Recent Developments on Primal–Dual Splitting Methods with Applications to Convex Minimization

This chapter presents a survey on primal–dual splitting methods for solving monotone inclusion problems involving maximally monotone operators, linear compositions of parallel sums of maximally monotone operators, and single-valued Lipschitzian or cocoercive monotone operators. The primal–dual algorithms have the remarkable property that the operators involved are evaluated separately in each iteration, either by forward steps in the case of the single-valued ones or by backward steps for the set-valued ones, by using the corresponding resolvents. In the hypothesis that strong monotonicity assumptions for some of the involved operators are fulfilled, accelerated algorithmic schemes are presented and analyzed from the point of view of their convergence. Finally, we discuss the employment of the primal–dual methods in the context of solving convex optimization problems arising in the fields of image denoising and deblurring, support vector machine learning, location theory, portfolio optimization and clustering.