Nonsmooth nonconvex optimization models have been widely used in the restoration and reconstruction of real images. In this paper, we consider a linearly constrained optimization problem with a non-Lipschitz regularization term in the objective function which includes the lp norm (0 < p < 1) of the gradient of the underlying image in the l2-lp problem as a special case. We prove that any cluster point of ε scaled first order stationary points satisfies a first order necessary condition for a local minimizer of the optimization problem as ε goes to 0. We propose a smoothing quadratic regularization (SQR) method for solving the problem. At each iteration of the SQR algorithm, a new iterate is generated by solving a strongly convex quadratic problem with linear constraints. Moreover, we show that the SQR algorithm can find an ε scaled first order stationary point in at most O(ε−2) iterations from any starting point. Numerical examples are given to show good performance of the SQR algorithm for image restoration.
- Image restoration
- Non-Lipschitz optimization
- Smoothing quadratic regularization method
- Total variation regularization
- Worst-case complexity
ASJC Scopus subject areas
- Applied Mathematics