A class of methods for unconstrained minimization based on stable numerical integration techniques

Abstract Recently a number of algorithms have been derived which minimize a nonlinear function by iteratively constructing a monotonically improving sequence of approximate minimizers along curvilinear search paths instead of rays. These curvilinear search paths were obtained by solving a first-order approximation to certain initial value systems of nonlinear differential equations. The simplest technique for solving some of the above systems numerically, viz., that of Euler, yields either the steepest-descent or Newton method, and this induced us to examine the possibility of modifying other, more sophisticated and stable numerical integration techniques for use in function minimization. In this paper are presented some theoretical as well as practical aspects of using numerical integration techniques in order to derive minimization algorithms. Results are also given and possible areas for future research are indicated.