A Smoothing Active Set Method for Linearly Constrained Non-Lipschitz Nonconvex Optimization

We propose a novel smoothing active set method for linearly constrained nonLipschitz nonconvex problems. At each step of the proposed method, we approximate the objective function by a smooth function with a fixed smoothing parameter and employ a new active set method for minimizing the smooth function over the original feasible set, until a special updating rule for the smoothing parameter meets. The updating rule is always satisfied within a finite number of iterations since the new active set method for smooth problems proposed in this paper forces at least one subsequence of projected gradients to zero. Any accumulation point of the smoothing active set method is a stationary point associated with the smoothing function used in the method, which is necessary for local optimality of the original problem. And any accumulation point for the \ell ₂ - \ell p (0 < p < 1) sparse optimization model is a limiting stationary point, which is a local minimizer under a certain second-order condition. Numerical experiments demonstrate the efficiency and effectiveness of our smoothing active set method for hyperspectral unmixing on a 3 dimensional image cube of large size.