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Characterizations and applications of prequasi-invex functions. (English) Zbl 1064.90038

Summary: In this paper, two new types of generalized convex functions are introduced. They are called strictly prequasi-invex functions and semistrictly prequasi-invex functions. Note that prequasi-invexity does not imply semistrict prequasi-invexity. The characterization of prequasi-invex functions is established under the condition of lower semicontinuity, upper semicontinuity, and semistrict prequasi-invexity, respectively. Furthermore, the characterization of semistrictly prequasi-invex functions is also obtained under the condition of prequasi-invexity and lower semicontinuity, respectively. A similar result is also obtained for strictly prequasi-invex functions. It is worth noting that these characterizations reveal various interesting relationships among prequasi-invex, semistrictly prequasi-invex, and strictly prequasi-invex functions. Finally, prequasi-invex, semistrictly prequasi-invex, and strictly prequasi-invex functions are used in the study of optimization problems.

MSC:

90C26 Nonconvex programming, global optimization
90C29 Multi-objective and goal programming
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[1] Weir, T., and Mond, B., Preinvex Functions in Multiple-Objective Optimization, Journal of Mathematical Analysis and Applications, Vol. 136, pp. 29–38, 1988. · Zbl 0663.90087
[2] Weir, T., and Jeyakumar, V., A Class of Nonconvex Functions and Mathematical Programming, Bulletin of the Australian Mathematical Society, Vol. 38, pp. 177–189, 1988. · Zbl 0639.90082
[3] Hanson, M. A., On Sufficiency of the Kuhn–Tucker Conditions, Journal of Mathematical Analysis and Applications, Vol. 80, pp. 545–550, 1981. · Zbl 0463.90080
[4] Ben-Israel, A., and Mond, B., What is Invexity? Journal of the Australian Mathematical Society, Vol. 28B, pp. 1–9, 1986. · Zbl 0603.90119
[5] Craven, B. D., Invex Functions and Constrained Local Minima, Bulletin of the Australian Mathematical Society, Vol. 24, pp. 357–366, 1981. · Zbl 0452.90066
[6] Pini, R., Invexity and Generalized Convexity, Optimization, Vol. 22, pp. 513–525, 1991. · Zbl 0731.26009
[7] Craven, B. D., Invex Functions and Duality, Journal of the Australian Mathematical Society, Vol. 39A, pp. 1–20, 1985. · Zbl 0565.90064
[8] Khan, Z. A., and Hanson, M. A., On Ratio Invexity in Mathematical Programming, Journal of Mathematical Analysis and Applications, Vol. 206, pp. 330–336, 1997. · Zbl 0872.90094
[9] Mohan, S. R., and Neogy, S. K., On Invex Sets and Preinvex Functions, Journal of Mathematical Analysis and Applications, Vol. 189, pp. 901–908, 1995. · Zbl 0831.90097
[10] Yang, X. Q., and Chen, G. Y., A Class of Nonconvex Functions and Prevariational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 169, pp. 359–373, 1992. · Zbl 0779.90067
[11] Roberts, A. W., and Varberg, D. E., Convex Functions, Academic Press, New York, NY, 1973. · Zbl 0271.26009
[12] Karamardian, S., Duality in Mathematical Programming, Journal of Mathematical Analysis and Applications, Vol. 20, pp. 344–358, 1967. · Zbl 0157.49603
[13] Avriel, M., Diewert, W. E., Schaible, S., and Zang, I., Generalized Concavity, Plenum Press, New York, NY, 1988.
[14] Yang, X. M., and Liu, S. Y., Three Kinds of Generalized Convexity, Journal of Optimization Theory and Applications, Vol. 86, pp. 501–513, 1995. · Zbl 0838.90117
[15] Mukherjee, R. M., and Reddy, L. V., Semicontinuity and Quasiconvex Functions, Journal of Optimization Theory and Applications, Vol. 94, pp. 715–726, 1997. · Zbl 0892.90145
[16] Wang, S. Y., Li, Z. F., and Craven, S. D., Global Efficiency in Multiobjective Programming, Optimization, Vol. 45, pp. 396–385, 1999. · Zbl 0955.90122
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