An Inner Approximation Algorithm for Dynamic Optimization with Strict Satisfaction of Path Constraints

An inner approximation algorithm is proposed for path-constrained dynamic optimization (PCDO) by iteratively solving restrictions of PCDO (RPCDO). Firstly, an upper bound function of the path constraint is designed based on interval analysis theory, which is utilized to construct RPCDO. Secondly, the algorithm is proposed based on iteratively approximating PCDO by dividing either all the time subintervals if RPCDO is infeasible or the active time subintervals if its solution does not satisfy the KKT conditions of PCDO. The algorithm locates a KKT point of PCDO with specified tolerances while strictly satisfying the path constraint. Thirdly, the finite convergence of the algorithm is proved theoretically. Finally, the numerical experiments show the effectiveness of the algorithm in terms of the computational time and strict satisfaction of the path constraint.