Polynomial approximation of inequality path constraints in dynamic optimization

Abstract

We propose an algorithm for dynamic optimization problems with inequality path constraints. It solves a sequence of approximated problems where the path constraint is imposed on a finite number of points. Between adjacent points, an approximating polynomial of the constraint value is calculated and an additional constraint is imposed on the maximum value of this polynomial. We consider Taylor and Hermite polynomials. New points are added based on constraint violations or large approximations errors of the approximating polynomials. We prove finite convergence to a feasible point assuming: (i) the dynamic optimization problem has a Slater point, (ii) pointwise constraints are respected at each iteration. We compare the performance of the algorithm with the algorithm by Fu et at. (Automatica 62, 2015, p. 184–192) for three small case studies and an up-to-date industrial application where we calculate optimal feed rates for a semi-batch emulsion polymerization reactor. The results show that our proposed algorithm needs to solve fewer subproblems, i.e. fewer iterations, at the cost of more constraints, resulting in smaller CPU times.