New algorithms for unconstrained optimization problems

The computation of an optimization problem is formulated as an optimal control problem and qualitative results on the nature of the trajectories are obtained. Generally, in order to compute a minimum point of a nonlinear function in finite time using a continuous time method one needs to use bang-bang and bang-intermediate trajectories. Using controllability conditions and the theory of Lyapunov functions the author develops a new continuous time method. A new iterative algorithm for computing the minimum point of a function which approximates the continuous time method is established and a simplified version of this algorithm is also developed. Generally one has bang-bang iterations, followed by partial Newton and bang-bang iterations and the full Newton iteration. In the simplified algorithm the partial Newton iterations are replaced by steepest descent iterations.