Globally convergent DC trust-region methods

In this paper, we investigate the use of DC (Difference of Convex functions) models and algorithms in the solution of nonlinear optimization problems by trust-region methods. We consider DC local models for the quadratic model of the objective function used to compute the trust-region step, and apply a primal-dual subgradient method to the solution of the corresponding trust-region subproblems. One is able to prove that the resulting scheme is globally convergent for first-order stationary points. The theory requires the use of exact second-order derivatives but, in turn, requires a minimum from the solution of the trust-region subproblems for problems where projecting onto the feasible region is computationally affordable. In fact, only one projection onto the feasible region is required in the computation of the trust-region step which seems in general less expensive than the calculation of the generalized Cauchy point. The numerical efficiency and robustness of the proposed new scheme when applied to bound-constrained problems is measured by comparing its performance against some of the current state-of-the-art nonlinear programming solvers on a vast collection of test problems.