Satisfaction of path chance constraints in dynamic optimization problems

Abstract

We propose an algorithm that calculates heuristically optimal solutions for dynamic optimization problems with path chance constraints. The solution is a feasible point in the chance constraint sense and an optimal point of an approximated problem. Uncertainty in parameters and initial conditions is modelled as Gaussian distributions. The method solves nonlinear programs (NLP) generated by replacing the probability constraint by a set of approximated deterministic pointwise constraints with a right-hand side restriction. For each NLP solution, the probability of constraint violation is calculated by Monte Carlo integration. When the NLP solution does not respect the chance constraint, new pointwise constraints are added, and we update the approximation and the restriction with the results from Monte Carlo integration. These steps are repeated until a feasible solution is found. The algorithm terminates after a finite number of iterations under mild assumptions. We demonstrate the algorithm in a fed-batch bioreactor case study, showing that it provides a solution in a shorter CPU time and fewer iterations when compared to using a fixed set of pointwise constraints where only the restriction is updated.