Combination of steepest descent and BFGS methods for nonconvex nonsmooth optimization

In this paper, a method is developed for solving nonsmooth nonconvex minimization problems. This method extends the classical BFGS framework. First, we generalize the Wolfe conditions for locally Lipschitz functions and prove that this generalization is well defined. Then, a line search algorithm is presented to find a step length satisfying the generalized Wolfe conditions. Next, the Goldstein e-subgradient is approximated by an iterative method and a descent direction is computed using a positive definite matrix. This matrix is updated using the BFGS method. Finally, a minimization algorithm based on the BFGS method is described. The algorithm is implemented in MATLAB and numerical results using it are reported.