Optimality conditions for semi-infinite programming problems involving generalized convexity. (English) Zbl 1462.90145
Summary: We apply some advanced tools of quasiconvex analysis to establish Karush-Kuhn-Tucker type necessary and sufficient optimality conditions for non-differentiable semi-infinite programming problems. In addition, we propose a linear characterization of optimality for the mentioned problems. Examples are also designed to analyze and illustrate the results obtained.
MSC:
| 90C34 | Semi-infinite programming |
| 90C46 | Optimality conditions and duality in mathematical programming |
Keywords:
semi-infinite programming; plastria function; Gutiérrez functions; optimality conditions; linear characterizationReferences:
| [1] | Cánovas, M.J., Hantoute, A., Parra, J., Toledo, F.J.: Boundary of subdifferentials and calmness moduli in linear semi-infinite optimization. Optim. Lett. 9, 513-521 (2015) · Zbl 1332.90319 · doi:10.1007/s11590-014-0767-1 |
| [2] | Gutiérrez Díez, J.M.: Infragradients and directions of decrease. Rev. Real Acad. Cienc. Exact. Fís. Natur. Madrid 78, 523-532 (1984) · Zbl 0576.49006 |
| [3] | Jeyakumar, V., Rubinov, A.M., Glover, B.M., Ishizuku, Y.: Inequality systems and global optimization. J. Math. Anal. Appl. 202, 900-919 (1996) · Zbl 0856.90128 · doi:10.1006/jmaa.1996.0353 |
| [4] | Kabgani, A., Soleimani-damaneh, M.: The relationships between convexificators and Greenberg-Pierskalla subdifferentials for quasiconvex functions. Numer. Funct. Anal. Optim. 38, 1548-1563 (2017) · Zbl 1383.49023 · doi:10.1080/01630563.2017.1349144 |
| [5] | Kanzi, N.: Necessary optimality conditions for nonsmooth semi-infinite programming problems. J. Glob. Optim. 49, 713-725 (2011) · Zbl 1254.90264 · doi:10.1007/s10898-010-9561-5 |
| [6] | Kanzi, N.: Constraint qualifications in semi-infinite systems and their applications in nonsmooth semi-infinite problems with mixed constraints. SIAM J. Optim. 24, 559-572 (2014) · Zbl 1297.90164 · doi:10.1137/130910002 |
| [7] | Kanzi, N., Nobakhtian, S.: Optimality conditions for nonsmooth semi-infinite programming. Optimization 59, 717-727 (2010) · Zbl 1195.90088 · doi:10.1080/02331930802434823 |
| [8] | Kanzi, N., Nobakhtian, S.: Nonsmooth semi-infinite programming problems with mixed constraints. J. Math. Anal. Appl. 351, 170-181 (2008) · Zbl 1172.90019 · doi:10.1016/j.jmaa.2008.10.009 |
| [9] | Kanzi, N., Soleimani-damaneh, M.: Slater CQ, optimality and duality for quasiconvex semi-infinite optimization problems. J. Math. Anal. Appl. 434, 638-651 (2016) · Zbl 1334.90183 · doi:10.1016/j.jmaa.2015.08.013 |
| [10] | Linh, N.T., Penot, J.P.: Optimality conditions for quasiconvex programming. SIAM J. Optim. 17, 500-510 (2006) · Zbl 1165.90679 · doi:10.1137/040621843 |
| [11] | Li, W., Nahak, C., Singer, I.: Constraint qualifications in semi-infinite systems of convex inequalities. SIAM J. Optim. 11, 31-52 (2000) · Zbl 0999.90045 · doi:10.1137/S1052623499355247 |
| [12] | Liu, Y., Goberna, M.A.: Asymptotic optimality conditions for linear semiinfinite programming. Optimization 65, 387-414 (2016) · Zbl 1332.90321 · doi:10.1080/02331934.2015.1051533 |
| [13] | López, M.A., Still, G.: Semi-infinite programming. Eur. J. Oper. Res. 180, 491-518 (2007) · Zbl 1124.90042 · doi:10.1016/j.ejor.2006.08.045 |
| [14] | López, M.A., Vercher, E.: Optimality conditions for nondifferentiable convex semi-infinite Programming. Math. Program. 27, 307-319 (1983) · Zbl 0527.49029 · doi:10.1007/BF02591906 |
| [15] | Martinez-Legaz, J.E.: Weak lower sbdifferentials and applications. Optimization 21, 321-341 (1990) · Zbl 0714.49018 · doi:10.1080/02331939008843551 |
| [16] | Mordukhovich, B.S., Nghia, T.T.A.: Constraint qualification and optimality conditions in semi-infinite and infinite programming. Math. Program. 139, 271-300 (2013) · Zbl 1292.90283 · doi:10.1007/s10107-013-0672-x |
| [17] | Mordukhovich, B.S., Nghia, T.T.A.: Nonsmooth cone-constrained optimization with applications to semi-infinite programming. Math. Oper. Res. 39, 301-324 (2014) · Zbl 1291.49014 · doi:10.1287/moor.2013.0622 |
| [18] | Mordukhovich, B.S., Nghia, T.T.A.: Subdifferentials of nonconvex supremum functions and their applications to semi-infinite and infinite programs with Lipschitzian data. SIAM J. Optim. 23, 406-431 (2013) · Zbl 1266.49027 · doi:10.1137/110857738 |
| [19] | Penot, JP; Crouzeix, J-P (ed.); Martinez-Legaz, JE (ed.); Volle, M. (ed.), Are generalized derivatives useful for generalized convex functions?, 3-59 (1998), Dordrecht · Zbl 0957.49013 · doi:10.1007/978-1-4613-3341-8_1 |
| [20] | Plastria, F.: Lower subdifferentiable functions and their minimization by cutting plane. J. Optim. Theory Appl. 46, 37-53 (1985) · Zbl 0542.90083 · doi:10.1007/BF00938758 |
| [21] | Rockafellar, R.T., Wets, J.B.: Variational Analysis. Springer, Berlin (1998) · Zbl 0888.49001 · doi:10.1007/978-3-642-02431-3 |
| [22] | Xu, M., Wu, S.Y., Ye, J.: Solving semi-infinite programs by smoothing projected gradient method. Comput. Optim. Appl. 59, 591-616 (2014) · Zbl 1308.65101 · doi:10.1007/s10589-014-9654-z |
| [23] | Zhao, X.: On constraints qualification for an infinite system of quasiconvex inequalities in normed linear space. Taiwan. J. Math. 20(206), 685-697 (2016) · Zbl 1357.90120 · doi:10.11650/tjm.20.2016.6713 |
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