Abstract
Nonlinear equality and inequality constrained optimization problems with uncertain parameters can be addressed by a robust worst-case formulation that is, however, difficult to treat computationally. In this paper we propose and investigate an approximate robust formulation that employs a linearization of the uncertainty set. In case of any norm bounded parameter uncertainty, this formulation leads to penalty terms employing the respective dual norm of first order derivatives of the constraints. The main advance of the paper is to present two sparsity preserving ways for efficient computation of these derivatives in the case of large scale problems, one similar to the forward mode, the other similar to the reverse mode of automatic differentiation. We show how to generalize the techniques to optimal control problems, and discuss how even infinite dimensional uncertainties can be treated efficiently. Finally, we present optimization results for an example from process engineering, a batch distillation.
Similar content being viewed by others
References
Ben-Tal, A., Nemirovskii, A.: Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. MPS-SIAM Series on Optimization, MPS-SIAM, Philadelphia 2001
Bock, H., E.Kostina: Robust experimental design. In: H.G.Bock, Project A4, Optimization methods for reactive flows of Sonderforshungsbereich 359 Reactive Flows, Diffusion and Transport, Report 1999-2001, University of Heidelberg, 2001, pp. 133–135
Bock, H., Plitt, K.: A multiple shooting algorithm for direct solution of optimal control problems. In: Proc. 9th IFAC World Congress Budapest, Pergamon Press, 1984, pp. 243–247
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, 2004
Correa, R., Ramírez, C.: A global algorithm for nonlinear semidefinite programming. SIAM Journal on Optimization 15 (1), 303–318 (2002)
Diehl, M., Leineweber, D., Schäfer, A., Bock, H., Schlöder, J.: Optimization of multiple-fraction batch distillation with recycled waste cuts. AIChE Journal 48 (12), 2869–2874 (2002)
Fares, B., Noll, D., Apkarian, P.: Robust control via sequential semidefinite programming. SIAM Journal on Control and Optimization 40 (6), 1791–1820 (2002)
Freund, R.W., Jarre, F., Vogelbusch, C.: A sequential semidefinite programming method and an application in passive reduced-order modeling, 2005, (submitted)
Griewank, A.: Evaluating derivatives: principles and techniques of algorithmic differentiation. SIAM, Philadelphia 2000
Körkel, S., Kostina, E., Bock, H., Schlöder, J.: Numerical methods for optimal control problems in design of robust optimal experiments for nonlinear dynamic processes. Optimization Methods and Software 19, 327–338 (2004)
Kočvara, M., Stingl, M.: Pennon - a generalized augmented lagrangian method for semidefinite programming. In: G.D. Pillo, A. Murli (eds.) High Performance Algorithms and Software for Nonlinear Optimization, Kluwer Academic Publishers, Dordrecht, 2003, pp. 297–315
Leibfritz, F., Mostafa, E.M.E.: An interior point constrained trust region method for a special class of nonlinear semidefinite programming problems. SIAM Journal on Optimization 12 (4), 1048–1074 (2002)
Leineweber, D.: Efficient reduced SQP methods for the optimization of chemical processes described by large sparse DAE models, Fortschr.-Ber. VDI Reihe 3, Verfahrenstechnik, vol. 613. VDI Verlag, Düsseldorf, 1999
Leineweber, D., Schäfer, A., Bock, H., Schlöder, J.: An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization. Part II: Software aspects and applications. Comp. & Chem. Eng. 27, 167–174 (2003)
Ma, D.L., Braatz, R.D.: Worst-case analysis of finite-time control policies. IEEE Transactions on Control Systems Technology 9 (5), 766–774 (2001)
Nagy, Z., Braatz, R.: Open-loop and closed-loop robust optimal control of batch proccesses using distributional and worst-case analysis. Journal of Process Control 14, 411–422 (2004)
Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, 1999
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Diehl, M., Bock, H. & Kostina, E. An approximation technique for robust nonlinear optimization. Math. Program. 107, 213–230 (2006). https://doi.org/10.1007/s10107-005-0685-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-005-0685-1