Summary and comparison of gradient-restoration algorithms for optimal control problems

This paper considers the problem of minimizing a functionalI which depends on the statex(t), the controlu(t), and the parameter . Here,I is a scalar,x ann-vector,u anm-vector, and ap-vector. At the initial point, the state is prescribed. At the final point, the state and the parameter are required to satisfyq scalar relations. Along the interval of integration, the state, the control, and the parameter are required to satisfyn scalar differential equations.Four types of gradient-restoration algorithms are considered, and their relative efficiency (in terms of number of iterations for convergence) is evaluated. The algorithms being considered are as follows: sequential gradient-restoration algorithm, complete restoration (SGRA-CR), sequential gradient-restoration algorithm, incomplete restoration (SGRA-IR), combined gradient-restoration algorithm, no restoration (CGRA-NR), and combined gradient-restoration algorithm, incomplete restoration (CGRA-IR).Evaluation of these algorithms is accomplished through six numerical examples. The results indicate that (i) the inclusion of a restoration phase is necessary for rapid convergence and (ii) while SGRA-CR is the most desirable algorithm if feasibility of the suboptimal solutions is required, rapidity of convergence to the optimal solution can be increased if one employs algorithms with incomplete restoration, in particular, CGRA-IR.This research was supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-72-2185.