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

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  • 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.
    Mathematics Subject Classification: Primary: 90C26, 90C46; Secondary: 90C30.

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  • [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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