American Institute of Mathematical Sciences

Advanced
2014, 4(1): 9-23. doi: 10.3934/naco.2014.4.9

The Toland-Fenchel-Lagrange duality of DC programs for composite convex functions

Yuying Zhou 1, and  Gang Li 2,

1. 

Department of Mathematics, Soochow University, Suzhou, 215006
2. 

School of Sciences, Zhejiang A & F University, Hangzhou 311300, China

Received  March 2013 Revised  October 2013 Published  December 2013

In this paper, by virtue of the epigraph technique, we construct a new kind of closedness-type constraint qualification, which is the sufficient and necessary condition to guarantee the strong duality between a cone constraint composite optimization problem and its dual problem holds. Under this closedness-type constraint qualification condition, we obtain a formula of subdifferential for composite functions and study a cone constraint composite DC optimization problem in locally convex Hausdorff topological vector spaces.
Keywords: constraint qualification., duality, Epigraph, composite.
Mathematics Subject Classification: Primary: 90C26, 90C46; Secondary: 90C3.
Citation: Yuying Zhou, Gang Li. The Toland-Fenchel-Lagrange duality of DC programs for composite convex functions. Numerical Algebra, Control & Optimization, 2014, 4 (1) : 9-23. doi: 10.3934/naco.2014.4.9
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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

B. D. Craven, Mathematical Programming and Control Theory, Chapman and Hall, London, 1978.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

S. P. Fitzpatrick and S. Simons, The conjugates, compositions and marginals of convex functions, J. Convex Anal., 8 (2001), 423-446.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

J. F. Toland, Duality in non-convex optimization, J. Math. Anal. Appl., 66 (1978), 399-415. Google Scholar

[25]

H. Tuy, Convex Analysis and Global Optimization, Kluwer Academic Publishers, Dordrecht, 1998.  Google Scholar

[26]

C. Zălinescu, Convex Analysis in General Vector Space, World Sciencetific Publishing, Singapore, 2002. doi: 10.1142/9789812777096.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

B. D. Craven, Mathematical Programming and Control Theory, Chapman and Hall, London, 1978.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

S. P. Fitzpatrick and S. Simons, The conjugates, compositions and marginals of convex functions, J. Convex Anal., 8 (2001), 423-446.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

J. F. Toland, Duality in non-convex optimization, J. Math. Anal. Appl., 66 (1978), 399-415. Google Scholar

[25]

H. Tuy, Convex Analysis and Global Optimization, Kluwer Academic Publishers, Dordrecht, 1998.  Google Scholar

[26]

C. Zălinescu, Convex Analysis in General Vector Space, World Sciencetific Publishing, Singapore, 2002. doi: 10.1142/9789812777096.  Google Scholar

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