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July  2013, 9(3): 621-630. doi: 10.3934/jimo.2013.9.621

## Generalized weak sharp minima of variational inequality problems with functional constraints

 1 School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, China 2 Rear services office, Chongqing Police College, Chongqing, China 3 College of Mathematics and Statistics, Chongqing University, Chongqing, 401331 4 School of Economics and Business Administration, Chongqing University, Chongqing, China

Received  April 2012 Revised  March 2013 Published  April 2013

In this paper, the notion of generalized weak sharp minima is introduced for variational inequality problems with functional constraints in finite-dimensional spaces by virtue of a dual gap function. Some equivalent and necessary conditions for the solution set of the variational inequality problems to be a set of generalized weak sharp minima are obtained.
Citation: Wenyan Zhang, Shu Xu, Shengji Li, Xuexiang Huang. Generalized weak sharp minima of variational inequality problems with functional constraints. Journal of Industrial and Management Optimization, 2013, 9 (3) : 621-630. doi: 10.3934/jimo.2013.9.621
##### References:
 [1] A. Auslender, Asymptotic analysis for penalty and barrier methods in variational inequalities, SIAM J. Control Optim., 37 (1999), 653-671. doi: 10.1137/S0363012996310909. [2] A. Auslender, Variational inequalities over the cone of semidefinite positive symmetric matrices and over the Lorentz cone, Optim. Methods Softw., 18 (2003), 359-376. doi: 10.1080/1055678031000122586. [3] J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problem," Springer-Verlag, New York, 2000. [4] J. V. Burke and S. Deng, Weak sharp minima revisited, part I: Basic theory, Control and Cybernetics, 31 (2002), 439-469. [5] J. V. Burke and S. Deng, Weak sharp minima revisited, part II: Application to linear regularity and error bounds, Math. Program., Ser. B, 104 (2005), 235-261. doi: 10.1007/s10107-005-0615-2. [6] J. V. Burke and S. Deng, Weak sharp minima revisited, part III: Error bounds for differentiable convex inclusions, Math. Program., Ser. B, 116 (2009), 37-56. doi: 10.1007/s10107-007-0130-8. [7] J. V. Burke and M. C. Ferris, Weak sharp minima in mathematical programming, SIAM J. Control and Optim., 31 (1993), 1340-1359. doi: 10.1137/0331063. [8] J. V. Burke and M. C. Ferris, A Gauss-Newton method for convex composite optimaztion, Math. Program., 71 (1995), 179-194. doi: 10.1007/BF01585997. [9] J. M. Danskin, The theory of min-max with applications, SIAM J. Appl. Math., 14 (1966), 641-664. doi: 10.1137/0114053. [10] S. Deng and X. Q. Yang, Weak sharp minima in multicriteria linear programming, SIAM J. Optim., 15 (2004), 456-460. doi: 10.1137/S1052623403434401. [11] S. Deng, Some remarks on finite termination of descent methods, Pacific Journal of Optimization, 1 (2005), 19-25. [12] M. C. Ferris, "Weak Sharp Minima and Penalty Functions in Mathematical Programming," Ph. D thesis, Universiy of Cambridge, Cambridge, 1988. [13] M. C. Ferris, Iterative linear programming solution of convex programs, J. Optim. Therory Appl., 65 (1990), 53-65. doi: 10.1007/BF00941159. [14] M. C. Ferris, Finite termination of the proximal point algorithm, Math. Program., 50 (1991), 359-366. doi: 10.1007/BF01594944. [15] R. Henrion and J. Outrata, A subdifferential condition for calmness of multifunctions, J. Math. Anal. Appl., 258 (2001), 110-103. doi: 10.1006/jmaa.2000.7363. [16] P. T. Harker and J. S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Math. Program., 48 (1990), 161-220. doi: 10.1007/BF01582255. [17] B. S. He, H. Yang and C. S. Zhang, A modified augmented Lagrangian method for a class of monotone variational inequalities, Eur. J. Oper. Res., 159 (2004), 35-51. doi: 10.1016/S0377-2217(03)00385-0. [18] X. X. Huang and X. Q. Yang, Levitin-Polyak well-posedness in generalized variational inequality problems with functional constraints, J. Ind. Manag. Optim., 3 (2007), 671-684. doi: 10.3934/jimo.2007.3.671. [19] A. S. Lewis and J. S. Pang, Error bounds for convex inequality systems, in "Proceedings of the Fifth Symposium on Generalized Convexity" (ed. J. P. Crouzeix), Luminy-Marseille, (1996). doi: 10.1007/978-1-4613-3341-8_3. [20] C. Li and X. Wang, On convergence of the gauss-Netow method of convex composite optimization, Math. Program., 91 (2002), 349-356. doi: 10.1007/s101070100249. [21] Laura J. Kettner and S. Deng, On well-posedness and hausdorff convergence of solution sets of vector optimization problems, J. Optim. Theory Appl., 153 (2012), 619-632. doi: 10.1007/s10957-011-9947-7. [22] M. Studniarski, Weak sharp minima in multiobjective optimiation, Control and Cybernetics, 36 (2007), 925-936. [23] M. Studniarski, Characterizations of weak sharp minima of order one in nonlinear programming, in "System Modelling and Optimization", Chapman Hall/CRC Res. Notes Math. 396 (1999), 207-215. [24] P. Marcotte and D. L. Zhu, Weak sharp solutions of variational inequalities, SIAM J. Optim., 9 (1998), 179-189. doi: 10.1137/S1052623496309867. [25] P. Marcotte and D. L. Zhu, Erratum: Weak sharp solutions of variational inequalities, SIAM J. Optim., 10 (2000), 942-942. doi: 10.1137/S1052623499360616. [26] B. T. Polyak and Sharp Minima, Institue of control sciences lecture notes, Moscow, USSR, 1979; Presented at the IIASA workshop om generalized lagrangians and their applications, IIASA, Laxenburg, Austria, 1979. [27] Z. L. Wu and S. Y. Wu, Weak sharp solutions of variational inequalities in Hilbert spaces, SIAM J. Optim., 14 (2004), 1011-1027. doi: 10.1137/S1052623403421486. [28] Z. L. Wu and S. Y. Wu, Gâteaux differentiability of the dual gap function of a variational inequality, Eur. J. Oper. Res., 190 (2008), 328-344. doi: 10.1016/j.ejor.2007.06.024. [29] X. Y. Zheng and X. Q. Yang, Weak sharp minima for semi-infinite optimization problems with applications, SIAM J. Optim., 18 (2007), 573-588. doi: 10.1137/060670213. [30] X. Y. Zheng and X. Q. Yang, Global weak sharp minima for convex (semi-)infinite optimization problems, J. Math. Anal. Appl., 348 (2008), 1021-1028. doi: 10.1016/j.jmaa.2008.07.052. [31] X. Y. Zheng and X. Q. Yang, Weak sharp minima for piecewise linear multiobjective optimization in normed spaces, Nonlinear Anal., 68 (2008), 3771-3779. doi: 10.1016/j.na.2007.04.018. [32] X. Y. Zheng and K. F. Ng, Strong KKT conditions and weak sharp minima in convex-composite optimization, Math. Program., Ser. A, published online, April 17, (2009). [33] J. Z. Zhang, C. Y. Wang and N. H. Xiu, The dual gap function for variational inequalities, Appl. Math. Optim., 48 (2003), 129-148. doi: 10.1007/s00245-003-0771-9. [34] T. Larsson and M. Patriksson, A class of gap functions for variational inequalities, Math. Program., 64 (1994), 53-80. doi: 10.1007/BF01582565. [35] P. Hartman and G. Stampacchia, On some nonlinear elliptic differential functional equations, Acta Math., 105 (1966), 271-310. doi: 10.1007/BF02392210. [36] F. H. Clarke, "Optimization and Nonsmooth Analysis," John Wiley Sons, New York, 1983. [37] R. T. Rockafellar, "Convex Analysis, Princeton University Press," Princeton, 1970. [38] R. T. Rockafellar, "Conjugate Duality and Optimization," SIAM, 1974. [39] S. S. Chang, "Variational Inequality and Complementarity Problem Theory with Applications," Shanghai Sci. and Tech. Literature Publishing House, Shanghai, 1991.

show all references

##### References:
 [1] A. Auslender, Asymptotic analysis for penalty and barrier methods in variational inequalities, SIAM J. Control Optim., 37 (1999), 653-671. doi: 10.1137/S0363012996310909. [2] A. Auslender, Variational inequalities over the cone of semidefinite positive symmetric matrices and over the Lorentz cone, Optim. Methods Softw., 18 (2003), 359-376. doi: 10.1080/1055678031000122586. [3] J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problem," Springer-Verlag, New York, 2000. [4] J. V. Burke and S. Deng, Weak sharp minima revisited, part I: Basic theory, Control and Cybernetics, 31 (2002), 439-469. [5] J. V. Burke and S. Deng, Weak sharp minima revisited, part II: Application to linear regularity and error bounds, Math. Program., Ser. B, 104 (2005), 235-261. doi: 10.1007/s10107-005-0615-2. [6] J. V. Burke and S. Deng, Weak sharp minima revisited, part III: Error bounds for differentiable convex inclusions, Math. Program., Ser. B, 116 (2009), 37-56. doi: 10.1007/s10107-007-0130-8. [7] J. V. Burke and M. C. Ferris, Weak sharp minima in mathematical programming, SIAM J. Control and Optim., 31 (1993), 1340-1359. doi: 10.1137/0331063. [8] J. V. Burke and M. C. Ferris, A Gauss-Newton method for convex composite optimaztion, Math. Program., 71 (1995), 179-194. doi: 10.1007/BF01585997. [9] J. M. Danskin, The theory of min-max with applications, SIAM J. Appl. Math., 14 (1966), 641-664. doi: 10.1137/0114053. [10] S. Deng and X. Q. Yang, Weak sharp minima in multicriteria linear programming, SIAM J. Optim., 15 (2004), 456-460. doi: 10.1137/S1052623403434401. [11] S. Deng, Some remarks on finite termination of descent methods, Pacific Journal of Optimization, 1 (2005), 19-25. [12] M. C. Ferris, "Weak Sharp Minima and Penalty Functions in Mathematical Programming," Ph. D thesis, Universiy of Cambridge, Cambridge, 1988. [13] M. C. Ferris, Iterative linear programming solution of convex programs, J. Optim. Therory Appl., 65 (1990), 53-65. doi: 10.1007/BF00941159. [14] M. C. Ferris, Finite termination of the proximal point algorithm, Math. Program., 50 (1991), 359-366. doi: 10.1007/BF01594944. [15] R. Henrion and J. Outrata, A subdifferential condition for calmness of multifunctions, J. Math. Anal. Appl., 258 (2001), 110-103. doi: 10.1006/jmaa.2000.7363. [16] P. T. Harker and J. S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Math. Program., 48 (1990), 161-220. doi: 10.1007/BF01582255. [17] B. S. He, H. Yang and C. S. Zhang, A modified augmented Lagrangian method for a class of monotone variational inequalities, Eur. J. Oper. Res., 159 (2004), 35-51. doi: 10.1016/S0377-2217(03)00385-0. [18] X. X. Huang and X. Q. Yang, Levitin-Polyak well-posedness in generalized variational inequality problems with functional constraints, J. Ind. Manag. Optim., 3 (2007), 671-684. doi: 10.3934/jimo.2007.3.671. [19] A. S. Lewis and J. S. Pang, Error bounds for convex inequality systems, in "Proceedings of the Fifth Symposium on Generalized Convexity" (ed. J. P. Crouzeix), Luminy-Marseille, (1996). doi: 10.1007/978-1-4613-3341-8_3. [20] C. Li and X. Wang, On convergence of the gauss-Netow method of convex composite optimization, Math. Program., 91 (2002), 349-356. doi: 10.1007/s101070100249. [21] Laura J. Kettner and S. Deng, On well-posedness and hausdorff convergence of solution sets of vector optimization problems, J. Optim. Theory Appl., 153 (2012), 619-632. doi: 10.1007/s10957-011-9947-7. [22] M. Studniarski, Weak sharp minima in multiobjective optimiation, Control and Cybernetics, 36 (2007), 925-936. [23] M. Studniarski, Characterizations of weak sharp minima of order one in nonlinear programming, in "System Modelling and Optimization", Chapman Hall/CRC Res. Notes Math. 396 (1999), 207-215. [24] P. Marcotte and D. L. Zhu, Weak sharp solutions of variational inequalities, SIAM J. Optim., 9 (1998), 179-189. doi: 10.1137/S1052623496309867. [25] P. Marcotte and D. L. Zhu, Erratum: Weak sharp solutions of variational inequalities, SIAM J. Optim., 10 (2000), 942-942. doi: 10.1137/S1052623499360616. [26] B. T. Polyak and Sharp Minima, Institue of control sciences lecture notes, Moscow, USSR, 1979; Presented at the IIASA workshop om generalized lagrangians and their applications, IIASA, Laxenburg, Austria, 1979. [27] Z. L. Wu and S. Y. Wu, Weak sharp solutions of variational inequalities in Hilbert spaces, SIAM J. Optim., 14 (2004), 1011-1027. doi: 10.1137/S1052623403421486. [28] Z. L. Wu and S. Y. Wu, Gâteaux differentiability of the dual gap function of a variational inequality, Eur. J. Oper. Res., 190 (2008), 328-344. doi: 10.1016/j.ejor.2007.06.024. [29] X. Y. Zheng and X. Q. Yang, Weak sharp minima for semi-infinite optimization problems with applications, SIAM J. Optim., 18 (2007), 573-588. doi: 10.1137/060670213. [30] X. Y. Zheng and X. Q. Yang, Global weak sharp minima for convex (semi-)infinite optimization problems, J. Math. Anal. Appl., 348 (2008), 1021-1028. doi: 10.1016/j.jmaa.2008.07.052. [31] X. Y. Zheng and X. Q. Yang, Weak sharp minima for piecewise linear multiobjective optimization in normed spaces, Nonlinear Anal., 68 (2008), 3771-3779. doi: 10.1016/j.na.2007.04.018. [32] X. Y. Zheng and K. F. Ng, Strong KKT conditions and weak sharp minima in convex-composite optimization, Math. Program., Ser. A, published online, April 17, (2009). [33] J. Z. Zhang, C. Y. Wang and N. H. Xiu, The dual gap function for variational inequalities, Appl. Math. Optim., 48 (2003), 129-148. doi: 10.1007/s00245-003-0771-9. [34] T. Larsson and M. Patriksson, A class of gap functions for variational inequalities, Math. Program., 64 (1994), 53-80. doi: 10.1007/BF01582565. [35] P. Hartman and G. Stampacchia, On some nonlinear elliptic differential functional equations, Acta Math., 105 (1966), 271-310. doi: 10.1007/BF02392210. [36] F. H. Clarke, "Optimization and Nonsmooth Analysis," John Wiley Sons, New York, 1983. [37] R. T. Rockafellar, "Convex Analysis, Princeton University Press," Princeton, 1970. [38] R. T. Rockafellar, "Conjugate Duality and Optimization," SIAM, 1974. [39] S. S. Chang, "Variational Inequality and Complementarity Problem Theory with Applications," Shanghai Sci. and Tech. Literature Publishing House, Shanghai, 1991.
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