(183f) A General Nonlinear Programming Algorithm for the Optimization of Problems with Dense Constraint Jacobians

Periodic Adsorption Processes (PAPs) have gained increasing commercial acceptance as an energy-efficient separation technique over the past two decades. These systems never reach steady state, instead they operate at cyclic steady state (CSS), where the bed conditions at the beginning of the cycle match with those at the end of the cycle. This CSS operation results in dense constraint Jacobians, where the time required for the computation of the Jacobian and its factorization dominates the overall optimization process.

Inorder to efficiently handle these dense constraint Jacobians during optimization and reduce the computation time, this work presents a novel trust-region based SQP algorithm for the solution of general non-linear programming problems with non-linear equality and inequality constraints. Instead of forming and factoring the dense constraint Jacobian, this algorithm approximates the Jacobian of constraints with a specialized quasi-Newton method [1]. Hence it is well suited to solve optimization problems related to PAPs. This algorithm represents a Byrd-Omojokun trust-region approach that takes the inexactness of the Jacobian and its null-space representation into account. In this approach, the SQP step is a combination of two subproblems, the normal subproblem which aims at reducing the infeasibility of the linearized constraints and the tangential subproblem which aims at reducing the objective function by maintaining linearized feasibility. Inequality constraints are converted to equality constraints by adding slacks with bounds on the slack variables. The augmented set of equality constraints is solved in the normal subproblem. The bounds on the free variables and the slacks are treated by solving a small scale quadratic program in the tangential problem by a calculation analogous to that performed in [2, 3]. The quality of the approximated constraint Jacobian can be adjusted by verifying two conditions that measure the inexactness of the null space representation [4]. The two required conditions on the inexactness can be easily verified during the optimization process. Furthermore, the derivative information required by the NLP is computed using a targeted approach [5] which combines automatic differentiation and more sophisticated integration algorithms, for example by CVODES, to evaluate the direct sensitivity equation, the adjoint equation and the second order adjoint equation.

We provide results of tests of our algorithm on problems from CUTE problem set. Furthermore, we present numerical results for dynamic optimization problems based on Simulated Moving Bed and Pressure Swing Adsorption processes, where the Jacobian of the constraints is dense.

References:

1. Griewank, A; Walther, A. On constrained optimization by adjoint-based quasi-Newton methods. Optim. Methods. Softw., 17: 869-889, 2002.

2. Gay, D. M. A trust-region approach to linearly constrained optimization. Numerical Analysis Proceedings, Springer-Verlag, 72, 1983.

3. Arora, N; Biegler, L. T. A Trust Region SQP algorithm for equality constrained parameter estimation with simple parameter bounds. Computational Optimization and Applications, 28, 51-86, 2004.

4. Walther, A. A First-Order Convergence analyis of Trust-Region methods with inexact Jacobians. SIAM J. of Optim, 2008; 19(1): 307-325.

5. Ozyurt, D.B.; Barton, P.I. Cheap Second Order Directional derivatives of stiff ODE Embedded Fuctionals. SIAM J. Sci. Comput. 2005; 26(5): 1725-1743.