An alternating linearization bundle method for a class of nonconvex nonsmooth optimization problems

In this paper, we propose an alternating linearization bundle method for minimizing the sum of a nonconvex function and a convex function, both of which are not necessarily differentiable. The nonconvex function is first locally “convexified” by imposing a quadratic term, and then a cutting-planes model of the local convexification function is generated. The convex function is assumed to be “simple” in the sense that finding its proximal-like point is relatively easy. At each iteration, the method solves two subproblems in which the functions are alternately represented by the linearizations of the cutting-planes model and the convex objective function. It is proved that the sequence of iteration points converges to a stationary point. Numerical results show the good performance of the method.