An adaptive competitive penalty method for nonsmooth constrained optimization

We present a competitive algorithm to minimize a locally Lipschitz function constrained with locally Lipschitz constraints. The approach is to use an ℓ1 nonsmooth penalty function. The method generates second order descent directions to minimize the ℓ1 penalty function. We introduce a new criterion to decide upon acceptability of a Goldstein subdifferential approximation. We show that the new criterion leads to an improvement of the Goldstein subdifferential approximation, as introduced by Mahdavi-Amiri and Yousefpour. Also, making use of our proposed line search strategy, the method always moves on differentiable points. Furthermore, the method has an adaptive behaviour in the sense that, when the iterates move on adequately smooth regions, the search directions switch exactly to the Shanno’s conjugate gradient directions and no subdifferential approximation is computed. The global convergence of the algorithm is established. Finally, an extensive comparison of the results obtained by our proposed algorithm with the ones obtained by some well-known methods shows significant reductions in number of function and gradient evaluations.