Summary
We consider design centering problems in their reformulation as general semi-infinite optimization problems. The main goal of the article is to show that the Reduction Ansatz of semi-infinite programming generically holds at each solution of the reformulated design centering problem. This is of fundamental importance for theory and numerical methods which base on the intrinsic bilevel structure of the problem.
For the genericity considerations we prove a new first order necessary optimality condition in design centering. Since in the course of our analysis also a certain standard semi-infinite programming problem turns out to be related to design centering, the connections to this problem are studied, too.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
E.W. Cheney, Introduction to Approximation Theory, McGraw-Hill, New York, 1966.
V. Chvatal, Linear Programming, Freeman, New York, 1983.
A.V. FIACCO, G.P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques, Wiley, New York, 1968.
T.J. Graettinger, B.H. Krogh, The acceleration radius: a global performance measure for robotic manipulators, IEEE Journal of Robotics and Automation, Vol. 4 (1988), 60–69.
P. Gritzmann, V. Klee, On the complexity of some basic problems in computational convexity. I. Containment problems, Discrete Mathematics, Vol. 136 (1994), 129–174.
R. Hettich, H.Th. Jongen, Semi-infinite programming: conditions of optimality and applications, in: J. Stoer (ed): Optimization Techniques, Part 2, Lecture Notes in Control and Information Sciences, Vol. 7, Springer, Berlin, 1978, 1–11.
R. Hettich, K.O. Kortanek, Semi-infinite programming: theory, methods, and applications, SIAM Review, Vol. 35 (1993), 380–429.
R. Hettich, G. Still, Semi-infinite programming models in robotics, in: J. Guddat, H.Th. Jongen, B. Kummer, F. Nožička (eds): Parametric Optimization and Related Topics II, Akademie Verlag, Berlin, 1991, 112–118.
R. hettich, G. Still, Second order optimality conditions for generalized semi-infinite programming problems, Optimization, Vol. 34 (1995), 195–211.
R. Hettich, P. Zencke, Numerische Methoden der Approximation und semiinfiniten Optimierung, Teubner, Stuttgart, 1982.
M.W. Hirsch, Differential Topology, Springer, New York, 1976.
R. Horst, H. Tuy, Global Optimization, Springer, Berlin, 1990.
F. John, Extremum problems with inequalities as subsidiary conditions, in: Studies and Essays, R. Courant Anniversary Volume, Interscience, New York, 1948, 187–204.
H.Th. Jongen, P. Jonker, F. Twilt, Critical sets in parametric optimization, Mathematical Programming, Vol. 34 (1986), 333–353.
H.Th. Jongen, P. Jonker, F. Twilt, Nonlinear Optimization in Finite Dimensions, Kluwer, Dordrecht, 2000.
H.Th. Jongen, J.-J. Rückmann, O. Stein, Generalized semi-infinite optimization: a first order optimality condition and examples, Mathematical Programming, Vol. 83 (1998), 145–158.
P.-J. Laurent, Approximation et Optimisation, Hermann, Paris, 1972.
V.H. Nguyen, J.J. Strodiot, Computing a global optimal solution to a design centering problem, Mathematical Programming, Vol. 53 (1992), 111–123.
O. Stein, On parametric semi-infinite optimization, Thesis, Shaker, Aachen, 1997.
O. Stein, On level sets of marginal functions, Optimization, Vol. 48 (2000), 43–67.
O. Stein, Bi-level Strategies in Semi-infinite Programming, Kluwer, Boston, 2003.
O. Stein, G. Still, On generalized semi-infinite optimization and bilevel optimization, European Journal of Operational Research, Vol. 142 (2002), 444–462.
O. Stein, G. Still, Solving semi-infinite optimization problems with interior point techniques, SIAM Journal on Control and Optimization, Vol. 42 (2003), 769–788.
W. Wetterling, Definitheitsbedingungen für relative Extrema bei Optimierungs-und Approximationsaufgaben, Numerische Mathematik, Vol. 15 (1970), 122–136.
G. Zwier, Structural Analysis in Semi-Infinite Programming, Thesis, University of Twente, 1987.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer Science + Business Media, LLC
About this chapter
Cite this chapter
Stein, O. (2006). A semi-infinite approach to design centering. In: Dempe, S., Kalashnikov, V. (eds) Optimization with Multivalued Mappings. Springer Optimization and Its Applications, vol 2. Springer, Boston, MA . https://doi.org/10.1007/0-387-34221-4_10
Download citation
DOI: https://doi.org/10.1007/0-387-34221-4_10
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-34220-7
Online ISBN: 978-0-387-34221-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)