A descent subgradient method using Mifflin's line search for nonsmooth nonconvex optimization

We propose a descent subgradient algorithm for minimizing a function $ f:\mathbb {R}<^>n\to \mathbb {R} $ f:Rn -> R, assumed to be locally Lipschitz, but not necessarily smooth or convex. To find an effective descent direction, the Goldstein epsilon-subdifferential is approximated through an iterative process. The method enjoys a new two-point variant of Mifflin's line search in which the subgradients are arbitrary. Thus, the line search procedure is easy to implement. Moreover, in comparison to bundle methods, the quadratic subproblems have a simple structure, and to handle nonconvexity the proposed method requires no algorithmic modification. We study the global convergence of the method and prove that any accumulation point of the generated sequence is Clarke stationary, assuming that the objective f is weakly upper semismooth. We illustrate the efficiency and effectiveness of the proposed algorithm on a collection of academic and semi-academic test problems.