Optimization by Way of the Trajectory Following Method

Abstract

The trajectory following method is a relatively old concept for solving the nonlinear programming problem. It is based on the idea of solving continuous differential equations whose equilibrium solutions satisfy the necessary conditions for a minimum. The method is “trajectory following” in the sense that an initial point is moved along a trajectory generated by the differential equations to the solution point. The fact that this method has not been popular is understandable. It may require many function evaluations for a solution and it may suffer from the same slow convergence difficulties that the “gradient method” is known to have. However with the advent of fast computers and efficient integration solvers, these difficulties may no longer exist for many problems of practical interest. Here the focus is on ease of use rather than number of function evaluations as a performance criteria. It is shown that the trajectory following method, used in conjunction with a good “stiff” integration routine, can provide a tool that is easy to use for solving a wide variety of optimization problems including linear programming, quadratic programming and global optimization problems.