Symbolic Preconditioning with Taylor Models: Some Examples
- 24 Downloads
Deterministic global optimization with interval analysis involves
• using interval enclosures for ranges of the constraints, objective, and gradient to reject infeasible regions, regions without global optima, and regions without critical points;
• using interval Newton methods to converge on optimum-containing regions and to verify global optima.
There are certain problems for which interval dependency leads to overestimation in the enclosures of the individual components, causing the optimization search to become prohibitively inefficient. As Hansen has observed earlier, in other problems, there is no overestimation in the individual components, but overestimation is introduced in the preconditioning in the interval Newton method.
We examine these issues for a particular nonlinear systems problem that, to date, has defied numerical solution. To reduce overestimation, we use Taylor models. The Taylor models sometimes reduce individual overestimation but, consistent with Hansen's observations, especially reduce the overestimation due to preconditioning. From numerical experiments, we conclude that, in certain instances, Taylor models can greatly reduce both the number of subregions necessary to complete an exhaustive search and the total computational effort.
Unable to display preview. Download preview PDF.
- 1.Berz, M.: COSY INFINIT web page, <http://cosy.pa.msu.edu/,2000>.Google Scholar
- 2.Berz, M. and Hoffstätter, G.: Computation and Application of Taylor Polynomials with Interval Remainder Bounds, Reliable Computing 4 (1) (1998), pp. 83-97.Google Scholar
- 3.Berz, M., Makino, K., Shamseddine, K., Hoffstätter, G. H., and Wan, W.: COSY INFINITY and Its Applications in Nonlinear Dynamics, in: Computational Differentiation, Techniques, Applications, and Tools, SIAM, Philadelphia, 1996, pp. 363-365.Google Scholar
- 4.Corliss, G. F.: GlobSol entry page, http://www.mscs.mu.edu/~globsol/,1998.Google Scholar
- 5.Corliss, G. F. and Kearfott, R. B.: Rigorous Global Search: Industrial Applications, in: Developments in Reliable Computing, Kluwer Academic Publishers, Dordrecht, 2000, pp. 1-16.Google Scholar
- 6.Hansen, E. R.: Preconditioning Linearized Equations, Computing 58 (1997), pp. 187-196.Google Scholar
- 7.Hertling, P.: A Limitation for Underestimation via Twin Arithmetic, Reliable Computing 7 (2) (2001), pp. 157-169.Google Scholar
- 8.Hoefkens, J.: Rigorous Numerical Analysis with High-Order Taylor Models, Ph.D. thesis, Department of Mathematics, Michigan State University, 2001.Google Scholar
- 9.Kearfott, R. B.: A Fortran 90 Environment for Research and Prototyping of Enclosure Algorithms for Nonlinear Equations and Global Optimization, ACM Trans. Math. Software 21 (1) (1995), pp. 63-78.Google Scholar
- 10.Kearfott, R. B.: Empirical Evaluation of Innovations in Interval Branch and Bound Algorithms for Nonlinear Algebraic Systems, SIAM J. Sci. Comput. 18 (2) (1997), pp. 574-594.Google Scholar
- 11.Kearfott, R. B.: Rigorous Global Search: Continuous Problems, Kluwer Academic Publishers, Dordrecht, 1996.Google Scholar
- 12.Kearfott, R. B. and Arazyan, A.: Taylor Series Models in Deterministic Global Optimization, in: Proceedings of Automatic Differentiation 2000: From Simulation to Optimization, Springer-Verlag, New York, 2000.Google Scholar
- 13.Kearfott, R. B., Dawande, M., Du, K.-S., and Hu, C.-Y.: Algorithm 737: INTLIB, A Portable FORTRAN 77 Interval Standard Function Library, ACM Trans. Math. Software 20 (4) (1994), pp. 447-459.Google Scholar
- 14.Makino, K. and Berz, M.: Efficient Control of the Dependency Problem Based on Taylor Model Methods, Reliable Computing 5 (1) (1999), pp. 3-12.Google Scholar
- 15.Makino, K. and Berz, M.: New Applications of Taylor Methods, in: Proceedings of Automatic Differentiation 2000: From Simulation to Optimization, Springer-Verlag, New York, 2000.Google Scholar
- 16.Muñoz, H.: Interval Slopes and Twin Slope Arithmetic in Nonsmooth Optimization, Ph.D. thesis, University of Louisiana at Lafayette, 2001.Google Scholar
- 17.Nesterov, V. M.: Interval and Twin Arithmetics, Reliable Computing 3 (4) (1997), pp. 369-380.Google Scholar
- 18.Neumaier, A.: Interval Methods for Systems of Equations, Cambridge University Press, Cambridge, 1990.Google Scholar
- 19.Pardalos, P. M. and Rosen, J. B.: Constrained Global Optimization: Algorithms and Applications, Lecture Notes in Computer Science 268, Springer-Verlag, New York, 1987.Google Scholar
- 20.Ratschek, H. and Rokne, J.: New Computer Methods for Global Optimization, Wiley, New York, 1988.Google Scholar
- 21.Ratz, D. and Csendes, T.: On the Selection of Subdivision Directions in Interval Branch-and-Bound Methods for Global Optimization, J. Global Optim. 7 (1995), pp. 183-207.Google Scholar
- 22.Van Hentenryck, P., Michel, L., and Deville, Y.: Numerica: A Modeling Language for Global Optimization, MIT Press, Cambridge, 1997.Google Scholar