Planar Quasi - Newton Algorithms for Unconstrained Saddlepoint Problems

A new class of quasi-Newton methods is introduced that can locate a unique stationary point of an n-dimensional quadratic function in at most n steps. When applied to positive-definite or negative-definite quadratic functions, the new class is identical to Huang's symmetric family of quasi-Newton methods (Ref. 1). Unlike the latter, however, the new family can handle indefinite quadratic forms and therefore is capable of solving saddlepoint problems that arise, for instance, in constrained optimization. The novel feature of the new class is a planar iteration that is activated whenever the algorithm encounters a near-singular direction of search, along which the objective function approaches zero curvature. In such iterations, the next point is selected as the stationary point of the objective function over a plane containing the problematic search direction, and the inverse Hessian approximation is updated with respect to that plane via a new four-parameter family of rank-three updates. It is shown that the new class possesses properties which are similar to or which generalize the properties of Huang's family. Furthermore, the new method is equivalent to Fletcher's (Ref. 2) modified version of Luenberger's (Ref. 3) hyperbolic pairs method, with respect to the metric defined by the initial inverse Hessian approximation. Several issues related to implementing the proposed method in nonquadratic cases are discussed.