Nonsmooth Optimization Method and Sparsity

Abstract

Nonsmooth variational problems are analyzed using the Lagrange multiplier theory. In particular, the sparsity optimization method has a multitude of important applications, i.e., in imaging analysis and friction contact and inverse problems, and can be cast as nonsmooth variational problems. The optimality condition is derived and it is of the form of the complementarity systems. An effective numerical optimization method using the semismooth Newton method is then developed and analyzed. The method takes the form of primal-dual active set methods and is much more efficient than numerical optimization algorithms based on first order methods. The l0 sparsity optimization for the linear least square problem is considered. The necessary optimality condition is derived and a numerical algorithm based on the Lagrange multiplier rule to determine a solution is developed and analyzed.