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A derivative for semipreinvex functions and its applications in semipreinvex programming. (English) Zbl 1247.90253

Mishra, Shashi Kant (ed.), Topics in nonconvex optimization. Theory and applications. Selected papers based on the presentations at the advanced training programme on nonconvex optimization and applications, Varanasi, India, March 22–26, 2010. New York, NY: Springer (ISBN 978-1-4419-9639-8/hbk; 978-1-4419-9640-4/ebook). Springer Optimization and Its Applications 50, 79-86 (2011).
Summary: A directional derivative concept is introduced to develop Fritz-John and Kuhn-Tucker conditions for the optimization of general semipreinvex functions. The relationship between the optimization problem and the corresponding semiprevariational inequality problem is also shown.
For the entire collection see [Zbl 1216.90003].

MSC:

90C30 Nonlinear programming
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[1] Craven, B. D.; Glover, B. M., Invex functions and duality, Austral. Math. Soc., 39, 1-20 (1985) · Zbl 0565.90064 · doi:10.1017/S1446788700022126
[2] Ng, C. T.; Nikodem, K., On approximately convex functions, J. Math. Anal. Appl., 118, 103-108 (1993) · Zbl 0823.26006
[3] Martin, D. H., The essence of invexity, J. Optim. Theory Appl., 47, 1, 65-76 (1985) · Zbl 0552.90077 · doi:10.1007/BF00941316
[4] Luo, H. Z.; Xu, Z. K., Note on characterizations of prequasi-invex functions, J. Optim. Theory Appl., 120, 2, 429-439 (2004) · Zbl 1100.90035 · doi:10.1023/B:JOTA.0000015930.47489.b7
[5] Fu, J. Y.; Wang, Y. H., Arcwise connected cone-convex functions and mathematical programming, J. Optim. Theory Appl., 118, 2, 339-352 (2003) · Zbl 1039.90064 · doi:10.1023/A:1025451422581
[6] Hayashi, M.; Komiya, H., Perfect duality for convexlike programs, J. Optim. Theory Appl., 38, 179-189 (1982) · Zbl 0471.49033 · doi:10.1007/BF00934081
[7] Hanson, M. A., On ratio invexity in mathematical programming, J. Math. Anal. Appl., 205, 330-336 (1997) · Zbl 0872.90094 · doi:10.1006/jmaa.1997.5180
[8] Rueda, N. G.; Hanson, M. A., Optimality criteria in mathematical programming involving generalized invexity, J. Math. Anal. Appl., 130, 375-385 (1988) · Zbl 0647.90076 · doi:10.1016/0022-247X(88)90313-7
[9] Mohan, S. R.; Neogy, S. K., Note on invex sets and preinvex functions, J. Math. Anal. Appl., 189, 901-908 (1995) · Zbl 0831.90097 · doi:10.1006/jmaa.1995.1057
[10] Weir, T., Pre-invex functions in multiple objective optimization, J. Math. Anal. Appl., 136, 29-38 (1988) · Zbl 0663.90087 · doi:10.1016/0022-247X(88)90113-8
[11] Yang, X. Q.; Chen, G. Y., A class of nonconvex functions and pre-variational inequalities, J. Math. Anal. Appl., 169, 359-373 (1992) · Zbl 0779.90067 · doi:10.1016/0022-247X(92)90084-Q
[12] Xu, Z. K., Necessary conditions for suboptimization over the weakly efficient set associated to generalized invex multiobjective programming, J. Math. Anal. Appl., 201, 502-515 (1996) · Zbl 0858.90113 · doi:10.1006/jmaa.1996.0270
[13] Yang, X. M.; Yang, X. Q.; Teo, K. L., Characterizations and applications of prequasi-invex functions, J. Optim. Theory Appl., 110, 3, 645-668 (2001) · Zbl 1064.90038 · doi:10.1023/A:1017544513305
[14] Xu, Z. K., On invexity-type nonlinear programming problems, J. Optim. Theory Appl., 80, 1, 135-148 (1994) · Zbl 0797.90102 · doi:10.1007/BF02196597
[15] Pini, R.; Singh, C., A survey of recent 1985-1995 advances in generalized convexity with applications to duality theory and optimality conditions, Optimization, 39, 311-360 (1997) · Zbl 0872.90074 · doi:10.1080/02331939708844289
[16] Lin, C. Y.; Dong, J. L., The Methods and Theory forMulti-objective Optimization (in Chinese), 62-63 (1992), Changchun, China: Jilin Education Press, Changchun, China
[17] Lai, H. C., Optimality conditions for semi-preinvex programming, Taiwanese J. Math., 1, 4, 389-404 (1997) · Zbl 0894.90164
[18] Yamashita, N.; Taji, K.; Fukushima, M., Unconstrained optimization reformulations of variational inequality problems, J. Optim. Theory Appl., 92, 3, 439-456 (1997) · Zbl 0879.90180 · doi:10.1023/A:1022660704427
[19] Chang, S. S.; Zhang, C. J., On a class of generalized variational inequalities and quasivariational inequalities, J. Math. Anal. Appl., 179, 250-259 (1993) · Zbl 0803.49011 · doi:10.1006/jmaa.1993.1348
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