A Curvilinear Method for Large Scale Optimization Problems

We present a new matrix-free method for the computation of the negative curvature direction in large scale unconstrained problems. We describe a curvilinear method which uses a combination of a quasi-Newton direction and a negative curvature direction. We propose an algorithm for the computation of the search directions which uses information of two specific L-BFGS matrices in such a way that avoids both the calculation and the storage of the approximate Hessian. Explicit forms for the eigenpair that corresponds to the most negative eigenvalue of the approximate Hessian are also presented. Numerical results show that the proposed approach is promising. Keywords— large scale unconstrained optimization, curvilinear search, negative curvature direction, eigenvalues, eigenvectors, power inverse method, quasi-Newton method