Alternating Direction Method of Multipliers for Separable Convex Optimization of Real Functions in Complex Variables

The alternating direction method of multipliers (ADMM) has been widely explored due to its broad applications, and its convergence has been gotten in the real field. In this paper, an ADMM is presented for separable convex optimization of real functions in complex variables. First, the convergence of the proposed method in the complex domain is established by using the Wirtinger Calculus technique. Second, the basis pursuit (BP) algorithm is given in the form of ADMM in which the projection algorithm and the soft thresholding formula are generalized from the real case. The numerical simulations on the reconstruction of electroencephalogram (EEG) signal are provided to show that our new ADMM has better behavior than the classic ADMM for solving separable convex optimization of real functions in complex variables

ON THE O(1/T) CONVERGENCE RATE OF ALTERNATING DIRECTION METHOD WITH LOGARITHMIC-QUADRATIC PROXIMAL REGULARIZATION

Abstract. It was recently shown that the alternating direction method with logarithmicquadratic proximal regularization can yield an efficient algorithm for a class of variational inequalities with separable structures. This paper further shows the O(1/t) convergence rate for this kind of algorithms. Both the cases with a simple or general Glowinski’s relaxation factor are discussed. Key words. Alternating direction method, logarithmic-quadratic proximal regularization, convergence rate, Glowinski’s relaxation factor, variational inequality, separable structure AMS subject classifications. 47J20, 90C25, 90C3