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We develop a trust-region method for minimizing the sum of a smooth term $f$and a nonsmooth term $h$), both of which can be nonconvex. Each iteration ofour method minimizes a possibly nonconvex model of $f + h$ in a trust region.The model coincides with $f + h$ in value and subdifferential at the center. Weestablish global convergence to a first-order stationary point when $f$satisfies a smoothness condition that holds, in particular, when it hasLipschitz-continuous gradient, and $h$ is proper and lower semi-continuous. Themodel of $h$ is required to be proper, lower-semi-continuous and prox-bounded.Under these weak assumptions, we establish a worst-case $O(1/\epsilon^2)$iteration complexity bound that matches the best known complexity bound ofstandard trust-region methods for smooth optimization. We detail a specialinstance, named TR-PG, in which we use a limited-memory quasi-Newton model of$f$ and compute a step with the proximal gradient method, resulting in apractical proximal quasi-Newton method. We establish similar convergenceproperties and complexity bound for a quadratic regularization variant, namedR2, and provide an interpretation as a proximal gradient method with adaptivestep size for nonconvex problems. R2 may also be used to compute steps insidethe trust-region method, resulting in an implementation named TR-R2. Wedescribe our Julia implementations and report numerical results on inverseproblems from sparse optimization and signal processing. Both TR-PG and TR-R2exhibit promising performance and compare favorably with two linesearchproximal quasi-Newton methods based on convex models.

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