Abstract
We study the first-order behaviour of the optimal value function associated to a convex parametric problem of calculus of variations. An important feature of this paper is that we do not assume the existence of optimal trajectories for the unperturbed problem. The concepts of approximate Euler-Lagrange inclusion and approximate transversality condition are key ingredients in the writing of our sensitivity results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Attouch, H. and Brezis, H.:Duality for the sum of convex functions in general Banach spaces, in J. Barrosso, (ed),Aspects of Mathematics and its Applications, North Holland, Amsterdam, 1986, pp. 125–133.
Bustos, M.: Conditions d'optimalité à ɛ-près dans un problème d'optimisation non différentiable, Thesis, Université Paul Sabatier, Toulouse, 1989.
Guerraggio, A. and Salsa, S.:Metodi matematici per l'economia e le scienze sociali, G.Giappichelli Editore, Torino, 1988.
Heukemes, N., Nguyen, V.H., and Strodiot, J.-J.: ɛ-optimal solutions in nondifferentiable convex programming and some related questions,Math. Programming 25 (1983), 307–328.
Hiriart-Urruty, J.-B.: Contributions à la programmation mathématique: cas déterministe et stochastique. Thesis, Université de Clermont-Ferrand, 1977.
Hiriart-Urruty, J.-B.: ɛ-subdifferential calculus, in J.-P. Aubin and R. B. Vinter (eds),Convex Analysis and Optimization, Notes in Math. 57, Pitman, Boston, (1982), pp. 43–92.
Hiriart-Urruty, J.-B.: From convex optimization to nonconvex optimization, in F. H. Clarke, V. F. Demyanov, and F. Giannessi (eds),Nonsmooth Optimizationr and Related Topics Plenum Press, New York, 1989, pp. 219–239.
Hiriart-Urruty, J.-B. and Lemarechal, C.: Testing necessary and sufficient conditions for global optimality in the problem of maximizing a convex quadratic function over a convex polyhedron, Preprint, Laboratoire d'Analyse Numérique, Université Paul Sabatier, Toulouse, 1990.
Kusraev, A. G. and Kutateladze, S. S.:Subdifferential Calculus, Nauka, Novosibirsk, 1987 (in Russian).
Kutateladze, S. S.: Convex ɛ-programming,Soviet Math. Dokl. 20 (1979), 391–393.
Martinez-Legaz, J.-E. and Seeger, A.: A formula on the approximate subdifferential of the difference of convex functions,Bull. Austral. Math. Soc. 45 (1992), 37–41.
Moussaoui, M. and Seeger, A.: Sensitivity analysis of optimal value functions of convex parametric problems with possibly empty solution sets, Preprint, Department of Mathematics, University of Avignon, March 1992, to appear inSIAM J. Optim.
Rockafellar, R. T.: Saddle points of Hamiltonian systems in convex problems of Lagrange,J. Optim. Theory Appl. 12 (1973), 367–390.
Rockafellar, R. T.: Conjugate convex functions in optimal control and the calculus of variation,J. Math. Anal. Appl. 32 (1970), 174–222.
Rockafellar, R. T.:Conjugate Duality and Optimization, Regional Conference Series in Applied Mathematics Vol. 16, SIAM Publications, 1973.
Rockafellar, R. T.: Dualisation of subgradient conditions for optimality, Preprint, Department of Mathematics, University of Washington, Seattle, 1991.
Thera, M.: Calcul ɛ-sous-différentiel des applications convexes vectorielles,C. R. Acad. Sci. Paris 290 (1980), 549–551.
Zalinescu, C.: Stability for a class of nonlinear optimization problems and applications, in F. H. Clarke, V. F. Demyanov, and F. Giannessi (eds),Nonsmooth Optimization and Related Topics, Plenum Press, New York, 1989, pp. 437–458.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Seeger, A. Approximate Euler-Lagrange inclusion, approximate transversality condition, and sensitivity analysis of convex parametric problems of calculus of variations. Set-Valued Anal 2, 307–325 (1994). https://doi.org/10.1007/BF01027108
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01027108