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Necessary conditions for weak sharp minima in cone-constrained optimization problems. (English) Zbl 1242.49052
Summary: We study weak sharp minima for optimization problems with cone constraints. Some necessary conditions for weak sharp minima of higher order are established by means of upper Studniarski or Dini directional derivatives. In particular, when the objective and constrained functions are strict derivative, a necessary condition is obtained by a normal cone.

49K27Optimal control problems in abstract spaces (optimality conditions)
Full Text: DOI
[1] M. C. Ferris, Weak sharp minima and penalty functions in mathematical programming, Ph.D. thesis, Universiy of Cambridge, Cambridge, UK, 1988.
[2] B. T. Polyak, “Sharp minima,” in Proceedings of the IIASA Workshop on Generalized Lagrangians and Their Applications, Institue of control sciences lecture notes, IIASA, Laxenburg, Austria, USSR, Moscow, Russia, 1979.
[3] R. Henrion and J. Outrata, “A subdifferential condition for calmness of multifunctions,” Journal of Mathematical Analysis and Applications, vol. 258, no. 1, pp. 110-130, 2001. · Zbl 0983.49010 · doi:10.1006/jmaa.2000.7363
[4] A. S. Lewis and J. S. Pang, “Error bounds for convex inequality systems,” in Proceedings of the 5th Symposium on Generalized Convexity, J. P. Crouzeix, Ed., Luminy, Marseille, France, 1996. · Zbl 0953.90048
[5] J. V. Burke and M. C. Ferris, “Weak sharp minima in mathematical programming,” SIAM Journal on Control and Optimization, vol. 31, no. 5, pp. 1340-1359, 1993. · Zbl 0791.90040 · doi:10.1137/0331063
[6] J. Burke and S.-P. Han, “A Gauss-Newton approach to solving generalized inequalities,” Mathematics of Operations Research, vol. 11, no. 4, pp. 632-643, 1986. · Zbl 0623.90072 · doi:10.1287/moor.11.4.632
[7] M. C. Ferris, “Iterative linear programming solution of convex programs,” Journal of Optimization Theory and Applications, vol. 65, no. 1, pp. 53-65, 1990. · Zbl 0672.90087 · doi:10.1007/BF00941159
[8] J. V. Burke and S. Deng, “Weak sharp minima revisited. I. Basic theory,” Control and Cybernetics, vol. 31, no. 3, pp. 439-469, 2002. · Zbl 1105.90356
[9] J. V. Burke and S. Deng, “Weak sharp minima revisited. II. Application to linear regularity and error bounds,” Mathematical Programming, vol. 104, no. 2-3, pp. 235-261, 2005. · Zbl 1124.90349 · doi:10.1007/s10107-005-0615-2
[10] J. V. Burke and S. Deng, “Weak sharp minima revisited. III. Error bounds for differentiable convex inclusions,” Mathematical Programming, vol. 116, no. 1-2, Ser. B, pp. 37-56, 2009. · Zbl 1163.90016 · doi:10.1007/s10107-007-0130-8
[11] S. Deng and X. Q. Yang, “Weak sharp minima in multicriteria linear programming,” SIAM Journal on Optimization, vol. 15, no. 2, pp. 456-460, 2004. · Zbl 1114.90111 · doi:10.1137/S1052623403434401
[12] X. Y. Zheng and X. Q. Yang, “Weak sharp minima for piecewise linear multiobjective optimization in normed spaces,” Nonlinear Analysis, vol. 68, no. 12, pp. 3771-3779, 2008. · Zbl 1191.90064 · doi:10.1016/j.na.2007.04.018
[13] X. Y. Zheng, X. M. Yang, and K. L. Teo, “Sharp minima for multiobjective optimization in Banach spaces,” Set-Valued Analysis, vol. 14, no. 4, pp. 327-345, 2006. · Zbl 1103.49009 · doi:10.1007/s11228-006-0023-7
[14] X. Y. Zheng and K. F. Ng, “Strong KKT conditions and weak sharp solutions in convex-composite optimization,” Mathematical Programming. Series A, vol. 126, no. 2, pp. 259-275, 2011. · Zbl 1229.90147 · doi:10.1007/s10107-009-0277-6
[15] X. Y. Zheng and X. Q. Yang, “Weak sharp minima for semi-infinite optimization problems with applications,” SIAM Journal on Optimization, vol. 18, no. 2, pp. 573-588, 2007. · Zbl 1190.90251 · doi:10.1137/060670213
[16] M. Studniarski, “On weak sharp minima for a special class of nonsmooth functions,” Discussiones Mathematicae. Differential Inclusions, Control and Optimization, vol. 20, no. 2, pp. 195-207, 2000. · Zbl 1012.49014 · doi:10.7151/dmdico.1012 · http://www.pz.zgora.pl/discuss/di/20_2/di20stu.htm
[17] M. Studniarski, “Characterizations of weak sharp minima of order one in nonlinear programming,” in Systems Modelling and Optimization, vol. 396 of Chapman & Hall/CRC Res. Notes Math., pp. 207-215, Chapman & Hall/CRC, Boca Raton, FL, 1999. · Zbl 0955.90130
[18] D. Klatte, “On quantitative stability for non-isolated minima,” Control and Cybernetics, vol. 23, no. 1-2, pp. 183-200, 1994. · Zbl 0808.90120
[19] J. F. Bonnans and A. Ioffe, “Second-order sufficiency and quadratic growth for nonisolated minima,” Mathematics of Operations Research, vol. 20, no. 4, pp. 801-817, 1995. · Zbl 0846.90095 · doi:10.1287/moor.20.4.801
[20] D. E. Ward, “Characterizations of strict local minima and necessary conditions for weak sharp minima,” Journal of Optimization Theory and Applications, vol. 80, no. 3, pp. 551-571, 1994. · Zbl 0797.90101 · doi:10.1007/BF02207780
[21] M. Studniarski and D. E. Ward, “Weak sharp minima: characterizations and sufficient conditions,” SIAM Journal on Control and Optimization, vol. 38, no. 1, pp. 219-236, 1999. · Zbl 0946.49011 · doi:10.1137/S0363012996301269
[22] J. P. Aubin and H. Frankowska, Set-Valued Analysis, vol. 2 of Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston, Mass, USA, 1990. · Zbl 0745.17013
[23] F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York, NY, USA, 1983. · Zbl 0582.49001
[24] Do Van Luu, “Higher-order necessary and sufficient conditions for strict local Pareto minima in terms of Studniarski’s derivatives,” Optimization, vol. 57, no. 4, pp. 593-605, 2008. · Zbl 1171.90012 · doi:10.1080/02331930601120086