Characterizations of strict local minima and necessary conditions for weak sharp minima. (English) Zbl 0797.90101

Summary: In nonlinear programming, sufficient conditions of order \(m\) usually identify a special type of local minimizers, here termed a strict local minimizer of order \(m\). It is demonstrated that, if a constraint qualification is satisfied, standard sufficient conditions often characterize this special sort of minimizer. The first- and second-order cases are treated in detail. Necessary conditions for weak sharp local minima of order \(m\), a larger class of local minima, are also presented.


90C30 Nonlinear programming
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[1] Cromme, L.,Strong Uniqueness: A Far Reaching Criterion for the Convergence of Iterative Procedures, Numerische Mathematik, Vol. 29, pp. 179–193, 1978. · Zbl 0352.65012 · doi:10.1007/BF01390337
[2] Auslender, A.,Stability in Mathematical Programming with Nondifferentiable Data, SIAM Journal on Control and Optimization, Vol. 22, pp. 239–254, 1984. · Zbl 0538.49020 · doi:10.1137/0322017
[3] Studniarski, M.,Necessary and Sufficient Conditions for Isolated Local Minima of Nonsmooth Functions, SIAM Journal on Control and Optimization, Vol. 24, pp. 1044–1049, 1986. · Zbl 0604.49017 · doi:10.1137/0324061
[4] Hiriart-Urruty, J. B., Strodiot, J. J., andNguyen, V. H.,Generalized Hessian Matrix and Second-Order Optimality Conditions for Problems with C 1,1 Data, Applied Mathematics and Optimization, Vol. 11, pp. 43–56, 1984. · Zbl 0542.49011 · doi:10.1007/BF01442169
[5] Klatte, D., andTammer, K.,On Second-Order Sufficient Optimality Conditions for C 1,1 Optimization Problems, Optimization, Vol. 19, pp. 169–179, 1988. · Zbl 0647.49014 · doi:10.1080/02331938808843333
[6] Burke, J. V., andFerris, M. C.,Weak Sharp Minima in Mathematical Programming, SIAM Journal on Control and Optimization, Vol. 31, pp. 1340–1359, 1993. · Zbl 0791.90040 · doi:10.1137/0331063
[7] Ward, D. E.,Isotone Tangent Cones and Nonsmooth Optimization, Optimization, Vol. 18, pp. 769–783, 1987. · Zbl 0633.49012 · doi:10.1080/02331938708843290
[8] Ward, D. E.,The Quantificational Tangent Cones, Canadian Journal of Mathematics, Vol. 40, pp. 666–694, 1988. · Zbl 0648.58004 · doi:10.4153/CJM-1988-029-6
[9] Ursescu, C.,Tangent Set’s Calculus and Necessary Conditions for Extremality, SIAM Journal on Control and Optimization, Vol. 20, pp. 563–574, 1982. · Zbl 0488.49009 · doi:10.1137/0320041
[10] Rockafellar, R. T.,First- and Second-Order Epidifferentiability in Nonlinear Programming, Transactions of the American Mathematical Society, Vol. 307, pp. 75–108, 1988. · Zbl 0655.49010 · doi:10.1090/S0002-9947-1988-0936806-9
[11] Ward, D. E.,Exact Penalties and Sufficient Conditions for Optimality in Nonsmooth Optimization, Journal of Optimization Theory and Applications, Vol. 57, pp. 485–499, 1988. · Zbl 0621.90081 · doi:10.1007/BF02346165
[12] Ward, D. E., andBorwein, J. M.,Nonsmooth Calculus in Finite Dimensons, SIAM Journal on Control and Optimization, Vol. 25, 1987, pp. 1312–1340. · Zbl 0633.46043 · doi:10.1137/0325072
[13] Fiacco, A. V., andMcCormick, G. P.,Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley, New York, New York, 1968. · Zbl 0193.18805
[14] Kyparisis, J.,On Uniqueness of Kuhn-Tucker Multipliers in Nonlinear Programming, Mathematical Programming, Vol. 32, pp. 242–246, 1985. · Zbl 0566.90085 · doi:10.1007/BF01586095
[15] Flett, T. M.,Differential Analysis, Cambridge University Press, Cambridge, England, 1980. · Zbl 0442.34002
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