Abstract
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.
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References
Weir, T., and Mond, B., Preinvex Functions in Multiple-Objective Optimization, Journal of Mathematical Analysis and Applications, Vol. 136, pp. 29–38, 1988.
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.
Hanson, M. A., On Sufficiency of the Kuhn–Tucker Conditions, Journal of Mathematical Analysis and Applications, Vol. 80, pp. 545–550, 1981.
Ben-Israel, A., and Mond, B., What is Invexity? Journal of the Australian Mathematical Society, Vol. 28B, pp. 1–9, 1986.
Craven, B. D., Invex Functions and Constrained Local Minima, Bulletin of the Australian Mathematical Society, Vol. 24, pp. 357–366, 1981.
Pini, R., Invexity and Generalized Convexity, Optimization, Vol. 22, pp. 513–525, 1991.
Craven, B. D., Invex Functions and Duality, Journal of the Australian Mathematical Society, Vol. 39A, pp. 1–20, 1985.
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.
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.
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.
Roberts, A. W., and Varberg, D. E., Convex Functions, Academic Press, New York, NY, 1973.
Karamardian, S., Duality in Mathematical Programming, Journal of Mathematical Analysis and Applications, Vol. 20, pp. 344–358, 1967.
Avriel, M., Diewert, W. E., Schaible, S., and Zang, I., Generalized Concavity, Plenum Press, New York, NY, 1988.
Yang, X. M., and Liu, S. Y., Three Kinds of Generalized Convexity, Journal of Optimization Theory and Applications, Vol. 86, pp. 501–513, 1995.
Mukherjee, R. M., and Reddy, L. V., Semicontinuity and Quasiconvex Functions, Journal of Optimization Theory and Applications, Vol. 94, pp. 715–726, 1997.
Wang, S. Y., Li, Z. F., and Craven, S. D., Global Efficiency in Multiobjective Programming, Optimization, Vol. 45, pp. 396–385, 1999.
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Yang, X.M., Yang, X.Q. & Teo, K.L. Characterizations and Applications of Prequasi-Invex Functions. Journal of Optimization Theory and Applications 110, 645–668 (2001). https://doi.org/10.1023/A:1017544513305
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DOI: https://doi.org/10.1023/A:1017544513305