A primal-dual splitting algorithm for composite monotone inclusions with minimal lifting

In this work, we study resolvent splitting algorithms for solving composite monotone inclusion problems. The objective of these general problems is finding a zero in the sum of maximally monotone operators composed with linear operators. Our main contribution is establishing the first primal-dual splitting algorithm for composite monotone inclusions with minimal lifting. Specifically, the proposed scheme reduces the dimension of the product space where the underlying fixed point operator is defined, in comparison to other algorithms, without requiring additional evaluations of the resolvent operators. We prove the convergence of this new algorithm and analyze its performance in a problem arising in image deblurring and denoising. This work also contributes to the theory of resolvent splitting algorithms by extending the minimal lifting theorem recently proved by Malitsky and Tam to schemes with resolvent parameters.FJAA and DTB were partially supported by the Ministry of Science, Innovation and Universities of Spain and the European Regional Development Fund (ERDF) of the European Commission, Grant PGC2018-097960-B-C22. FJAA was partially supported by the Generalitat Valenciana (AICO/2021/165). RIB was partially supported by FWF (Austrian Science Fund), project P 34922-N. DTB was supported by MINECO and European Social Fund (PRE2019-090751) under the program “Ayudas para contratos predoctorales para la formación de doctores” 2019