| This dissertation investigates the use of ordinary differential equations (ODEs) to determine a search path to find a solution of an unconstrained optimization problem. Two strategies, linesearch and trust-region, are the basis of current optimization algorithms. They are affected either by the definiteness of the Hessian matrix at any iteration or by the scaling of the optimization problem. A third strategy to solve unconstrained optimization problems is the method of gradients. It was originally proposed in 1941 by Courant as a variational method for numerically solving partial differential equations (PDEs) arising in problems of equilibrium and vibrations. While historically much of the motivation for the method of gradients and the mathematical analysis behind it came from the desire to solve PDEs, the function it seeks to minimize need not come from a PDE. Indeed, the method of gradients is a general-purpose method to solve unconstrained optimization problems. Its convergence theory requires knowledge of the objective function at a continuum of points. The method has not found general acceptance because of the difficulties inherent in solving a system of nonlinear ODEs. Recently, Behrman proposed a method of gradients based upon the solution of a system of n linear ODEs, where n denotes the number of independent variables of the objective function. Here we propose a method similar in spirit to the original method of gradients, but one that requires the solution of a system of only two linear ODEs. These equations are definable and solvable regardless of the nature of the Hessian matrix, and their solutions are unaffected by poor scaling. We show that the new algorithm converges to a point satisfying the first and second-order necessary conditions for local unconstrained optimality. This approach is also efficient for large-scale problems. Numerical testing was performed on a Silicon Graphics ORIGIN 2000 running IRIX64 release 6.5.6m. |
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