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Generalized Directional Derivatives and Subgradients of Nonconvex Functions

Published online by Cambridge University Press:  20 November 2018

R. T. Rockafellar
R. T. Rockafellar*
Affiliation:
University of Washington, Seattle, Washington
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Studies of optimization problems and certain kinds of differential equations have led in recent years to the development of a generalized theory of differentiation quite distinct in spirit and range of application from the one based on L. Schwartz's “distributions.” This theory associates with an extended-real-valued function ƒ on a linear topological space E and a point x ∈ E certain elements of the dual space E* called subgradients or generalized gradients of ƒ at x. These form a set ∂ƒ(x) that is always convex and weak*-closed (possibly empty). The multifunction ∂ƒ: x →∂ƒ(x) is the sub differential of ƒ.

Rules that relate ∂ƒ to generalized directional derivatives of ƒ, or allow ∂ƒ to be expressed or estimated in terms of the subdifferentials of other functions (whenƒ = ƒ1 + ƒ2,ƒ = g o A, etc.), comprise the sub differential calculus.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 2 , February 1980 , pp. 257 - 280
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Aubin, J.-P., Gradients generalises de Clarke, lecture notes, Centre de Recherches Mathématiques, Université de Montréal, 1977.Google Scholar
2. Aubin, J.-P. and Clarke, F. H., Multiplicateur de Lagrange en optimisation non convexe et applications, C.R. Acad. Sci. Pari. 285 (1977), 451–454.Google Scholar
3. Berge, C., Espaces topologiques, fonctions multivoques (Dunod, Paris, 1959).Google Scholar
4. Browder, F. E., Multivalued monotone nonlinear mappings and duality mappings in Banach spaces, Trans. A.M.S. 118 (1965), 338–351.Google Scholar
5. Chaney, R. and Goldstein, A. A., On directions of steepest descent (to appear).Google Scholar
6. Clarke, F. H., Necessary conditions for nonsmooth problems in optimal control and the calculus of variations, Thesis, University of Washington, Seattle (1973).Google Scholar
7. Clarke, F. H., Generalized gradients and applications, Trans. M.M.S. 205 (1975), 247–262.Google Scholar
8. Clarke, F. H., Generalized gradients of Lipschitz Junctionals, Advances in Math, (to appear). (Available as Tech. Report 1687, Math. Research Center, University of Wisconsin, Madison, 1976.)Google Scholar
9. Clarke, F. H., On the inverse function theorem, Pacific J. Math. 64 (1976), 97–102.Google Scholar
10. Clarke, F. H., A new approach to Lagrange multipliers, Math, of Op. Researc. 1 (1976), 165–174.Google Scholar
11. Clarke, F. H., La condition hamiltonienne d'optimalité, C. R. Acad. Sci. Pari. 280 (1975), 1205–1207.Google Scholar
12. Clarke, F. H., The Euler-Lagrange differential inclusion, J. Diff. Eq. 19 (1975), 80–90.Google Scholar
13. Clarke, F. H., Optimal solutions to differential inclusions, J. Opt. Theory Appl. 19 (1976), 469–478.Google Scholar
14. Clarke, F. H., The generalized problem of Boka, SIAM J. Control. Opt. 14 (1976), 682–699.Google Scholar
15. Clarke, F. H., The maximum principle under minimal hypotheses, SIAM J. Control Opt. 14 (1976), 1078–1091.Google Scholar
16. Clarke, F. H., Necessary conditions for a general control problem, Calculus of variations and control theory (Academic Press, 1976), 257–278.Google Scholar
17. Clarke, F. H., Inequality constraints in the calculus of variations, Can. J. Math. 39 (1977), 528–540.Google Scholar
18. Clarke, F. H., Multiple integrals of Lipschitz functions in the calculus of variations, Proc. A.M.S. 64 (1977), 260–264.Google Scholar
19. Clarke, F. H., The Erdmann condition and Hamiltonian inclusions in optimal control and the calculus of variations, SIAM J. Control Opt.Google Scholar
20. Clarke, F. H., Optimal control and the true Hamiltonian, SIAM Review (to appear).Google Scholar
21. Feuer, A., An implementable mathematical programming algorithm for admissible fundamental functions, Thesis, Columbia University (1974).Google Scholar
22. Feuer, A., Minimizing well-behaved functions, Proceedings 12th Annual Alberton Conference on Circuit and System Theory (University of Illinois Press, 1974), 25–33.Google Scholar
23. Gauvin, J., The generalized gradient of a marginal function in mathematical programming, Math, of Op. Research (to appear).Google Scholar
24. Goldstein, A. A., Optimization of Lipschitz continuous functions, Math. Prog. 13 (1977), 14–22.Google Scholar
25. Hausdorff, F., Mengenlehre, 3rd edition (Springer, Berlin, 1927).Google Scholar
26. Hiriart-Urruty, J.-B., Contributions a la programmation mathématique: Cas déterministe et stochastique, Thèse, Université de Clermont-Ferrand II (1977).Google Scholar
27. Hiriart-Urruty, J.-B., Tangent cones, generalized gradients and mathematical programming in Banach spaces, Math, of Op. Research (to appear).Google Scholar
28. Hiriart-Urruty, J.-B., Conditions nécessaires d'optimalité en programmation non diffêrentiable, C. R. Acad. Sci. Pari. 283 (1976), 843–845.Google Scholar
29. Hiriart-Urruty, J.-B., On optimality conditions in nondifferentiable programming, Math. Prog, (to appear).Google Scholar
30. Hiriart-Urruty, J.-B., Gradients généralisés de fonctions marginales, SIAM J. Control Opt. (to appear).Google Scholar
31. Hôrmander, L., Sur la fonction d'appui des ensembles convexes dans un espace localement convexe, Arkiv for Math. 3 (1954), 181–186.Google Scholar
32. Iofïe, A. D., Estimates of the distance to level sets, SIAM J. Control Opt. (to appear).Google Scholar
33. Iofïe, A. D., Regular points of Lipschitz mappings, Trans. A.M.S. (to appear).Google Scholar
34. Iofïe, A. D., Necessary and sufficient conditions for a local minimum. 1. A reduction theorem and first-order conditions, SIAM J. Control Opt. 17 (1979), 245–250.Google Scholar
35. Iofïe, A. D., Necessary and sufficient conditions for a local minimum. 2. Conditions of Levitin- Miljutin-Osmolovskii-type, SIAM J. Control Opt. (to appear).Google Scholar
36. Iofïe, A. D., Necessary and sufficient conditions for a local minimum. 3. Second-order conditions and augmented duality, SIAM J. Control Opt. (to appear).Google Scholar
37. Lebourg, G., Valeur moyenne pour gradient généralisé, C. R. Acad. Aci. Paris, Ser. A. 281 (1975), 795–797.Google Scholar
38. McLinden, L., Dual operations on saddle functions, Trans. A.M.S. 179 (1973), 363–381.Google Scholar
39. Mifflin, R., Semismooth and semiconvex functions in constrained optimization, SIAM J. Control Opt. 15 (1977), 959–972.Google Scholar
40. Mifflin, R., An algorithm for constrained optimization with semismooth functions, Math, of Op. Researc. 2 (1977), 191–207.Google Scholar
41. Minty, G. J., Monotone (nonlinear) operators in Hilbert space, Duke Math. J. 29 (1962), 341–346.Google Scholar
42. Moreau, J. J., Fonctionelles convexes, lecture notes, Séminaire “Equations aux Dérivées Partielles,” Collège de France, 1966.Google Scholar
43. Rockafellar, R. T., Convex functions and dual extremum problems, thesis, Harvard (1963).Google Scholar
44. Rockafellar, R. T., Convex analysis (Princeton University Press, 1970).Google Scholar
45. Rockafellar, R. T., Conjugate duality and optimization, Regional Conference Series in Applied Math. 16 (S.I.A.M. Publications, Philadelphia, 1974).Google Scholar
46. Rockafellar, R. T., On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970), 209–216.Google Scholar
47. Rockafellar, R. T., Monotone operators associated with saddle functions and minimax problems, Nonlinear functional analysis, Part 1, Symposia in Pure Math. 18 (Amer. Math. Soc, Providence, R.I., 1970), 397–407.Google Scholar
48. Rockafellar, R. T., On the maximality of sums of nonlinear monotone operators, Trans. A.M.S. 149 (1970), 75–88.Google Scholar
49. Rockafellar, R. T., Clarke's tangent cones and the boundaries of closed sets in Rz, Nonlin. Analysis. Th. Meth. Appl. 3 (1979), 145–154.Google Scholar
50. Rockafellar, R. T., Directionally Lipschitzian functions and subdifferential calculus, Proc. London Math. Soc. 39 (1979), 331–355.Google Scholar
51. Rockafellar, R. T., Dual problems of Lagrange for arcs of bounded variation, Calculus of variations and control theory (Academic Press, 1976), 155–192.Google Scholar
52. Rockafellar, R. T., Conjugate convex functions in optimal control and the calculus of variations, J. Math. Anal. Appl. 32 (1970), 174–222.Google Scholar
53. Rockafellar, R. T. and Wets, R. J.-B., The optimal recourse problem in discrete time: Ll-multipliers for inequality constraints, SIAM J. Control Opt. 16 (1978).Google Scholar
54. Saks, S., Theory of the integral (Hafner Publishing Co., New York, second revised edition, 1937).Google Scholar
55. Thibault, L., Propriétés des s ou s-différentiel s de fonctions localement Lip s chiliennes définies sur un espace de Banach separable. Applications, Thèse, Université des Sciences et Techniques du Languedoc, Monpellier (1976).Google Scholar