Gill-Murray’s method

Gill-Murray s method has the further advantage of making it easy to evaluate a ( ) direction with negative curvature when x is a saddle point... 

In this case, it is necessary to detect a direction with negative curvature to get a direction of function decrease. If E is collected during factorization using Gill-Murray s method, it is possible to get such a direction with the procedure described below. 

The function BuildGradientAndPositiveHessian ensures that the Hessian is positive definite by adequately increasing diagonal elements similarly to Gill-Murray s method (see Section 3.6.1). 

The function BuildGradientAndUpdatePositiveHessian is used when we need the Hessian positive definite. It updates the Hessian with the aforementioned technique to preserve its sparsity and, analogously to BuildGradien-tAndPositiv6H6ssian, it makes the Hessian positive definite by adequately increasing diagonal elements similarly to Gill-Murray s method (see Section 3.6.1). 

The Gill-Murray modified Newton s method uses a Cholesky factorization of the Hessian matrix (Gill and Murray, 1974). The method is described in detail by Scales (1985). 

Gill, P. E. W. Murray M. A. Saunders J. A. Tomlin and M. H. Wright. On Projected Newton Barrier Methods for Linear Programming and an Equivalence to Karmarkar s Projective Method. Math Program 36 183-191 (1986). 

---