Programming with semilocally convex functions. (English) Zbl 0762.90064

Summary: A theorem of the alternatives is derived for semilocally convex functions defined on locally starshaped sets. This result is applied to constrained minimization problems to obtain optimality conditions and duality theorems.


90C26 Nonconvex programming, global optimization
Full Text: DOI


[1] Avriel, M.; Diewert, W. E.; Schaible, S.; Zang, I., Generalized Concavity, (Mathematical Concepts and Methods in Science and Engineering, Vol. 36 (1988), Plenum: Plenum New York) · Zbl 0483.26007
[2] Craven, B. D., Mathematical Programming and Control Theory (1978), Chapman & Hall: Chapman & Hall London · Zbl 0431.90039
[3] Ewing, G. M., Sufficient conditions for global minima of suitably convex functions from variational and control theory, SIAM Rev., 19, 202-220 (1977) · Zbl 0361.49011
[4] Kaul, R. N.; Kaur, S., Generalisations of convex and related functions, European J. Oper. Res., 9, 369-377 (1982) · Zbl 0501.90090
[5] Kaur, S., Theoretical Studies in Mathematical Programming, (Ph.D. thesis (1983), University of Delhi)
[6] Kuhn, H. W.; Tucker, A. W., Nonlinear programming, (Neyman, J., Proceedings of the Second Berkely Symposium on Mathematical Statistics and Probability (1951), Univ. of California Press: Univ. of California Press Berkeley, CA), 481-493 · Zbl 0044.05903
[7] John, F., Extremum problems with inequalities as subsidiary condtions, (Friedrichs, K. O.; Neugebauer, O. E.; Stoker, J. J., Studies and Essays, Courant Aniversary Volume (1948), Wiley-Interscience: Wiley-Interscience New York)
[8] Werner, J., Optimization Theory and Applications (1984), Vieweg: Vieweg Braunschweig
[9] Wolfe, P., A duality theorem for nonlinear programming, Quart. Appl. Math., 19, 239-244 (1961) · Zbl 0109.38406
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