Abstract
We show that a familiar constraint qualification of differentiable programming has “nonsmooth” counterparts. As a result, necessary optimality conditions of Kuhn—Tucker type can be established for inequality-constrained mathematical programs involving functions not assumed to be differentiable, convex, or locally Lipschitzian. These optimality conditions reduce to the usual Karush—Kuhn—Tucker conditions in the differentiable case and sharpen previous results in the locally Lipschitzian case.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J.-P. Aubin, “Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions,” in: L. Nachbin, ed.,Mathematical Analysis and Applications, Part A, Advances in Mathematics, Vol. 7A (Academic Press, New York, 1981) pp. 159–229.
M.S. Bazaraa, J.J. Goode and Z. Nashed, “On the cones of tangents with applications to mathematical programming,”Journal of Optimization Theory and Applications 13 (1974) 389–426.
M.S. Bazaraa and C.M. Shetty,Nonlinear Programming: Theory and Algorithms (Wiley, New York, 1979).
J.M. Borwein and H.M. Strojwas, “Directionally Lipschitzian mappings on Baire spaces,”Canadian Journal of Mathematics 36 (1984) 95–130.
F.H. Clarke,Optimization and Nonsmooth Analysis (Wiley, New York, 1983).
S. Dolecki, “Tangency and differentiation: some applications of convergence theory,”Annali di Matematica pura ed applicata 130 (1982) 223–255.
H. Frankowska, “Necessary conditions for the Bolza problem,”Mathematics of Operations Research 10 (1985) 361–366.
M. Guignard, “Generalized Kuhn-Tucker conditions for mathematical programming problems in a Banach space,”SIAM Journal on Control and Optimization 7 (1969) 232–241.
J.-B. Hiriart-Urruty, “On optimality conditions in nondifferentiable programming,”Mathematical Programming 14 (1978) 73–86.
J.-B. Hiriart-Urruty, “Tangent cones, generalized gradients and mathematical programming in Banach space,”Mathematics of Operations Research 4 (1979) 79–97.
A.D. Ioffe, “Necessary conditions for a local minimum. 1. A reduction theorem and first-order conditions,”SIAM Journal on Control and Optimization 17 (1979) 245–250.
V. Jeyakumar, “On optimality conditions in nonsmooth inequality-constrained minimization,”Numerical Functional Analysis and Optimization 9 (1987) 535–546.
D.H. Martin and G.G. Watkins, “Cores of tangent cones and Clarke's tangent cone,”Mathematics of Operations Research 10 (1985) 565–575.
R.R. Merkovsky and D.E. Ward, “Upper D.S.L. approximates and nonsmooth optimization,”Optimization 21 (1990), to appear.
V.H. Nguyen, J.-J. Strodiot and R. Mifflin, “On conditions to have bounded multipliers in locally Lipschitz programming,”Mathematical Programming 18 (1980) 100–106.
J.-P. Penot, “Calcul sous-differentiel et optimisation,”Journal of Functional Analysis 27 (1978) 248–276.
J.-P. Penot, “Variations on the theme of nonsmooth analysis: another subdifferential,” in: V.F. Demyanov and D. Pallaschke, eds.,Nondifferentiable Optimization: Motivations and Applications (Springer, Berlin, 1985) pp. 41–54.
R. T. Rockafellar, “Directionally Lipschitzian functions and subdifferential calculus,”Proceedings of the London Mathematical Society 39 (1979) 331–355.
R.T. Rockafellar, “Generalized directional derivatives and subgradients of nonconvex functions,”Canadian Journal of Mathematics 32 (1980) 257–280.
R.T. Rockafellar,The Theory of Subgradients and its Applications to Problems of Optimization: Convex and Nonconvex Functions (Heldermann, Berlin, 1981).
J.-J. Strodiot and V.H. Nguyen, “Kuhn-Tucker multipliers and nonsmooth programs,”Mathematical Programming Study 19 (1982) 222–240.
C. Ursescu, “Tangent sets' calculus and necessary conditions for extremality,”SIAM Journal on Control and Optimization 20 (1982) 563–574.
D.E. Ward, “Convex subcones of the contingent cone in nonsmooth calculus and optimization,”Transactions of the AMS 302 (1987) 661–682.
D.E. Ward, “Isotone tangent cones and nonsmooth optimization,”Optimization 18 (1987) 769–783.
D.E. Ward, “Directional derivative calculus and optimality conditions in nonsmooth mathematical programming,”Journal of Information and Optimization Sciences 10 (1989) 81–96.
D.E. Ward, “The quantificational tangent cones,”Canadian Journal of Mathematics 40 (1988) 666–694.
D.E. Ward and J.M. Borwein, “Nonsmooth calculus in finite dimensions,”SIAM Journal on Control and Optimization 25 (1987) 1312–1340.
C. Zalinescu, “Solvability results for sublinear functions and operators,”Zeitschrift für Operations Research 31 (1987) A79-A101.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Merkovsky, R.R., Ward, D.E. General constraint qualifications in nondifferentiable programming. Mathematical Programming 47, 389–405 (1990). https://doi.org/10.1007/BF01580871
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01580871