Primal-dual properties of sequential gradient-restoration algorithms for optimal control problems 2. General problem

The problem of minimizing a functional, subject to differential constraints, nondifferential constraints, initial constraints, and final constraints, is considered in connection with sequential gradient-restoration algorithms (SGRA) for optimal control problems. Both the primal formulation and the dual formulation are presented. Depending on whether the primal formulation is used or the dual formulation is used, one obtains a primal sequential gradient-restoration algorithm (PSGRA) or a dual sequential gradient-restoration algorithm (DSGRA). For the problem under consideration, it is found convenient to split the control vector into an independent control vector and a dependent control vector, the latter having the same dimension as the nondifferential constraint vector. This modification enhances the computational efficiency of both the primal formulation and the dual formulation. The system of Lagrange multipliers associated with (i) the gradient phase of SGRA and (ii) the restoration phase of SGRA is examined. For each phase, it is shown that the Lagrange multipliers are endowed with a duality property: they minimize a special functional, quadratic in the multipliers, subject to the multiplier differential equations and boundary conditions, for given state, control, and parameter. These duality properties have considerable computational implications: they allow one to reduce the auxiliary optimal control problems associated with (i) and (ii) to mathematical programming problems involving a finite number of parameters as unknowns.