(135f) Deterministic Global Optimization with Differential-Algebraic Equations Embedded | AIChE

(135f) Deterministic Global Optimization with Differential-Algebraic Equations Embedded

Authors 

Barton, P. I. - Presenter, Massachusetts Institute of Technology


A fully deterministic method is proposed for the global
solution of optimization problems with nonlinear, semi-explicit, index-one
differential-algebraic equations (DAEs) embedded. Detailed models of a wide
variety dynamic chemical processes, which may include material and energy
balances, chemical reaction kinetics, thermodynamic relationships, and
semi-empirical process models, are naturally described by systems of nonlinear
DAEs. Accordingly, many important process engineering problems take the form of
optimization problems with DAEs embedded, including determination of optimal
operating conditions and/or control profiles, as well as safety verification
problems. Another important application is parameter estimation for
differential-algebraic models, which takes the form of a least-squares
minimization  with DAEs embedded.

Research on dynamic optimization in the chemical engineering
literature alone has produced a number of representative examples which are nonconvex and display multiple suboptimal local minima.
Moreover, the absence of an analytical solution to the embedded dynamic system
in all but trivial cases makes it extremely difficult to determine whether a
given problem is convex or not.  These
considerations motivate the development of global optimization algorithms for
problems with dynamic systems embedded. This need is further compounded by the
observation that local optima do not provide meaningful information for many
important applications, including safety verification problems and parameter
estimation problems in the context of model falsification.

Previous literature has primarily treated optimization
problems with DAEs embedded using local methods or stochastic global methods
which cannot guarantee global optimality. Another simple approach is to discretize the embedded DAEs  in order to reduce the original dynamic optimization
problem to a standard nonlinear program, which can be solved to global
optimality using existing methods. However, an accurate representation of the embedded
dynamics requires a fine discretization, which results in the addition of too
many variables for practical global optimization, even for very simple
problems.  A more customized approach, which
does not require discretization, uses a branch-and-bound optimization strategy
in conjunction with a theory for constructing convex underestimating programs
using second order sensitivities of the embedded DAEs.  Though this method appears to produce the
global solution for simple test cases, it employs a sampling procedure to
obtain global information about the second order sensitivities, and hence
cannot guarantee global optimality in the general case.

The method presented in this work uses recent extensions of
McCormick's relaxation theory to construct convex and concave relaxations of
the solutions of nonlinear systems of DAEs. Using these relaxations, existing
techniques can be used to construct convex underestimating programs, which in
turn enables the use of branch-and-bound global optimization techniques.
Because no discretization is required, the proposed algorithm operates in the
original decision space, without the introduction of additional variables.
Moreover, global information concerning solutions of the dynamic system is
obtained via the proposed relaxation theory, rather than through sampling.
Accordingly, the proposed algorithm is fully deterministic.

This presentation describes the proposed algorithm, with
particular focus on the construction of convex and concave relaxations of the
solutions of the embedded DAEs, and performance is analyzed through chemical
engineering examples.