A primal-dual algorithm for nonlinear programming exploiting negative curvature directions

In this paper we propose a primal-dual algorithm for the solution of inequality constrained optimization problems. The distinguishing feature of the proposed algorithm is that of exploiting as much as possible the local non-convexity of the problem to the aim of producing a sequence of points converging to second order stationary points. In the unconstrained case this task is accomplished by computing a suitable negative curvature direction of the objective function. In the constrained case it is possible to gain analogous information by exploiting the non-convexity of a particular exact merit function. The algorithm hinges on the idea of comparing, at every iteration, the relative effects of two directions and then selecting the more promising one. The first direction conveys first order information on the problem and can be used to define a sequence of points converging toward a KKT pair of the problem. Whereas, the second direction conveys information on the local non-convexity of the problem and can be used to drive the algorithm away from regions of non-convexity. We propose a proper selection rule for these two directions which, under suitable assumptions, is able to generate a sequence of points that is globally convergent to KKT pairs that satisfy the second order necessary optimality conditions, with superlinear convergence rate if the KKT pair satisfies also the strong second order sufficiency optimality conditions.