A direct search procedure utilizing pseudo random numbers over a region is presented to solve nonlinear programming problems. After each iteration the size of the region is reduced so that the optimum can be found as accurately as desired. The ease of programming, the speed of convergence, and the reliability of results make the procedure very attractive for solving nonlinear programming problems.
In using dynamic programming, by taking only accessible states for the x-grid and using an iterative procedure employing region contraction, only a small number of grid points are required at each iteration to yield very good accuracy even if the dimension of the system is high. The effect of the number of grid points and the choice of the contraction factor are analysed by considering a non-linear system consisting of eight ordinary differential equations and four control variables.No difficulties were encountered in convergence to the optimal solution in no more than 20 iterations. The proposed procedure overcomesthe curse of dimensionality that has discouraged the use of dynamic programming in the past to solve high-dimensional non-linear optimal control problems, and provides an attractive means of solving optimal control problems in general.
The use of a penalty function to handle final-state constraints is incorporated into the iterative dynamic programming algorithm employing accessible grid points and region contraction. By using three examples, the importance of the size of the penalty function is illustrated. If the penalty function is very large, the rate of convergence to the opimum is slow and a large number of allowable values for control are required. If, however, the penalty function is reasonably small, the rate of convergence is rapid and a very small number of allowable control values are required for convergence to the optimum.
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