Abstract
The object of this paper is to present new theorems on the differentiability of an Infimum, Supremum, Min Max, or saddle point with respect to a parameter t≥0 at t=0 under relaxed assumptions. Those technical results have a wide spectrum of applications: Control Theory, Shape Sensitivity Analysis, Game Theory, etc... In non-differentiable situations they provide an interesting description of the non-differentiability.
The research of the first author has been supported by a Killam Fellowship from Canada Council, Natural Sciences and Engineering Research Council of Canada operating grant A-8730, and a FCAR grant from the “Ministère de l'éducation du Québec”.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
R. Correa and A. Seeger, Directional derivatives of a minimax function, Nonlinear Analysis, Theory, Methods and Applications 9 (1985), 13–22.
M.C. Delfour and J. Morgan, “A complement to the differentiability of saddle points and Min Max,” CRM Report CRM-1682, Université de Montréal, Canada, July 1990.
M.C. Delfour and J.-P. Zolésio, Dérivation d'un MinMax et application à la dérivation par rapport au contrôle d'une obscrvation non-différentiable de l'état, C.R. Acad. Sc. Paris, t.302, Sér. I, no. 16 (1986), 571–574.
, Shape Sensitivity Analysis via MinMax Differentiability, SIAM J. on Control and Optimization 26 (1988), 834–862.
, Differentiability of a MinMax and Application to Optimal Control and Design Problems, Part I, in “Control Problems for Systems Described as Partial Differential Equations and Applications,” I. Lasiecka and R. Triggiani, eds., Springer-Verlag, New York, 1987, pp. 204–219.
, Differentiability of a MinMax and Application to Optimal Control and Design Problems, Part II, in “Control Problems for Systems Described as Partial Differential Equations and Applications,” I. Lasiecka and R. Triggiani, eds., Springer-Verlag, New York, 1987, pp. 220–229.
, Further Developments in Shape Sensitivity Analysis via a Penalization Method, in “Boundary Control and Boundary Variations,” J. P. Zolésio, ed., Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1988, pp. 153–191.
, Shape Sensitivity Analysis via a Penalization Method, Annali di Matematica Pura ed Applicata CLI (1988), 179–212.
, Analyse des problèmes de forme par la dérivation des Min Max, in “Analyse Non Linéaire,” H. Attouch, J.P. Aubin, F.H. Clarke and I. Ekeland, eds, Série Analyse Non Linéaire, Annales de l'Institut Henri-Poincaré, Special volume in honor of J.-J. Moreau, Gauthier-Villars, Bordas, Paris, France, 1989, pp. 211–228.
M.C. Delfour J.-P. Zolésio, Anatomy of the shape Hessian, Annali di Matematica Pura et Applicata CLVIII (1989).
M.C. Delfour J.-P. Zolésio, Computation of the shape Hessian by a Lagrangian method, in “Fifth Symp. on Control of Distributed Parameter Systems,” A. El Jai and M. Amouroux, eds., Pergamon Press, 1989, pp. 85–90.
, Shape Hessian by the velocity method: a Lagrangian approach, in “Stabilization of Flexible Structures,” J.P. Zolésio, ed., Springer-Verlag, Berlin, New-York, 1990, pp. 255–279.
M.C. Delfour J.-P. Zolésio, Velocity method and Lagrangian formulation in the computation of the shape Hessian, SIAM J. on Control and Optimization 29 6 (1991) (to appear).
B. Lemaire, “Problèmes Min-Max et applications au contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles linéaires,” Thèse de doctorat d'état, Montpellier, France, 1970.
Lignola and J. Morgan, Topological existence and stability for MinSup problems, J. Math. Anal. Appl. (to appear).
J. Morgan, Constrained well posed two level optimization problems, in “E. Majorana School (Erice) on Non Smooth Optimization,” Plenum Press, 1989.
J. P. Zolésio, Domain variational formulation for free boundary problems, in “Optimization of Distributed Parameter Structures, vol II,” E.J. Haug and J. Céa, eds., Sijhofff and Nordhoff, Alphen aan den Rijn, The Netherlands, 1981, pp. 1089–1151.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1992 International Federation for Information Processing
About this paper
Cite this paper
Delfour, M.C., Morgan, J. (1992). Differentiability of Min Max and saddle points under relaxed assumptions. In: Zoléesio, J.P. (eds) Boundary Control and Boundary Variation. Lecture Notes in Control and Information Sciences, vol 178. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0006694
Download citation
DOI: https://doi.org/10.1007/BFb0006694
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-55351-9
Online ISBN: 978-3-540-47029-8
eBook Packages: Springer Book Archive