Abstract
This paper studies properties of a subdifferential defined using a generalized conjugation scheme. We relate this subdifferential together with the domain of an appropriate conjugate function and the ε-directional derivative. In addition, we also present necessary conditions for ε-optimality and global optimality in optimization problems involving the difference of two convex functions. These conditions will be written via this generalized notion of subdifferential studied in the first sections of the paper.
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Horst, R., Thoai, N.V.: DC Programming: Overview. J. Optim. Theory Appl. 103(1), 1–43 (1999)
Carrizosa, E., Guerrero, V., Romero-Morales, D.: Visualizing data as objects by DC (difference of convex) optimization. Mathematical Programming Ser. B 169, 119–140 (2018). https://doi.org/10.1007/s10107-017-1156-1
An, L.T.H., Tao, P.D.: The DC (Difference of Convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems. Annals Operations Research 133, 23–46 (2005). https://doi.org/10.1007/s10479-004-5022-1
Tao, P.H., An, L.T.H.: Convex analysis approach to D. C programming: theory, algorithms and applications. Acta Mathematica Vietnamica 22(1), 289–355 (1997)
Correa, R., López, M.A., Pérez-Aros, P.: Necessary and sufficient optimality conditions in DC semi-infinite programming. SIAM J. Optimization 31 (1), 837–865 (2021). https://doi.org/10.1137/19M1303320
Dolgopolik, M.V.: New global optimality conditions for nonsmooth DC optimization problems. J. Glob. Optim. 76, 25–55 (2020)
Jeyakumar, V., Glover, B.M.: Characterizing global optimality for DC optimization problems under convex inequality constraints. J. Glob. Optim. 8, 171–187 (1996)
Ernst, E., Théra, M.: Necessary and sufficient conditions for the existence of a global maximun for convex functions in reflexive Banach spaces. Journal of Convex Analysis 13(3-4), 687–694 (2006)
Dinh, N., Mordukhovich, B.S., Nguia, T.T.A.: Qualification and optimality conditions for DC programs with infinite constraints. Acta Mathematica Vietnamica 34(1), 125–155 (2009)
Fang, D.H., Zhao, X.P.: Local and global optimality conditions for DC infinite optimization problems. Taiwan. J. Math. 18(3), 817–834 (2014). https://doi.org/10.11650/tjm.18.2014.3888
Jeyakumar, V., Li, G.Y.: Necessary global optimality conditions for nonlinear programming problems with polynomial constraints. Mathematical Programming Ser. A 126, 393–399 (2011)
Sun, X. -K., Fu, H.Y.: A note on optimality conditions for DC programs involving composite functions. Abstract and Applied Analysis (203467), 6 (2014)
Zhang, Q.: A new necessary and sufficient global optimality condition for canonical DC problems. J. Glob. Optim. 55, 559–577 (2013). https://doi.org/10.1007/s10898-012-9908-1
Flores-Bazán, F., Martínez-Legaz, J.: Simplified global optimality conditions in generalized conjugation theory. In: Generalized Convexity, Generalized Monotonicity: Recent Results. Nonconvex Optimization and Its Applications. https://doi.org/10.1007/978-1-4613-3341-8_14, vol. 27, pp 237–292. Springer (1998)
Hiriart-Urruty, J.B.: Global optimality conditions in maximizing a convex quadratic function under convex quadratic constraints. J. Glob. Optim. 21, 445–455 (2001)
Dür, M., Horst, R., Locatelli, M.: Necessary and sufficient global optimality conditions for convex maximization revisited. J. Math. Anal. Appl. 217, 637–649 (1998)
Hiriart-Urruty, J.B.: From convex optimization to nonconvex optimization. Necessary and sufficient conditions for global optimality. In: Nonsmooth Optimization and Related Topics. Ettore Majorana International Sciences. https://doi.org/10.1007/978-1-4757-6019-4_13, vol. 43, pp 219–239. Springer (1989)
Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, New Jersey (2002)
Rodríguez, M.-M.-L., Vicente-Pérez, J.: On evenly convex functions. J. Convex Anal. 18, 721–736 (2011)
Fenchel, W.: A remark on convex sets and polarity. Comm. Sè,m. Math. Univ. Lund (Medd. Lunds Algebra Univ. Math. Sem.) (Tome Suppl’ementaire), 82–89 (1952)
Goberna, M. -A., Jornet, V., Rodríguez, M.-M.-L.: On linear systems containing strict inequalities. Linear Algebra Appl. 360, 151–171 (2003). https://doi.org/10.1016/S0024-3795(02)00445-7
Goberna, M. -A., Rodríguez, M.-M.-L.: Analyzing linear systems containing strict inequalities via evenly convex hulls. Eur. J. Oper. Res. 169, 1079–1095 (2006)
Klee, V., Maluta, E., Zanco, C.: Basic properties of evenly convex sets. J. Convex Anal. 14(1), 137–148 (2006)
Martínez-Legaz, J.-E., Vicente-Pérez, J.: The e-support function of an e-convex set and conjugacy for e-convex functions. J. Math. Anal. Appl. 376, 602–612 (2011). https://doi.org/10.1016/j.jmaa.2010.10.058
Fajardo, M.-D., Grad, S., Vidal, J.: New duality results for evenly convex optimization problems. Optimization 70(9), 1837–1858 (2021). https://doi.org/10.1080/02331934.2020.1756287
Fajardo, M.D., Goberna, M.A., Rodríguez, M. M. L., Vicente-Pérez, J.: Even Convexity and Optimization. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-53456-1
Daniilidis, A., Martínez-legaz, J.-E.: Characterizations of evenly convex sets and evenly quasiconvex functions. J. Math. Anal. Appl. 273, 58–66 (2002). https://doi.org/10.1016/S0022-247X(02)00206-8
Fajardo, M. -D., Vidal, J.: A comparison of alternative c-conjugate dual problems in infinite convex optimization. Optimization 66(5), 705–722 (2017). https://doi.org/10.1080/02331934.2017.1295046
Moreau, J. -J.: Inf-convolution, sous-additivité, convexité des fonctions numériques. J. Math. Pures Appl. 49, 109–154 (1970)
Fajardo, M. -D., Vicente-Pérez, J., Rodríguez, M.-M.-L.: Infimal convolution, c-subdifferentiability and Fenchel duality in evenly convex optimization. TOP 20(2), 375–396 (2012). https://doi.org/10.1007/s11750-011-0208-6
Fajardo, M.D., Vidal, J.: E\(^{\prime }\)-convex sets and functions: Properties and characterizations. Vietnam J. Math. 48(3), 407–423 (2020). https://doi.org/10.1007/s10013-020-00414-2
Martínez-Legaz, J.-E.: Generalized convex duality and its economic applications. In: Handbook of Generalized Convexity and Generalized Monotonicity. Nonconvex Optim. Appl, vol. 76, pp 237–292. Springer
Balder, E.J.: An extension of duality-stability relations to non-convex optimization problems. SIAM J. Control Optimization 15 (2), 329–343 (1977). https://doi.org/10.1137/0315022
Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland publishing company, Amsterdam (1976)
Rockafellar, R.-T.: Convex Analysis. Princeton University Press, Princeton (1970)
Martínez-Legaz, J.-E., Seeger, A.: A formula on the approximate subdifferential set of the difference of convex functions. Bull. Austral Math. Soc. 45, 37–41 (1992)
Hiriart-Urruty, J.B.: Generalized differentiability duality and optimization for problems dealing with differences of convex functions. In: Convexity and Duality in Optimization. Lectures Notes in Econom. and Math. Systems. https://doi.org/10.1007/978-3-642-45610-7_3, vol. 256, pp 37–70. Springer (1985)
Martínez-Legaz, J.-E.: Generalized Conjugation and Related Topics. In: Generalized Convexity and Fractional Programming with Economic Applications, pp 168–197. Springer, Berlin (1990)
Dür, M.: A parametric characterization of local optimality. Mathematical Methods of Operations Research 57, 101–109 (2003). https://doi.org/10.1007/s001860200232
Bomze, I.M., Lemaréchal, C.: Necessary conditions for local optimality in difference-of-convex programming. Journal of Convex Analysis 17, 673–680 (2010)
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The authors wish to thank the anonymous referees for their valuable critical comments which have definitely improved the first version of the manuscript.
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Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Research partially supported by MICIIN of Spain and ERDF of EU, Grant PGC2018 097960-B-C22.
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Fajardo, M., Vidal, J. On Subdifferentials Via a Generalized Conjugation Scheme: An Application to DC Problems and Optimality Conditions. Set-Valued Var. Anal 30, 1313–1331 (2022). https://doi.org/10.1007/s11228-022-00644-1
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DOI: https://doi.org/10.1007/s11228-022-00644-1