Generalized derivatives of dynamic systems with a linear program embedded

Dynamic systems with a linear program (LP) embedded can be found in control and optimization of bioreactor models based on dynamic flux balance analysis (DFBA). Derivatives of the dynamic states with respect to a parameter vector are essential for open and closed-loop dynamic optimization and parameter estimation of such systems. These derivatives, given by a forward sensitivity system, may not exist because the optimal value of a linear program as a function of the right-hand side of the constraints is not continuously differentiable. Therefore, nonsmooth analysis must be applied which provides optimality conditions in terms of subgradients, for convex functions, or Clarke’s generalized gradient, for nonconvex functions. This work presents an approach to compute the necessary information for nonsmooth optimization, i.e., an element of the generalized gradient. Moreover, a numerical implementation of the results is introduced. The approach is illustrated through a large-scale dynamic flux balance analysis example.