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

Abstract

In this paper we propose a primal-dual algorithm for the solution of inequality constrained optimization problems. Thedistinguishing feature of the proposed algorithm is that of exploiting as much as possible the local non-convexity ofthe problem to the aim of producing a sequence of points converging to second order stationary points. In theunconstrained case this task is accomplished by computing a suitable negative curvature direction of the objectivefunction. In the constrained case it is possible to gain analogous information by exploiting the non-convexity of aparticular exact merit function. The algorithm hinges on the idea of comparing, at every iteration, the relativeeffects of two directions and then selecting the more promising one. The first direction conveys first orderinformation 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 drivethe 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 thatsatisfy the second order necessary optimality conditions, with superlinear convergence rate if the KKT pair satisfiesalso the strong second order sufficiency optimality conditions.