(216e) Deterministic Global Optimization of Processes Described by Nonlinear Differential-Algebraic Equations

By combining nonlinear reaction kinetics models, thermodynamic relationships, and material and energy balances between various process units, first principles dynamic models for a wide variety of chemical processes result in nonlinear systems of differential-algebraic equations (DAEs). Through recent advances in dynamic simulation techniques, detailed dynamic models of this type can be solved very efficiently. Moreover, efficient methods for simultaneously solving parametric sensitivity or adjoint equations have made local optimization of DAE models a ubiquitous and indispensible tool for the design and control of dynamic chemical processes. Unfortunately, it is well known that optimization problems with dynamic process models embedded are pathologically nonconvex and have multiple sub-optimal local minima. At the same time, there are many problems in chemical process engineering which require the global solution of such optimization problems. For example, safety verification and robust feasibility problems for dynamic chemical processes subject to variable inputs and /or uncertain model parameters can be reduced to checking that the "worst-case" state of the system satisfies some given constraints, where the "worst-case" is defined as the global solution of an optimization problem constrained by the process model. For such problems, feasibility of a locally optimal solution cannot guarantee that the system remains feasible for inputs and/ or uncertain model parameters far from the local optimum. A second example is parameter estimation for dynamic model verification and discrimination, where the "best fit" of a dynamic model to a set of experimental data can only be obtained through the global solution of an optimization problem with the dynamic model embedded. Again, a locally optimal solution is insufficient for this application because there is no guarantee that the "best-fit" has been found and therefore statistical significance tests cannot be meaningfully applied.

This work presents a deterministic global optimization algorithm for optimization problems with nonlinear index-one semi-explicit DAEs embedded. This algorithm does not involve any a priori discretization of the state variables, and hence no additional decision variables are added to the optimization problem. Furthermore, no convexity assumptions are required whatsoever. The presented algorithm uses a standard spatial branch-and-bound framework in conjunction with a novel method for constructing convex underestimating programs for problems with DAEs embedded. Standard methods for constructing convex underestimating programs are only applicable to optimization problems where the objective function and constraints can be expressed algebraically, i.e., they do not embed the solutions of differential or differential-algebraic equations. Thus, the key theoretical advance which enables the use of a branch-and-bound algorithm without discretization of the state variables is the ability to construct convex and concave relaxations of the solutions of the embedded DAEs with respect to the decision variables.

In recent years, a few methods for constructing convex and concave relaxations of the solutions of nonlinear ordinary differential equations (ODEs) with respect to initial conditions and model parameters have been presented in the literature. However, no method has so far been proposed for computing convex and concave relaxations for the solutions of differential-algebraic systems. Even in the simpler case of ODEs, all of these methods can be unsatisfactory because they either result in relaxations which poorly approximate the true ODE solutions or because they are very expensive to compute. Moreover, most of these methods can only compute relaxations which are constant or affine with respect to the initial conditions and parameters, which is a significant source of weakness for these relaxations when applied to sufficiently nonlinear problems.

This work presents a novel technique for computing nonlinear convex and concave relaxations for the solutions of nonlinear ODEs, and then extends this result in order to provide convex and concave relaxations for the solutions of nonlinear index-one semi-explicit DAEs. To reiterate, this technique does not require any discretization or approximation of the original system of DAEs. The proposed method is based on a novel use of McCormick's relaxation technique to construct an auxiliary system of differential-algebraic equations whose solutions are the desired convex and concave relaxations of the original DAE solutions. This auxiliary system is four times the size of the original model and can be constructed automatically from computer code formulating the original DAEs. Thus, evaluating the proposed relaxations is comparable in cost to a single dynamic simulation of the original model. For ODEs, this technique typically provides tighter relaxations than other methods in the literature, while for DAEs, no other method can provide the desired relaxations. Most importantly, these relaxations enable the construction of convex underestimating programs for optimization problems with DAEs embedded. Therefore, optimization problems of this type can be solved to within any nonzero tolerance of global optimality in finitely many iterations using standard spatial branch-and-bound global optimization techniques.