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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

Wenyan Zhang 1, , Shu Xu 2, , Shengji Li 3, and  Xuexiang Huang 4,

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.
Keywords: Robinson's constraint qualification., dual gap function, Variational inequality problems, generalized weak sharp minima.
Mathematics Subject Classification: 49J40, 49K9.
Citation: Wenyan Zhang, Shu Xu, Shengji Li, Xuexiang Huang. Generalized weak sharp minima of variational inequality problems with functional constraints. Journal of Industrial & 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.  doi: 10.1137/S0363012996310909.  Google Scholar

[2]

A. Auslender, Variational inequalities over the cone of semidefinite positive symmetric matrices and over the Lorentz cone,, Optim. Methods Softw., 18 (2003), 359.  doi: 10.1080/1055678031000122586.  Google Scholar

[3]

J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problem,", Springer-Verlag, (2000).   Google Scholar

[4]

J. V. Burke and S. Deng, Weak sharp minima revisited, part I: Basic theory,, Control and Cybernetics, 31 (2002), 439.   Google Scholar

[5]

J. V. Burke and S. Deng, Weak sharp minima revisited, part II: Application to linear regularity and error bounds,, Math. Program., 104 (2005), 235.  doi: 10.1007/s10107-005-0615-2.  Google Scholar

[6]

J. V. Burke and S. Deng, Weak sharp minima revisited, part III: Error bounds for differentiable convex inclusions,, Math. Program., 116 (2009), 37.  doi: 10.1007/s10107-007-0130-8.  Google Scholar

[7]

J. V. Burke and M. C. Ferris, Weak sharp minima in mathematical programming,, SIAM J. Control and Optim., 31 (1993), 1340.  doi: 10.1137/0331063.  Google Scholar

[8]

J. V. Burke and M. C. Ferris, A Gauss-Newton method for convex composite optimaztion,, Math. Program., 71 (1995), 179.  doi: 10.1007/BF01585997.  Google Scholar

[9]

J. M. Danskin, The theory of min-max with applications,, SIAM J. Appl. Math., 14 (1966), 641.  doi: 10.1137/0114053.  Google Scholar

[10]

S. Deng and X. Q. Yang, Weak sharp minima in multicriteria linear programming,, SIAM J. Optim., 15 (2004), 456.  doi: 10.1137/S1052623403434401.  Google Scholar

[11]

S. Deng, Some remarks on finite termination of descent methods,, Pacific Journal of Optimization, 1 (2005), 19.   Google Scholar

[12]

M. C. Ferris, "Weak Sharp Minima and Penalty Functions in Mathematical Programming,", Ph. D thesis, (1988).   Google Scholar

[13]

M. C. Ferris, Iterative linear programming solution of convex programs,, J. Optim. Therory Appl., 65 (1990), 53.  doi: 10.1007/BF00941159.  Google Scholar

[14]

M. C. Ferris, Finite termination of the proximal point algorithm,, Math. Program., 50 (1991), 359.  doi: 10.1007/BF01594944.  Google Scholar

[15]

R. Henrion and J. Outrata, A subdifferential condition for calmness of multifunctions,, J. Math. Anal. Appl., 258 (2001), 110.  doi: 10.1006/jmaa.2000.7363.  Google Scholar

[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.  doi: 10.1007/BF01582255.  Google Scholar

[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.  doi: 10.1016/S0377-2217(03)00385-0.  Google Scholar

[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.  doi: 10.3934/jimo.2007.3.671.  Google Scholar

[19]

A. S. Lewis and J. S. Pang, Error bounds for convex inequality systems,, in, (1996).  doi: 10.1007/978-1-4613-3341-8_3.  Google Scholar

[20]

C. Li and X. Wang, On convergence of the gauss-Netow method of convex composite optimization,, Math. Program., 91 (2002), 349.  doi: 10.1007/s101070100249.  Google Scholar

[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.  doi: 10.1007/s10957-011-9947-7.  Google Scholar

[22]

M. Studniarski, Weak sharp minima in multiobjective optimiation,, Control and Cybernetics, 36 (2007), 925.   Google Scholar

[23]

M. Studniarski, Characterizations of weak sharp minima of order one in nonlinear programming,, in, 396 (1999), 207.   Google Scholar

[24]

P. Marcotte and D. L. Zhu, Weak sharp solutions of variational inequalities,, SIAM J. Optim., 9 (1998), 179.  doi: 10.1137/S1052623496309867.  Google Scholar

[25]

P. Marcotte and D. L. Zhu, Erratum: Weak sharp solutions of variational inequalities,, SIAM J. Optim., 10 (2000), 942.  doi: 10.1137/S1052623499360616.  Google Scholar

[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, (1979).   Google Scholar

[27]

Z. L. Wu and S. Y. Wu, Weak sharp solutions of variational inequalities in Hilbert spaces,, SIAM J. Optim., 14 (2004), 1011.  doi: 10.1137/S1052623403421486.  Google Scholar

[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.  doi: 10.1016/j.ejor.2007.06.024.  Google Scholar

[29]

X. Y. Zheng and X. Q. Yang, Weak sharp minima for semi-infinite optimization problems with applications,, SIAM J. Optim., 18 (2007), 573.  doi: 10.1137/060670213.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2008.07.052.  Google Scholar

[31]

X. Y. Zheng and X. Q. Yang, Weak sharp minima for piecewise linear multiobjective optimization in normed spaces,, Nonlinear Anal., 68 (2008), 3771.  doi: 10.1016/j.na.2007.04.018.  Google Scholar

[32]

X. Y. Zheng and K. F. Ng, Strong KKT conditions and weak sharp minima in convex-composite optimization,, Math. Program., (2009).   Google Scholar

[33]

J. Z. Zhang, C. Y. Wang and N. H. Xiu, The dual gap function for variational inequalities,, Appl. Math. Optim., 48 (2003), 129.  doi: 10.1007/s00245-003-0771-9.  Google Scholar

[34]

T. Larsson and M. Patriksson, A class of gap functions for variational inequalities,, Math. Program., 64 (1994), 53.  doi: 10.1007/BF01582565.  Google Scholar

[35]

P. Hartman and G. Stampacchia, On some nonlinear elliptic differential functional equations,, Acta Math., 105 (1966), 271.  doi: 10.1007/BF02392210.  Google Scholar

[36]

F. H. Clarke, "Optimization and Nonsmooth Analysis,", John Wiley Sons, (1983).   Google Scholar

[37]

R. T. Rockafellar, "Convex Analysis, Princeton University Press,", Princeton, (1970).   Google Scholar

[38]

R. T. Rockafellar, "Conjugate Duality and Optimization,", SIAM, (1974).   Google Scholar

[39]

S. S. Chang, "Variational Inequality and Complementarity Problem Theory with Applications,", Shanghai Sci. and Tech. Literature Publishing House, (1991).   Google Scholar

show all references

References:
[1]

A. Auslender, Asymptotic analysis for penalty and barrier methods in variational inequalities,, SIAM J. Control Optim., 37 (1999), 653.  doi: 10.1137/S0363012996310909.  Google Scholar

[2]

A. Auslender, Variational inequalities over the cone of semidefinite positive symmetric matrices and over the Lorentz cone,, Optim. Methods Softw., 18 (2003), 359.  doi: 10.1080/1055678031000122586.  Google Scholar

[3]

J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problem,", Springer-Verlag, (2000).   Google Scholar

[4]

J. V. Burke and S. Deng, Weak sharp minima revisited, part I: Basic theory,, Control and Cybernetics, 31 (2002), 439.   Google Scholar

[5]

J. V. Burke and S. Deng, Weak sharp minima revisited, part II: Application to linear regularity and error bounds,, Math. Program., 104 (2005), 235.  doi: 10.1007/s10107-005-0615-2.  Google Scholar

[6]

J. V. Burke and S. Deng, Weak sharp minima revisited, part III: Error bounds for differentiable convex inclusions,, Math. Program., 116 (2009), 37.  doi: 10.1007/s10107-007-0130-8.  Google Scholar

[7]

J. V. Burke and M. C. Ferris, Weak sharp minima in mathematical programming,, SIAM J. Control and Optim., 31 (1993), 1340.  doi: 10.1137/0331063.  Google Scholar

[8]

J. V. Burke and M. C. Ferris, A Gauss-Newton method for convex composite optimaztion,, Math. Program., 71 (1995), 179.  doi: 10.1007/BF01585997.  Google Scholar

[9]

J. M. Danskin, The theory of min-max with applications,, SIAM J. Appl. Math., 14 (1966), 641.  doi: 10.1137/0114053.  Google Scholar

[10]

S. Deng and X. Q. Yang, Weak sharp minima in multicriteria linear programming,, SIAM J. Optim., 15 (2004), 456.  doi: 10.1137/S1052623403434401.  Google Scholar

[11]

S. Deng, Some remarks on finite termination of descent methods,, Pacific Journal of Optimization, 1 (2005), 19.   Google Scholar

[12]

M. C. Ferris, "Weak Sharp Minima and Penalty Functions in Mathematical Programming,", Ph. D thesis, (1988).   Google Scholar

[13]

M. C. Ferris, Iterative linear programming solution of convex programs,, J. Optim. Therory Appl., 65 (1990), 53.  doi: 10.1007/BF00941159.  Google Scholar

[14]

M. C. Ferris, Finite termination of the proximal point algorithm,, Math. Program., 50 (1991), 359.  doi: 10.1007/BF01594944.  Google Scholar

[15]

R. Henrion and J. Outrata, A subdifferential condition for calmness of multifunctions,, J. Math. Anal. Appl., 258 (2001), 110.  doi: 10.1006/jmaa.2000.7363.  Google Scholar

[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.  doi: 10.1007/BF01582255.  Google Scholar

[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.  doi: 10.1016/S0377-2217(03)00385-0.  Google Scholar

[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.  doi: 10.3934/jimo.2007.3.671.  Google Scholar

[19]

A. S. Lewis and J. S. Pang, Error bounds for convex inequality systems,, in, (1996).  doi: 10.1007/978-1-4613-3341-8_3.  Google Scholar

[20]

C. Li and X. Wang, On convergence of the gauss-Netow method of convex composite optimization,, Math. Program., 91 (2002), 349.  doi: 10.1007/s101070100249.  Google Scholar

[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.  doi: 10.1007/s10957-011-9947-7.  Google Scholar

[22]

M. Studniarski, Weak sharp minima in multiobjective optimiation,, Control and Cybernetics, 36 (2007), 925.   Google Scholar

[23]

M. Studniarski, Characterizations of weak sharp minima of order one in nonlinear programming,, in, 396 (1999), 207.   Google Scholar

[24]

P. Marcotte and D. L. Zhu, Weak sharp solutions of variational inequalities,, SIAM J. Optim., 9 (1998), 179.  doi: 10.1137/S1052623496309867.  Google Scholar

[25]

P. Marcotte and D. L. Zhu, Erratum: Weak sharp solutions of variational inequalities,, SIAM J. Optim., 10 (2000), 942.  doi: 10.1137/S1052623499360616.  Google Scholar

[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, (1979).   Google Scholar

[27]

Z. L. Wu and S. Y. Wu, Weak sharp solutions of variational inequalities in Hilbert spaces,, SIAM J. Optim., 14 (2004), 1011.  doi: 10.1137/S1052623403421486.  Google Scholar

[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.  doi: 10.1016/j.ejor.2007.06.024.  Google Scholar

[29]

X. Y. Zheng and X. Q. Yang, Weak sharp minima for semi-infinite optimization problems with applications,, SIAM J. Optim., 18 (2007), 573.  doi: 10.1137/060670213.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2008.07.052.  Google Scholar

[31]

X. Y. Zheng and X. Q. Yang, Weak sharp minima for piecewise linear multiobjective optimization in normed spaces,, Nonlinear Anal., 68 (2008), 3771.  doi: 10.1016/j.na.2007.04.018.  Google Scholar

[32]

X. Y. Zheng and K. F. Ng, Strong KKT conditions and weak sharp minima in convex-composite optimization,, Math. Program., (2009).   Google Scholar

[33]

J. Z. Zhang, C. Y. Wang and N. H. Xiu, The dual gap function for variational inequalities,, Appl. Math. Optim., 48 (2003), 129.  doi: 10.1007/s00245-003-0771-9.  Google Scholar

[34]

T. Larsson and M. Patriksson, A class of gap functions for variational inequalities,, Math. Program., 64 (1994), 53.  doi: 10.1007/BF01582565.  Google Scholar

[35]

P. Hartman and G. Stampacchia, On some nonlinear elliptic differential functional equations,, Acta Math., 105 (1966), 271.  doi: 10.1007/BF02392210.  Google Scholar

[36]

F. H. Clarke, "Optimization and Nonsmooth Analysis,", John Wiley Sons, (1983).   Google Scholar

[37]

R. T. Rockafellar, "Convex Analysis, Princeton University Press,", Princeton, (1970).   Google Scholar

[38]

R. T. Rockafellar, "Conjugate Duality and Optimization,", SIAM, (1974).   Google Scholar

[39]

S. S. Chang, "Variational Inequality and Complementarity Problem Theory with Applications,", Shanghai Sci. and Tech. Literature Publishing House, (1991).   Google Scholar

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