(264g) On the Global Optimality of a Class of Multilevel QP Problems in Dynamic Operability Applications

The steady-state economic optimum of process plants generally lies at the intersection of constraints (Lee and Weekman, 1976). However, in order for operation to remain feasible in the face of disturbances, the steady-state operating point needs to be backed-off from the constraints into the feasible region. The calculation of the optimal back-off amount has been the subject of several studies, typically involving the solution of a dynamic optimization problem of some form (Figueroa et al., 1996). This required amount of back-off can be related to the plant economics, and can also be used as a measure of dynamic operability (Narraway et. al, 1991; Figueroa et al., 1996).

Most studies considered either no control or linear controllers in back-off calculations. Recent studies by Soliman et al. (2008) and Baker and Swartz (2008) considered constrained model predictive control (MPC) within this framework. The resulting optimization problem is multilevel in nature, since the control input at each sampling interval over the closed-loop horizon requires the solution of a quadratic programming (QP) problem. In both studies, the inner level QPs were replaced by their equivalent Karush-Kuhn-Tucker (KKT) optimality conditions to give a single-level mathematical program with complementarity constraints (MPCC). In Soliman et al. (2008) the complementarity constraints are replaced by mixed-integer linear constraints, similar to the approach in Fortuny-Amat and McCarl (1981), resulting in a mixed-integer quadratic programming problem. Baker et al. (2008) instead solved the MPCC problem directly, using an interior-point algorithm specially tailored to handle complementarity constraints (Raghunathan and Biegler, 2003). Solution times were found to be significantly faster than the MIQP approach, and moreover, the solution was found to coincide with that of the MIQP when a solution to the MIQP was obtained (guaranteed to be the global optimum). Since the MPCC formulation is nonlinear and nonconvex, some analysis is required to determine whether the global optimum can be reliably obtained for these problems using this approach. This is the subject of the present study.

Small-scale numerical experiments were performed with a global optimizer to detect the presence of multiple local optima; only single optima were found. We considered then a bilevel LP problem, and recognized a condition under which only a single local optimum is obtained. Key to this is that the upper level objective function does not involve lower-level variables. By extending bilevel optimization theory to the multilevel case, we explore the optimality characteristics of the SISO back-off problem under constrained MPC. We argue that convergence to a stationary point of an appropriate reformulation of the MPCC as an NLP yields the optimal solution to the multilevel problem (Fletcher et al., 2006).

Case studies include the demonstration of the existence of a single optimum for a small example problem, as well as a counter-example that demonstrates that this result does not apply to all multilevel quadratic problems. Additional case studies demonstrating the difference in performance of solving the problem as a MPCC compared to solving it as a MIQP are also presented. Since the results apply to a general class of multilevel quadratic programs rather than the specific case of back-off problems with constrained MPC, additional chemical engineering applications that can be reformulated in this fashion are being explored.

References

Baker, R. and Swartz, C. L. E. (2008). Interior point solution of multilevel quadratic programming (QP) problems in constrained model predictive control (MPC) applications. Ind. Eng. Chem. Res., 47 (1), 81-91.

Figueroa, J.L., Bahri, P.A., Bandoni, J.A., and Romagnoli, J.A. (1996). Economic impact of disturbances and uncertain parameters in chemical processes ? A dynamic back-off analysis. Comput. Chem. Eng., 20(4), 453-461.

Fletcher, R., Leyffer, S., and Ralph, D. (2006). Local Convergence of SQP methods for Mathematical Programs with Equilibrium Constraints. SIAM Journal Optimization, 17 (1), 259-286.

Fortuny-Amat, J. and McCarl, B. (1981). A representation and economic interpretation of a two-level programming problem. J. Operational Research Society, 32(9), 783-792.

Lee, W. and Weekman, W.V. (1976). Advanced control practice in the chemical process industry: A view from industry. AIChE J., 22(1), 27-38.

Narraway, L.T., Perkins, J.D., and Barton, G.W. (1991). Interactions between process design and process control: economic analysis of process dynamics. J. Proc. Control, 1, 243-250.

Raghunathan, A.U. and Biegler, L.T. (2003). Mathematical programs with equilibrium constraints (MPECs) in process engineering. Comput. Chem. Eng., 27 (10), 1381-1392.

Soliman, M., Swartz, C. L. E., and Baker, R. (2008). A mixed-integer formulation for back-off under constrained predictive control. In press, Comput. Chem. Eng. doi:10.1016/j.compchemeng.2008.01.004