Splitting methods for separable convex programming

Many applications arising in various fields can be modeled into such a separable convex programming problem whose objective function is formed as a sum of m individual functions without crossed variables.

The special case with m=2 has been well studied in the literature, while the research for the general case with m>=3 is in complete infancy. The classical augmented Lagrangian method (ALM) is obviously applicable for this separable convex programming problem. But, the direct application of ALM treats such a well-structured model as a generic convex programming without any consideration of the particular separable structure, and thus ignores completely the nice separable structure which might be very beneficial for algorithmic design. Realizing the fact that each function component might be simple for many concrete applications of this model, the idea stimulating this project is to decompose the iterative subroutine generated by ALM into finitely many easier and smaller subproblems, with the purpose of exploring the favorable structure of the involved function components individually. Some augmented-Lagrangian-based splitting methods for separable convex programming are thus proposed. The principles of designing these new algorithms are fully structure- exploiting, easily implementable and numerically efficient.