×

General constraint qualifications in nondifferentiable programming. (English) Zbl 0708.90078

A general inequality-constrained minimization program is considered on a real normed space E. The constraints are given by a finite number of inequalities and an explicit set C. The paper deals with necessary conditions of Kuhn-Tucker type for a local minimum. The essential classical cases are mentioned and well-known constraints qualifications are compared. The authors present nonsmooth analogs of the Cottle constraint qualification in which gradients are replaced by directional derivatives associated with closed tangent cones. Using these constraint qualifications, they get the optimality conditions involving subgradients associated with closed convex tangent cones. The latter may be Clarke tangent or Penot’s prototangent or some other cones. The authors define the class of directionally Lipschitzian functions containing the known class of locally Lipschitzian ones. This allows to improve results of some other authors by obtaining sharper optimality conditions. In case \(C=E\) the equivalence between the constraint qualification offered in the paper and the boundedness and nonemptiness of the Kuhn-Tucker multipliers set is established under locally Lipschitzian conditions.
Reviewer: N.Novikova

MSC:

90C30 Nonlinear programming
49J52 Nonsmooth analysis
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] 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. · Zbl 0484.47034
[2] 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. · Zbl 0259.90037 · doi:10.1007/BF00934938
[3] M.S. Bazaraa and C.M. Shetty,Nonlinear Programming: Theory and Algorithms (Wiley, New York, 1979).
[4] J.M. Borwein and H.M. Strojwas, ”Directionally Lipschitzian mappings on Baire spaces,”Canadian Journal of Mathematics 36 (1984) 95–130. · Zbl 0534.46031 · doi:10.4153/CJM-1984-008-7
[5] F.H. Clarke,Optimization and Nonsmooth Analysis (Wiley, New York, 1983). · Zbl 0582.49001
[6] S. Dolecki, ”Tangency and differentiation: some applications of convergence theory,”Annali di Matematica pura ed applicata 130 (1982) 223–255. · Zbl 0518.49009 · doi:10.1007/BF01761497
[7] H. Frankowska, ”Necessary conditions for the Bolza problem,”Mathematics of Operations Research 10 (1985) 361–366. · Zbl 0589.49011 · doi:10.1287/moor.10.2.361
[8] M. Guignard, ”Generalized Kuhn-Tucker conditions for mathematical programming problems in a Banach space,”SIAM Journal on Control and Optimization 7 (1969) 232–241. · Zbl 0182.53101 · doi:10.1137/0307016
[9] J.-B. Hiriart-Urruty, ”On optimality conditions in nondifferentiable programming,”Mathematical Programming 14 (1978) 73–86. · Zbl 0373.90071 · doi:10.1007/BF01588951
[10] J.-B. Hiriart-Urruty, ”Tangent cones, generalized gradients and mathematical programming in Banach space,”Mathematics of Operations Research 4 (1979) 79–97. · Zbl 0409.90086 · doi:10.1287/moor.4.1.79
[11] 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. · Zbl 0417.49027 · doi:10.1137/0317019
[12] V. Jeyakumar, ”On optimality conditions in nonsmooth inequality-constrained minimization,”Numerical Functional Analysis and Optimization 9 (1987) 535–546. · Zbl 0611.90081 · doi:10.1080/01630568708816246
[13] D.H. Martin and G.G. Watkins, ”Cores of tangent cones and Clarke’s tangent cone,”Mathematics of Operations Research 10 (1985) 565–575. · Zbl 0584.49008 · doi:10.1287/moor.10.4.565
[14] R.R. Merkovsky and D.E. Ward, ”Upper D.S.L. approximates and nonsmooth optimization,”Optimization 21 (1990), to appear. · Zbl 0711.90086
[15] 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. · Zbl 0437.90075 · doi:10.1007/BF01588302
[16] J.-P. Penot, ”Calcul sous-differentiel et optimisation,”Journal of Functional Analysis 27 (1978) 248–276. · Zbl 0404.90078 · doi:10.1016/0022-1236(78)90030-7
[17] 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. · Zbl 0598.49008
[18] R. T. Rockafellar, ”Directionally Lipschitzian functions and subdifferential calculus,”Proceedings of the London Mathematical Society 39 (1979) 331–355. · Zbl 0413.49015 · doi:10.1112/plms/s3-39.2.331
[19] R.T. Rockafellar, ”Generalized directional derivatives and subgradients of nonconvex functions,”Canadian Journal of Mathematics 32 (1980) 257–280. · Zbl 0447.49009 · doi:10.4153/CJM-1980-020-7
[20] R.T. Rockafellar,The Theory of Subgradients and its Applications to Problems of Optimization: Convex and Nonconvex Functions (Heldermann, Berlin, 1981). · Zbl 0462.90052
[21] J.-J. Strodiot and V.H. Nguyen, ”Kuhn-Tucker multipliers and nonsmooth programs,”Mathematical Programming Study 19 (1982) 222–240. · Zbl 0501.90074
[22] C. Ursescu, ”Tangent sets’ calculus and necessary conditions for extremality,”SIAM Journal on Control and Optimization 20 (1982) 563–574. · Zbl 0488.49009 · doi:10.1137/0320041
[23] D.E. Ward, ”Convex subcones of the contingent cone in nonsmooth calculus and optimization,”Transactions of the AMS 302 (1987) 661–682. · Zbl 0629.58007 · doi:10.1090/S0002-9947-1987-0891640-2
[24] D.E. Ward, ”Isotone tangent cones and nonsmooth optimization,”Optimization 18 (1987) 769–783. · Zbl 0633.49012 · doi:10.1080/02331938708843290
[25] D.E. Ward, ”Directional derivative calculus and optimality conditions in nonsmooth mathematical programming,”Journal of Information and Optimization Sciences 10 (1989) 81–96. · Zbl 0682.49016
[26] D.E. Ward, ”The quantificational tangent cones,”Canadian Journal of Mathematics 40 (1988) 666–694. · Zbl 0648.58004 · doi:10.4153/CJM-1988-029-6
[27] D.E. Ward and J.M. Borwein, ”Nonsmooth calculus in finite dimensions,”SIAM Journal on Control and Optimization 25 (1987) 1312–1340. · Zbl 0633.46043 · doi:10.1137/0325072
[28] C. Zalinescu, ”Solvability results for sublinear functions and operators,”Zeitschrift für Operations Research 31 (1987) A79-A101. · Zbl 0695.49009 · doi:10.1007/BF01259338
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.