A proximal alternating minimization method for ℓ0-regularized nonlinear optimization problems: application to state estimation

In this paper we consider the minimization of ℓ
₀
-regularized nonlinear optimization problems, where the objective function is the sum of a smooth convex term and the ℓ
₀
quasi-norm of the decision variable. We introduce the class of coordinatewise minimizers and prove that any point in this class is a local minimum for our ℓ
₀
-regularized problem. Then, we devise a random proximal alternating minimization method, which has a simple iteration and is suitable for solving this class of optimization problems. Under convexity and coordinatewise Lipschitz gradient assumptions, we prove that any limit point of the sequence generated by our new algorithm belongs to the class of coordinatewise minimizers almost surely. We also show that the state estimation of dynamical systems with corrupted measurements can be modeled in our framework. Numerical experiments on state estimation of power systems, using IEEE bus test case, show that our algorithm performs favorably on solving such problems.