A Proximal Quasi-Newton Trust-Region Method for Nonsmooth Regularized Optimization

We develop a trust-region method for minimizing the sum of smooth term (f) and nonsmooth (h), both which can be nonconvex. Each iteration our minimizes possibly nonconvex model (f + h) in trust region. The coincides with value subdifferential at center. establish global convergence to first-order stationary point when satisfies smoothness condition that holds, particular, it has Lipschitz-continuous gradient, (h) is proper lower semicontinuous. required proper, semi-continuous prox-bounded. Under these weak assumptions, we worst-case (O(1/\epsilon^2)) complexity bound matches best known standard methods optimization. detail special instance, named TR-PG, use limited-memory quasi-Newton compute step proximal gradient method, resulting practical method. similar properties quadratic regularization variant, R2, provide an interpretation as adaptive size problems. R2 may also used steps inside implementation TR-R2. describe Julia implementations report numerical results on inverse problems from sparse optimization signal processing. Both TR-PG TR-R2 exhibit promising performance compare favorably two linesearch based convex models.