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Nonlinear conjugate gradient methods with sufficient descent properties for unconstrained optimization
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 |
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. |
[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. |
[3] |
J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problem,", Springer-Verlag, (2000).
|
[4] |
J. V. Burke and S. Deng, Weak sharp minima revisited, part I: Basic theory,, Control and Cybernetics, 31 (2002), 439.
|
[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. |
[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. |
[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. |
[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. |
[9] |
J. M. Danskin, The theory of min-max with applications,, SIAM J. Appl. Math., 14 (1966), 641.
doi: 10.1137/0114053. |
[10] |
S. Deng and X. Q. Yang, Weak sharp minima in multicriteria linear programming,, SIAM J. Optim., 15 (2004), 456.
doi: 10.1137/S1052623403434401. |
[11] |
S. Deng, Some remarks on finite termination of descent methods,, Pacific Journal of Optimization, 1 (2005), 19.
|
[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. |
[14] |
M. C. Ferris, Finite termination of the proximal point algorithm,, Math. Program., 50 (1991), 359.
doi: 10.1007/BF01594944. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
M. Studniarski, Weak sharp minima in multiobjective optimiation,, Control and Cybernetics, 36 (2007), 925.
|
[23] |
M. Studniarski, Characterizations of weak sharp minima of order one in nonlinear programming,, in, 396 (1999), 207.
|
[24] |
P. Marcotte and D. L. Zhu, Weak sharp solutions of variational inequalities,, SIAM J. Optim., 9 (1998), 179.
doi: 10.1137/S1052623496309867. |
[25] |
P. Marcotte and D. L. Zhu, Erratum: Weak sharp solutions of variational inequalities,, SIAM J. Optim., 10 (2000), 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, (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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[34] |
T. Larsson and M. Patriksson, A class of gap functions for variational inequalities,, Math. Program., 64 (1994), 53.
doi: 10.1007/BF01582565. |
[35] |
P. Hartman and G. Stampacchia, On some nonlinear elliptic differential functional equations,, Acta Math., 105 (1966), 271.
doi: 10.1007/BF02392210. |
[36] |
F. H. Clarke, "Optimization and Nonsmooth Analysis,", John Wiley Sons, (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, (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. |
[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. |
[3] |
J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problem,", Springer-Verlag, (2000).
|
[4] |
J. V. Burke and S. Deng, Weak sharp minima revisited, part I: Basic theory,, Control and Cybernetics, 31 (2002), 439.
|
[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. |
[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. |
[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. |
[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. |
[9] |
J. M. Danskin, The theory of min-max with applications,, SIAM J. Appl. Math., 14 (1966), 641.
doi: 10.1137/0114053. |
[10] |
S. Deng and X. Q. Yang, Weak sharp minima in multicriteria linear programming,, SIAM J. Optim., 15 (2004), 456.
doi: 10.1137/S1052623403434401. |
[11] |
S. Deng, Some remarks on finite termination of descent methods,, Pacific Journal of Optimization, 1 (2005), 19.
|
[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. |
[14] |
M. C. Ferris, Finite termination of the proximal point algorithm,, Math. Program., 50 (1991), 359.
doi: 10.1007/BF01594944. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
M. Studniarski, Weak sharp minima in multiobjective optimiation,, Control and Cybernetics, 36 (2007), 925.
|
[23] |
M. Studniarski, Characterizations of weak sharp minima of order one in nonlinear programming,, in, 396 (1999), 207.
|
[24] |
P. Marcotte and D. L. Zhu, Weak sharp solutions of variational inequalities,, SIAM J. Optim., 9 (1998), 179.
doi: 10.1137/S1052623496309867. |
[25] |
P. Marcotte and D. L. Zhu, Erratum: Weak sharp solutions of variational inequalities,, SIAM J. Optim., 10 (2000), 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, (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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[34] |
T. Larsson and M. Patriksson, A class of gap functions for variational inequalities,, Math. Program., 64 (1994), 53.
doi: 10.1007/BF01582565. |
[35] |
P. Hartman and G. Stampacchia, On some nonlinear elliptic differential functional equations,, Acta Math., 105 (1966), 271.
doi: 10.1007/BF02392210. |
[36] |
F. H. Clarke, "Optimization and Nonsmooth Analysis,", John Wiley Sons, (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, (1991). Google Scholar |
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