Unconstrained Minimization Algorithms for Functions with Singular Or Ill-Conditioned Hessian.

The minimization of an unconstrained, real and twice differentiable function f:R('n) (--->) R is considered. Interest is focused on situations when the Hessian matrix of f(x) is either singular or ill-conditioned at or near the minimum. Two algorithms are presented for h and ling such problems. The first uses the pseudoinverse concept together with a finite scheme developed for computing the pseudoinverse or any square matrix. Convergence of the algorithm is proved, and some of its properties are also discussed. The second algorithm is based upon the transformation of the original minimization problem into one of finding the asymptotic solution of a related differential system. This leads to a "duality" between ill-conditioned minimization problems and stiff systems of ordinary differential equations. This duality is investigated and techniques for integrating stiff systems of ordinary differential equations are used for implementing a viable minimization algorithm. A study of the tradeoffs between speed and accuracy shows that stiff methods with only moderate levels of accuracy are required. This is an encouraging note since it eliminates the need for methods with high accuracy which tend to be computationally expensive due to the small stepsize that is often associated with them. Also since the Hessian matrix is real and symmetric, its eigenvalues are always real. Thus the stability problems, associated with the presence of some complex eigenvalues, do not exist here. This makes stiff integration schemes even more effective. Finally, a parameter identification problem, that is associated with an adaptive controller for the path control of surface ships in restricted waters, is solved using the two developed algorithms. Comparisons with the conjugate gradient method indicate the viability of the new algorithms.Ph.D.Industrial engineeringUniversity of Michiganhttp://deepblue.lib.umich.edu/bitstream/2027.42/158389/1/8125055.pd