58
CHAPITRE 2. CONDITIONS D’OPTIMALITÉ APPROCHÉE
conditions), et bornée inférieurement sur H . Montrer qu’il existe alors une
suite (x k ) telle que :
f (x k ) → inf
H
f et ∇ f (x k ) → 0 quand k → +∞.
Références
[E1] I. Ekeland. "On the variational principle". J. Math. Anal. Appl. 47
(1974), p. 324–353.
[E2] I. Ekeland. "Nonconvex minimization problems". Bull. Amer. Math.
Soc. 1 (1979), p. 443–474.
[F] D.G. De Figueiredo. Lectures on the Ekeland Variational Principle with
Applications and Detours. Tata Institute of Fundamental Research, Bombay, 1989.
[BP] J.M. Borwein and D. Preiss. "A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions". Trans. Amer. Math. Soc. 303 (1987), p. 517–527.
[L] P.D. Loewen. Optimal Control Via Nonsmooth Analysis. CRM Proceedings & Lecture notes, American Mathematical Society, 1993.
[CLSW] F.H. Clarke, Yu.S. Ledyaev, R.J. Stern and P.R. Wolenski. Nonsmooth Analysis and Control Theory. Graduate texts in mathematics,
Springer Verlag, 1998.
[FHV] M. Fabian, P. Hájek and J. Vanderwerff. "On smooth variational principles in Banach spaces". J. Math. Anal. Appl. 197 (1996), p. 153–173.
[St] C. Stegall. "Optimization of functions on certain subsets of Banach
spaces". Math. Ann. 236 (1978), p. 171–176.
[DGZ] R. Deville, G. Godefroy and V.E. Zizler. "A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions". J. Funct. Anal. 111 (1993), p. 192–212.
[Sc] W. Schirotzek. Nonsmooth Analysis. Universitext, Springer Verlag,
2007.
[BZ] J.M. Borwein and Q.J. Zhu. Techniques of Variational Analysis. CMB
books in mathematics, Springer Verlag, 2005.
Nous signalons les articles d’origine... il vaut mieux souvent revenir aux
sources. L’article-revue [E2] reste, trente après sa publication, une très
bonne référence pour l’énoncé et quelques-unes des premières applications
du principe variationnel d’Ekeland. Notre § 2, sur le principe variationnel
de Borwein-Preiss est tiré de ([L], Chap. 3).
CHAPITRE 2. CONDITIONS D’OPTIMALITÉ APPROCHÉE
conditions), et bornée inférieurement sur H . Montrer qu’il existe alors une
suite (x k ) telle que :
f (x k ) → inf
H
f et ∇ f (x k ) → 0 quand k → +∞.
Références
[E1] I. Ekeland. "On the variational principle". J. Math. Anal. Appl. 47
(1974), p. 324–353.
[E2] I. Ekeland. "Nonconvex minimization problems". Bull. Amer. Math.
Soc. 1 (1979), p. 443–474.
[F] D.G. De Figueiredo. Lectures on the Ekeland Variational Principle with
Applications and Detours. Tata Institute of Fundamental Research, Bombay, 1989.
[BP] J.M. Borwein and D. Preiss. "A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions". Trans. Amer. Math. Soc. 303 (1987), p. 517–527.
[L] P.D. Loewen. Optimal Control Via Nonsmooth Analysis. CRM Proceedings & Lecture notes, American Mathematical Society, 1993.
[CLSW] F.H. Clarke, Yu.S. Ledyaev, R.J. Stern and P.R. Wolenski. Nonsmooth Analysis and Control Theory. Graduate texts in mathematics,
Springer Verlag, 1998.
[FHV] M. Fabian, P. Hájek and J. Vanderwerff. "On smooth variational principles in Banach spaces". J. Math. Anal. Appl. 197 (1996), p. 153–173.
[St] C. Stegall. "Optimization of functions on certain subsets of Banach
spaces". Math. Ann. 236 (1978), p. 171–176.
[DGZ] R. Deville, G. Godefroy and V.E. Zizler. "A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions". J. Funct. Anal. 111 (1993), p. 192–212.
[Sc] W. Schirotzek. Nonsmooth Analysis. Universitext, Springer Verlag,
2007.
[BZ] J.M. Borwein and Q.J. Zhu. Techniques of Variational Analysis. CMB
books in mathematics, Springer Verlag, 2005.
Nous signalons les articles d’origine... il vaut mieux souvent revenir aux
sources. L’article-revue [E2] reste, trente après sa publication, une très
bonne référence pour l’énoncé et quelques-unes des premières applications
du principe variationnel d’Ekeland. Notre § 2, sur le principe variationnel
de Borwein-Preiss est tiré de ([L], Chap. 3).