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Upper Hölder estimates of solutions to parametric primal and dual vector quasi-equilibria
On an exact penalty function method for semi-infinite programming problems
1. | Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China, China |
2. | Department of Mathematics, Shanghai University, 99, Shangda Road, Shanghai, China |
3. | Department of Mathematics, Shanghai University, 99, Shangda Road, 200444, Shanghai |
References:
[1] |
J. V. Burke, An exact penalization viewpoint of constrained optimization, SIAM J. Control and Optimization, 29 (1991), 968-998.
doi: 10.1137/0329054. |
[2] |
H. Cheng and B. Özçam, A discretization based smoothing method for solving semi-infinite variational inequalities, Journal of Industrial and Management Optimization, 1 (2005), 219-233. |
[3] |
A. R. Conn and N. I. M. Gould, An exact penalty function for semi-infinite programming, Mathematical Programming, 37 (1987), 19-40.
doi: 10.1007/BF02591681. |
[4] |
R. DeVore and G. Lorentz, "Constructive Approximation," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 303, Springer-Verlag, Berlin, 1993. |
[5] |
Z. G. Feng, K. L. Teo and Y. Zhao, Branch and bound method for sensor scheduling in discrete time, Journal of Industrial and Management Optimization, 1 (2005), 499-512. |
[6] |
P. Gribik, A central-cutting-plane algorithm for semi-infinite programming problems, in "Semi-Infinite Programming" (ed. R. Hettich) (Proc. Workshop, Bad Honnef, 1978), Lecture Notes in Control and Information Sci., 15, Springer, Berlin-New York, (1979), 66-82. |
[7] |
R. Hettich, An implementation of a discretization method for semi-infinite programming, Mathematical Programming, 34 (1986), 354-361.
doi: 10.1007/BF01582235. |
[8] |
R. Hettich and G. Gramlich, A note on an implementation of a method for quadratic semi-infinite programming, Mathematical Programming, 46 (1990), 249-254.
doi: 10.1007/BF01585742. |
[9] |
W. Huyer and A. Neumair, A new exact penalty function, SIAM J. Optimization, 13 (2003), 1141-1159.
doi: 10.1137/S1052623401390537. |
[10] |
B. Jerez, General equilibrium with asymmetric information: A dual approach, Journal of Economic Theory, in press. |
[11] |
H. T. Jongen, F. Twilt and G.-W. Weber, Semi-infinite optimization: Structure and stability of the feasible set, Journal of Optimization Theory and Application, 72 (1992), 529-552.
doi: 10.1007/BF00939841. |
[12] |
B. Li, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty function method for continuous inequality constrained optimal control problem, Journal of Optimization Theory and Applications, published online, 2011. |
[13] |
D.-H. Li, L. Qi, J. Tam and S.-Y. Wu, A smoothing Newton method for semi-infinite programming, Journal of Global Optimization, 30 (2004), 169-194.
doi: 10.1007/s10898-004-8266-z. |
[14] |
C. Ling, Q. Ni, L. Qi and S.-Y. Wu, A new smoothing Newton-type algorithm for semi-infinite programming, Journal of Global Optimization, 47 (2010), 133-159.
doi: 10.1007/s10898-009-9462-7. |
[15] |
Y. Liu and K. L. Teo, An adaptive dual parametrization algorithm for quadratic semi-infinite programming problems, Journal of Global Optimization, 24 (2002), 205-217.
doi: 10.1023/A:1020234019886. |
[16] |
Y. Liu, K. L. Teo and S.-Y. Wu, An new quadratic semi-infinite programming algorithm based on dual parameterization, Journal of Global Optimization, 29 (2005), 401-413.
doi: 10.1023/B:JOGO.0000047910.80739.95. |
[17] |
C. Ma and C. Y. Wang, A nonsmooth Levenberg-Marquardt method for solving semi-infinite programming problems, Journal of Computational and Applied Mathematics, 230 (2009), 633-642.
doi: 10.1016/j.cam.2009.01.004. |
[18] |
Q. Ni, C. Ling, L. Qi and K. L. Teo, A truncated projected Newton-type algorithm for large-scale semi-infinite programming, SIAM J. Optimization, 16 (2006), 1137-1154.
doi: 10.1137/040619867. |
[19] |
J.-S. Pang, Error bound in mathematical programming, Mathematical Programming, 79 (1997), 299-332.
doi: 10.1007/BF02614322. |
[20] |
C. Price and I. Coope, Numerical experienments in semi-infinite programming, Computational Optimization and Application, 6 (1996), 169-189. |
[21] |
L. Qi, S.-Y. Wu and G. L. Zhou, Semismooth newton methods for solving semi-infinite programming problems, Journal of Global Optimization, 27 (2003), 215-232.
doi: 10.1023/A:1024814401713. |
[22] |
R. Reemtsen, Discretization methods for the solution of semi-infinite programming problems, Journal of Optimization Theory and Application, 71 (1991), 85-103.
doi: 10.1007/BF00940041. |
[23] |
R. T. Rockafellar and R. J.-B. Wets, "Variational Analysis," Grundlehren der Math. Wissenschaften, 317, Springer-Verlag, Berlin, 1998. |
[24] |
G. Still, Discretization in semi-infinite programming: The rate of convergence, Mathematical Programming, 91 (2001), 53-69. |
[25] |
H. Voigt, Semi-infinite transportation problems, Zeitschrift für Analysis und ihre Anwendungen, 17 (1998), 729-741. |
[26] |
J.-P. Z. Wang and B. G. Lindsay, A penalized nonparametric maximum likelihood approach to species richness estimation, Journal of the American Statistical Association, 100 (2005), 942-959.
doi: 10.1198/016214504000002005. |
[27] |
G. A. Watson, Globally convergent methods for semi-infinite programming, BIT, 21 (1981), 362-373.
doi: 10.1007/BF01941472. |
[28] |
G. W. Weber and Ö. Uǧur, Optimization and dynamics of gene-environment networks with intervals, Journal of Industrial and Management Optimization, 3 (2007), 357-379. |
[29] |
S.-Y. Wu, S.-C. Fang and C.-J. Lin, Solving quadratic semi-infinite programming problems by using relaxed cutting-plane scheme, Journal of Computational and Applied Mathematics, 129 (2001), 89-104.
doi: 10.1016/S0377-0427(00)00544-6. |
[30] |
Z. L. Wu and J. J. Ye, First-order and second-order conditions for error bounds, SIAM J. Optimization, 14 (2003), 621-645.
doi: 10.1137/S1052623402412982. |
[31] |
K. F. C. Yiu, X. Q. Yang, S. Nordholm and K. L. Teo, Near-field broadband beamformer design via multidimensional semi-infinite linear programming techniques, IEEE Transactions on Speech and Audio Processing, 11 (2003), 725-732.
doi: 10.1109/TSA.2003.815527. |
[32] |
C. J. Yu, K. L. Teo, L. S. Zhang and Y. Q. Bai, A new exact penalty function method for continuous inequality constrained optimization problems, Journal of Industrial and Management Optimization, 6 (2010), 895-910. |
[33] |
K. Zhang and S. Wang, Convergence property of an interior penalty approach to pricing American option, Journal of Industrial and Management Optimization, 7 (2011), 435-447. |
[34] |
H. J. Zhou, K. F. C. Yiu and L. K. Li, Evaluating American put options on zero-coupon bonds by a penalty method, Journal of Computational and Applied Mathematics, 235 (2011), 3921-3931.
doi: 10.1016/j.cam.2011.01.038. |
show all references
References:
[1] |
J. V. Burke, An exact penalization viewpoint of constrained optimization, SIAM J. Control and Optimization, 29 (1991), 968-998.
doi: 10.1137/0329054. |
[2] |
H. Cheng and B. Özçam, A discretization based smoothing method for solving semi-infinite variational inequalities, Journal of Industrial and Management Optimization, 1 (2005), 219-233. |
[3] |
A. R. Conn and N. I. M. Gould, An exact penalty function for semi-infinite programming, Mathematical Programming, 37 (1987), 19-40.
doi: 10.1007/BF02591681. |
[4] |
R. DeVore and G. Lorentz, "Constructive Approximation," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 303, Springer-Verlag, Berlin, 1993. |
[5] |
Z. G. Feng, K. L. Teo and Y. Zhao, Branch and bound method for sensor scheduling in discrete time, Journal of Industrial and Management Optimization, 1 (2005), 499-512. |
[6] |
P. Gribik, A central-cutting-plane algorithm for semi-infinite programming problems, in "Semi-Infinite Programming" (ed. R. Hettich) (Proc. Workshop, Bad Honnef, 1978), Lecture Notes in Control and Information Sci., 15, Springer, Berlin-New York, (1979), 66-82. |
[7] |
R. Hettich, An implementation of a discretization method for semi-infinite programming, Mathematical Programming, 34 (1986), 354-361.
doi: 10.1007/BF01582235. |
[8] |
R. Hettich and G. Gramlich, A note on an implementation of a method for quadratic semi-infinite programming, Mathematical Programming, 46 (1990), 249-254.
doi: 10.1007/BF01585742. |
[9] |
W. Huyer and A. Neumair, A new exact penalty function, SIAM J. Optimization, 13 (2003), 1141-1159.
doi: 10.1137/S1052623401390537. |
[10] |
B. Jerez, General equilibrium with asymmetric information: A dual approach, Journal of Economic Theory, in press. |
[11] |
H. T. Jongen, F. Twilt and G.-W. Weber, Semi-infinite optimization: Structure and stability of the feasible set, Journal of Optimization Theory and Application, 72 (1992), 529-552.
doi: 10.1007/BF00939841. |
[12] |
B. Li, C. J. Yu, K. L. Teo and G. R. Duan, An exact penalty function method for continuous inequality constrained optimal control problem, Journal of Optimization Theory and Applications, published online, 2011. |
[13] |
D.-H. Li, L. Qi, J. Tam and S.-Y. Wu, A smoothing Newton method for semi-infinite programming, Journal of Global Optimization, 30 (2004), 169-194.
doi: 10.1007/s10898-004-8266-z. |
[14] |
C. Ling, Q. Ni, L. Qi and S.-Y. Wu, A new smoothing Newton-type algorithm for semi-infinite programming, Journal of Global Optimization, 47 (2010), 133-159.
doi: 10.1007/s10898-009-9462-7. |
[15] |
Y. Liu and K. L. Teo, An adaptive dual parametrization algorithm for quadratic semi-infinite programming problems, Journal of Global Optimization, 24 (2002), 205-217.
doi: 10.1023/A:1020234019886. |
[16] |
Y. Liu, K. L. Teo and S.-Y. Wu, An new quadratic semi-infinite programming algorithm based on dual parameterization, Journal of Global Optimization, 29 (2005), 401-413.
doi: 10.1023/B:JOGO.0000047910.80739.95. |
[17] |
C. Ma and C. Y. Wang, A nonsmooth Levenberg-Marquardt method for solving semi-infinite programming problems, Journal of Computational and Applied Mathematics, 230 (2009), 633-642.
doi: 10.1016/j.cam.2009.01.004. |
[18] |
Q. Ni, C. Ling, L. Qi and K. L. Teo, A truncated projected Newton-type algorithm for large-scale semi-infinite programming, SIAM J. Optimization, 16 (2006), 1137-1154.
doi: 10.1137/040619867. |
[19] |
J.-S. Pang, Error bound in mathematical programming, Mathematical Programming, 79 (1997), 299-332.
doi: 10.1007/BF02614322. |
[20] |
C. Price and I. Coope, Numerical experienments in semi-infinite programming, Computational Optimization and Application, 6 (1996), 169-189. |
[21] |
L. Qi, S.-Y. Wu and G. L. Zhou, Semismooth newton methods for solving semi-infinite programming problems, Journal of Global Optimization, 27 (2003), 215-232.
doi: 10.1023/A:1024814401713. |
[22] |
R. Reemtsen, Discretization methods for the solution of semi-infinite programming problems, Journal of Optimization Theory and Application, 71 (1991), 85-103.
doi: 10.1007/BF00940041. |
[23] |
R. T. Rockafellar and R. J.-B. Wets, "Variational Analysis," Grundlehren der Math. Wissenschaften, 317, Springer-Verlag, Berlin, 1998. |
[24] |
G. Still, Discretization in semi-infinite programming: The rate of convergence, Mathematical Programming, 91 (2001), 53-69. |
[25] |
H. Voigt, Semi-infinite transportation problems, Zeitschrift für Analysis und ihre Anwendungen, 17 (1998), 729-741. |
[26] |
J.-P. Z. Wang and B. G. Lindsay, A penalized nonparametric maximum likelihood approach to species richness estimation, Journal of the American Statistical Association, 100 (2005), 942-959.
doi: 10.1198/016214504000002005. |
[27] |
G. A. Watson, Globally convergent methods for semi-infinite programming, BIT, 21 (1981), 362-373.
doi: 10.1007/BF01941472. |
[28] |
G. W. Weber and Ö. Uǧur, Optimization and dynamics of gene-environment networks with intervals, Journal of Industrial and Management Optimization, 3 (2007), 357-379. |
[29] |
S.-Y. Wu, S.-C. Fang and C.-J. Lin, Solving quadratic semi-infinite programming problems by using relaxed cutting-plane scheme, Journal of Computational and Applied Mathematics, 129 (2001), 89-104.
doi: 10.1016/S0377-0427(00)00544-6. |
[30] |
Z. L. Wu and J. J. Ye, First-order and second-order conditions for error bounds, SIAM J. Optimization, 14 (2003), 621-645.
doi: 10.1137/S1052623402412982. |
[31] |
K. F. C. Yiu, X. Q. Yang, S. Nordholm and K. L. Teo, Near-field broadband beamformer design via multidimensional semi-infinite linear programming techniques, IEEE Transactions on Speech and Audio Processing, 11 (2003), 725-732.
doi: 10.1109/TSA.2003.815527. |
[32] |
C. J. Yu, K. L. Teo, L. S. Zhang and Y. Q. Bai, A new exact penalty function method for continuous inequality constrained optimization problems, Journal of Industrial and Management Optimization, 6 (2010), 895-910. |
[33] |
K. Zhang and S. Wang, Convergence property of an interior penalty approach to pricing American option, Journal of Industrial and Management Optimization, 7 (2011), 435-447. |
[34] |
H. J. Zhou, K. F. C. Yiu and L. K. Li, Evaluating American put options on zero-coupon bonds by a penalty method, Journal of Computational and Applied Mathematics, 235 (2011), 3921-3931.
doi: 10.1016/j.cam.2011.01.038. |
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