Abstract
In this paper we consider a nonconvex and nondifferentiable (but locally Lipschitz) program with inequality constraints. We study the question of existence of bounded Kuhn-Tucker multipliers for this program in not assuming that a constraint qualification such as Hiriart-Urruty’s condition (U) is satisfied. The present work may be viewed as a complement of the investigations concerned with optimality conditions in nonsmooth optimization by Clarke and Hiriart-Urruty. Our results are shown, when specialized to the case of convexity, to yield naturally a complete characterization of optimality of Ben-Israel, Ben-Tal and Zlobec.
Our approach here is new and enables us to treat the general class of locally Lipschitz problems. The newness of the approach is demonstrated to be effective for the convex case.
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References
R. Abrams and L. Kerzner, “A simplified test for optimality”, Journal of Optimization Theory and Applications 25 (1978) 161–170.
A. Ben-Israel, A. Ben-Tal and S. Zlobec, “Optimality conditions in convex programming”, in: A. Prékopa, ed., Survey of mathematical programming (North-Holland, Amsterdam, 1979) pp. 153–169.
A. Ben-Tal and A. Ben-Israel, “Characterizations of optimality in convex programming: The nondifferentiable case”, Applicable Analysis 9 (1979) 137–156.
A. Ben-Tal, A. Ben-Israel and S. Zlobec, “Characterization of optimality in convex programming without a constraint qualification”, Journal of Optimization Theory and Applications 20 (1976) 417–437.
F.H. Clarke, “Generalized gradients and applications”, Transactions of the American Mathematical Society 205 (1975) 247–262.
F.H. Clarke, “A new approach to Lagrange multipliers”, Mathematics of Operations Research 1 (1976) 165–174.
F.H. Clarke, “Generalized gradients of Lipschitz functionals”, Mathematics Research Center Technical Summary Report No. 1687, University of Wisconsin, Madison (1976).
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 spaces”, Mathematics of Operations Research 4 (1979) 79–97.
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.
R.T. Rockafellar, Convex analysis (Princeton University Press, Princeton, NJ, 1970).
R.T. Rockafellar, “Clarke’s tangent cones and the boundaries of closed sets in R n”, Nonlinear Analysis 3 (1979) 145–154.
R.T. Rockafellar, “Directionally Lipschitzian functions and subdifferential calculus”, Proceedings of the London Mathematical Society 39 (1979) 331–355.
H. Wolkowicz, “Geometry of optimality conditions and constraint qualifications: The convex case”, Mathematical Programming 19 (1980) 32–60.
S. Zlobec and B. Craven, “Stabilization and calculation of the minimal index of binding constraints”, Mathematische Operationsforschung und Statistik, Series Optimization 12 (1981) 203–220.
S. Zlobec and D.H. Jacobson, “Minimizing an arbitrary function subject to convex constraints”, Utilitas Mathematica 17 (1980) 239–257.
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© 1982 The Mathematical Programming Society, Inc.
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Strodiot, JJ., Hien Nguyen, V. (1982). Kuhn-tucker multipliers and nonsmooth programs. In: Guignard, M. (eds) Optimality and Stability in Mathematical Programming. Mathematical Programming Studies, vol 19. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0120990
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DOI: https://doi.org/10.1007/BFb0120990
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