  In this paper, we introduce a class of nonsmooth nonconvex least square
optimization problem using convex analysis tools and we propose to use the
iterative minimization-majorization (MM) algorithm on a convex set with
initializer away from the origin to find an optimal point for the optimization
problem. For this, first we use an approach to construct a class of convex
majorizers which approximate the value of non-convex cost function on a convex
set. The convergence of the iterative algorithm is guaranteed when the initial
point $x^{(0)}$ is away from the origin and the iterative points $x^{(k)}$ are
obtained in a ball centred at $x^{(k-1)}$ with small radius. The algorithm
converges to a stationary point of cost function when the surregators are
strongly convex. For the class of our optimization problems, the proposed
penalizer of the cost function is the difference of $\ell_1$-norm and the
Moreau envelope of a convex function, and it is a generalization of GMC
non-separable penalty function previously introduced by Ivan Selesnick in
\cite{IS17}.

Optimization and Control;Numerical Analysis;Classical Analysis and ODEs;Numerical Analysis