balanced augmented Lagrangian method. MPD3O does not require knowledge of the spectral norm of the linear operator and solves the primal and dual problems simultaneously without iteratively solving subproblems or linear system of equations. We establish global pointwise convergence using the Krasnoselskii–Mann theorem, as well as ergodic and nonergodic \(\mathcal {O}(1/k)\) $\mathcal{O}(1/k)$ sublinear convergence rate results, where k denotes the iteration number. Finally, we conduct numerical experiments on the constrained LASSO problem to demonstrate the efficiency of the proposed algorithm.