The focus of this special issue is optimization methods for problems whose solutions have simple structures.These structures include sparsity (the solution has very few nonzero entries),low-rankness (the solution is a matrix of very low rank),consensus (the solution is a set of identical vectors),and beyond.The authors of the articles in this special issue use certain functions and constraints to ensure their solutions to have these structures.Such functions are typically nonsmooth,and such constraints involve all components of variables,thus posing a challenge to algorithm design.Classic algorithms using (sub)gradients and projections are either non-applicable or performing poorly.Therefore,the authors study new methods for better efficiency.