A Projected Lagrangian Algorithm for Nonlinear $l_1 $ Optimization

The nonlinear $l_1 $ problem is an unconstrained optimization problem whose objective function is not differentiable everywhere, and hence cannot be solved efficiently using standard techniques for unconstrained optimization. The problem can be transformed into a nonlinearly constrained optimization problem, but it involves many extra variables. We show how to construct a method based on projected Lagrangian methods for constrained optimization which requires successively solving quadratic programs in the same number of variables as that of the original problem. Special Lagrange multiplier estimates are used to form an approximation to the Hessian of the Lagrangian function, which appears in the quadratic program. A special line search algorithm is used to obtain a reduction in the $l_1 $ objective function at each iteration. Under certain conditions the method is locally quadratically convergent if analytical Hessians are used.