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On the Hermite--Hadamard inequality for convex functions of two variables
The Toland-Fenchel-Lagrange duality of DC programs for composite convex functions
1. | Department of Mathematics, Soochow University, Suzhou, 215006 |
2. | School of Sciences, Zhejiang A & F University, Hangzhou 311300, China |
References:
[1] |
L. T. H. An, An efficient algorithm for globally minimizing a quadratic function under convex quadratic constraints, Math. Program., 87 (2000), 401-426.
doi: 10.1007/s101070050003. |
[2] |
L. T. H. An and P. D. Tao, The DC (difference of convex functions) programming and DCA revisited with DC models of real world non-convex optimization problems, Ann. Oper. Res., 133 (2005), 23-46.
doi: 10.1007/s10479-004-5022-1. |
[3] |
J. M. Borwein and A. S. Lewis, Partially finite convex programming, part I: Quasi relative interiors and duality theory, Math. Program., 57 (1992), 15-48.
doi: 10.1007/BF01581072. |
[4] |
R. I. Boţ, E. R. Csetnek and G. Wanka, Regularity conditions via quasi-relative interior in convex programming, SIAM. J. Optim., 19 (2008), 217-233.
doi: 10.1137/07068432X. |
[5] |
R. I. Boţ, S. M. Grad and G. Wanka, A new constraint qualification for the formula of the subdifferential of composed convex functions in infinite dimensional spaces, Math. Nachr., 281 (2008), 1088-1107.
doi: 10.1002/mana.200510662. |
[6] |
R. I. Boţ, S. M. Grad and G. Wanka, Generalized Moreau-Rockafellar results for composed convex functions, Optimization, 58 (2009), 917-933.
doi: 10.1080/02331930902945082. |
[7] |
R. I. Boţ, S. M. Grad and G. Wanka, On strong and total Lagrange duality for convex optimization problems, J. Math. Anal. Appl., 337 (2008), 1315-1325.
doi: 10.1016/j.jmaa.2007.04.071. |
[8] |
R. I. Boţ, I. B. Hodrea and G. Wanka, Farkas-type results for inequality systems with composed convex functions via conjugate duality, J. Math. Anal. Appl., 322 (2006), 316-328.
doi: 10.1016/j.jmaa.2005.09.007. |
[9] |
R. I. Boţ and G. Wanka, A weaker regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces, Nonlinear Anal., 64 (2006), 2787-2804.
doi: 10.1016/j.na.2005.09.017. |
[10] |
R. I. Boţ and G. Wanka, An alternative formulation for a new closed cone constraint qualification, Nonlinear Anal., 64 (2006), 1367-1381.
doi: 10.1016/j.na.2005.06.041. |
[11] |
R. S. Burachik and V. Jeyakumar, A dual condition for the convex subdifferential sum formula with applications, J. Convex Anal., 12 (2005), 279-290. |
[12] |
R. S. Burachik and V. Jeyakumar, A new geometric condition for Fenchel duality in infinite dimensional spaces, Math. Program., 104 (2005), 229-233.
doi: 10.1007/s10107-005-0614-3. |
[13] |
B. D. Craven, Mathematical Programming and Control Theory, Chapman and Hall, London, 1978. |
[14] |
N. Dinh, M. A. Goberna and M. A. López, From linear to convex systems: consistency, Farkas lemma and applications, J. Convex Anal., 13 (2006), 113-133. |
[15] |
N. Dinh, M. A. Goberna and M. A. López and T. Q. Son, New Farkas-type constraint qualifications in convex infinite programming, ESAIM Control Optim. Calc. Var., 13 (2007), 580-597.
doi: 10.1051/cocv:2007027. |
[16] |
N. Dinh, T. T. A. Nghia and G. Vallet, A closedness condition and its applications to DC programs with convex constraints, Optimization, 59 (2010), 541-560.
doi: 10.1080/02331930801951348. |
[17] |
N. Dinh, G. Vallet and T. T. A. Nghia, Farkas-type results and duality for DC programs with convex constraints, J. Convex Anal., 15 (2008), 253-262. |
[18] |
D. H. Fang, C. Li and X. Q. Yang, Stable and total Fenchel duality for DC optimization problems in locally convex spaces, SIAM J. Optim., 21 (2011), 730-760.
doi: 10.1137/100789749. |
[19] |
S. P. Fitzpatrick and S. Simons, The conjugates, compositions and marginals of convex functions, J. Convex Anal., 8 (2001), 423-446. |
[20] |
M. S. Gowda and M. Teboulle, A comparison of constraint qualifications in infinite dimensional convex programming, SIAM J. Control Optim., 28 (1990), 925-935.
doi: 10.1137/0328051. |
[21] |
C. Li, D. H. Fang, G. López and M. A. López, Stable and total Fenchel duality for convex optimization problems in locally convex spaces, SIAM, J. Optim., 20 (2009), 1032-1051.
doi: 10.1137/080734352. |
[22] |
C. Li, F. Ng and T. K. Pong, The SECQ, linear regularity and the strong CHIP for infinite system of closed convex sets in normed linear space, SIAM J. Optim., 18 (2007), 643-665.
doi: 10.1137/060652087. |
[23] |
G. Li, X. Q. Yang and Y. Y. Zhou, Stable strong and total parametrized dualities for DC optimization problems in locally convex spaces, J. Ind. Manag. Optim., 9 (2013), 669-685.
doi: 10.3934/jimo.2013.9.669. |
[24] |
J. F. Toland, Duality in non-convex optimization, J. Math. Anal. Appl., 66 (1978), 399-415. |
[25] |
H. Tuy, Convex Analysis and Global Optimization, Kluwer Academic Publishers, Dordrecht, 1998. |
[26] |
C. Zălinescu, Convex Analysis in General Vector Space, World Sciencetific Publishing, Singapore, 2002.
doi: 10.1142/9789812777096. |
show all references
References:
[1] |
L. T. H. An, An efficient algorithm for globally minimizing a quadratic function under convex quadratic constraints, Math. Program., 87 (2000), 401-426.
doi: 10.1007/s101070050003. |
[2] |
L. T. H. An and P. D. Tao, The DC (difference of convex functions) programming and DCA revisited with DC models of real world non-convex optimization problems, Ann. Oper. Res., 133 (2005), 23-46.
doi: 10.1007/s10479-004-5022-1. |
[3] |
J. M. Borwein and A. S. Lewis, Partially finite convex programming, part I: Quasi relative interiors and duality theory, Math. Program., 57 (1992), 15-48.
doi: 10.1007/BF01581072. |
[4] |
R. I. Boţ, E. R. Csetnek and G. Wanka, Regularity conditions via quasi-relative interior in convex programming, SIAM. J. Optim., 19 (2008), 217-233.
doi: 10.1137/07068432X. |
[5] |
R. I. Boţ, S. M. Grad and G. Wanka, A new constraint qualification for the formula of the subdifferential of composed convex functions in infinite dimensional spaces, Math. Nachr., 281 (2008), 1088-1107.
doi: 10.1002/mana.200510662. |
[6] |
R. I. Boţ, S. M. Grad and G. Wanka, Generalized Moreau-Rockafellar results for composed convex functions, Optimization, 58 (2009), 917-933.
doi: 10.1080/02331930902945082. |
[7] |
R. I. Boţ, S. M. Grad and G. Wanka, On strong and total Lagrange duality for convex optimization problems, J. Math. Anal. Appl., 337 (2008), 1315-1325.
doi: 10.1016/j.jmaa.2007.04.071. |
[8] |
R. I. Boţ, I. B. Hodrea and G. Wanka, Farkas-type results for inequality systems with composed convex functions via conjugate duality, J. Math. Anal. Appl., 322 (2006), 316-328.
doi: 10.1016/j.jmaa.2005.09.007. |
[9] |
R. I. Boţ and G. Wanka, A weaker regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces, Nonlinear Anal., 64 (2006), 2787-2804.
doi: 10.1016/j.na.2005.09.017. |
[10] |
R. I. Boţ and G. Wanka, An alternative formulation for a new closed cone constraint qualification, Nonlinear Anal., 64 (2006), 1367-1381.
doi: 10.1016/j.na.2005.06.041. |
[11] |
R. S. Burachik and V. Jeyakumar, A dual condition for the convex subdifferential sum formula with applications, J. Convex Anal., 12 (2005), 279-290. |
[12] |
R. S. Burachik and V. Jeyakumar, A new geometric condition for Fenchel duality in infinite dimensional spaces, Math. Program., 104 (2005), 229-233.
doi: 10.1007/s10107-005-0614-3. |
[13] |
B. D. Craven, Mathematical Programming and Control Theory, Chapman and Hall, London, 1978. |
[14] |
N. Dinh, M. A. Goberna and M. A. López, From linear to convex systems: consistency, Farkas lemma and applications, J. Convex Anal., 13 (2006), 113-133. |
[15] |
N. Dinh, M. A. Goberna and M. A. López and T. Q. Son, New Farkas-type constraint qualifications in convex infinite programming, ESAIM Control Optim. Calc. Var., 13 (2007), 580-597.
doi: 10.1051/cocv:2007027. |
[16] |
N. Dinh, T. T. A. Nghia and G. Vallet, A closedness condition and its applications to DC programs with convex constraints, Optimization, 59 (2010), 541-560.
doi: 10.1080/02331930801951348. |
[17] |
N. Dinh, G. Vallet and T. T. A. Nghia, Farkas-type results and duality for DC programs with convex constraints, J. Convex Anal., 15 (2008), 253-262. |
[18] |
D. H. Fang, C. Li and X. Q. Yang, Stable and total Fenchel duality for DC optimization problems in locally convex spaces, SIAM J. Optim., 21 (2011), 730-760.
doi: 10.1137/100789749. |
[19] |
S. P. Fitzpatrick and S. Simons, The conjugates, compositions and marginals of convex functions, J. Convex Anal., 8 (2001), 423-446. |
[20] |
M. S. Gowda and M. Teboulle, A comparison of constraint qualifications in infinite dimensional convex programming, SIAM J. Control Optim., 28 (1990), 925-935.
doi: 10.1137/0328051. |
[21] |
C. Li, D. H. Fang, G. López and M. A. López, Stable and total Fenchel duality for convex optimization problems in locally convex spaces, SIAM, J. Optim., 20 (2009), 1032-1051.
doi: 10.1137/080734352. |
[22] |
C. Li, F. Ng and T. K. Pong, The SECQ, linear regularity and the strong CHIP for infinite system of closed convex sets in normed linear space, SIAM J. Optim., 18 (2007), 643-665.
doi: 10.1137/060652087. |
[23] |
G. Li, X. Q. Yang and Y. Y. Zhou, Stable strong and total parametrized dualities for DC optimization problems in locally convex spaces, J. Ind. Manag. Optim., 9 (2013), 669-685.
doi: 10.3934/jimo.2013.9.669. |
[24] |
J. F. Toland, Duality in non-convex optimization, J. Math. Anal. Appl., 66 (1978), 399-415. |
[25] |
H. Tuy, Convex Analysis and Global Optimization, Kluwer Academic Publishers, Dordrecht, 1998. |
[26] |
C. Zălinescu, Convex Analysis in General Vector Space, World Sciencetific Publishing, Singapore, 2002.
doi: 10.1142/9789812777096. |
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