# A Levenberg-Marquardt Method for Nonsmooth Regularized Least Squares


We develop a Levenberg-Marquardt method for minimizing the sum of a smooth nonlinear least-squares term $$f(x) = \frac{1}{2} \|F(x)\|_2^2$$ and a nonsmooth term $$h$$.
Both $$f$$ and $$h$$ may be nonconvex.
Steps are computed by minimizing the sum of a regularized linear least-squares model and a model of $$h$$ using a first-order method such as the proximal gradient method.
We establish global convergence to a first-order stationary point of both a trust-region and a regularization variant of the Levenberg-Marquardt method under the assumptions that $$F$$ and its Jacobian are Lipschitz continuous and $$h$$ is proper and lower semi-continuous.
In the worst case, both methods perform $$O(\epsilon^{-2})$$ iterations to bring a measure of stationarity below $$\epsilon \in (0, 1)$$.
We report numerical results on three examples: a group-lasso basis-pursuit denoise example, a nonlinear support vector machine, and parameter estimation in neuron firing.
For those examples to be implementable, we describe in detail how to evaluate proximal operators for separable $$h$$ and for the group lasso with trust-region constraint.
In all cases, the Levenberg-Marquardt methods perform fewer outer iterations than a proximal-gradient method with adaptive step length and a quasi-Newton trust-region method, neither of which exploit the least-squares structure of the problem.
Our results also highlight the need for more sophisticated subproblem solvers than simple first-order methods.