Abstract
The issue of the pseudoconvexity of a function on a closed set is addressed. It is proved that if a function has no critical points on the boundary of a convex set, then the pseudoconvexity on the interior guarantees the pseudoconvexity on the closure of the set. This result holds even when the boundary of the set contains line segments, and it is used to characterize the pseudoconvexity, on the nonnegative orthant, of a wide class of generalized fractional functions, namely the sum between a linear one and a ratio which has an affine function as numerator and, as denominator, the p-th power of an affine function. The relationship between quasiconvexity and pseudoconvexity is also investigated.
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References
Avriel, M., Schaible, S.: Second order characterizations of pseudoconvex functions. Math. Program. 14(1), 170–185 (1978)
Avriel, M., Diewert, W.E., Schaible, S., Zang, I.: Generalized concavity, vol. 63. SIAM (2010). (originally published by Plenum Press, New York, 1988)
Cambini, A., Martein, L.: Generalized Convexity and Optimization: Theory and Applications. Lecture Notes in Economics and Mathematical Systems, vol. 616. Springer, Berlin (2009)
Cambini, R., Sodini, C.: A unifying approach to solve some classes of rank-three multiplicative and fractional programs involving linear functions. Eur. J. Oper. Res. 207(1), 25–29 (2010)
Carosi, L., Martein, L.: A sequential method for a class of pseudoconcave fractional problems. Cent. Eur. J. Oper. Res. 16(2), 153–164 (2008)
Carosi, L., Martein, L.: Characterizing the pseudoconvexity of a wide class of generalized fractional functions. Technical report, Department of Economics and Management, University of Pisa, Italy, (2013)
Cottle, R.W., Ferland, J.A.: Matrix-theoretic criteria for the quasi-convexity and pseudo-convexity of quadratic functions. Linear Algebra Appl. 5(2), 123–136 (1972)
Crouzeix, J.P., Ferland, J.A.: Criteria for quasi-convexity and pseudo-convexity: relationships and comparisons. Math. Program. 23(1), 193–205 (1982)
Martein, L., Carosi, L.: The sum of a linear and a linear fractional function: pseudoconvexity on the nonnegative orthant and solution methods. Bull. Malaysian Math. Sci. Soc. 35(2A), 591–599 (2012)
Acknowledgements
The author would like to thank the anonymous referees for their valuable remarks and suggestions. Moreover, the author would like to express her sincere appreciation and gratitude to Professors Laura Martein and Alberto Cambini for their comments and fruitful support.
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Carosi, L. Pseudoconvexity on a closed convex set: an application to a wide class of generalized fractional functions. Decisions Econ Finan 40, 145–158 (2017). https://doi.org/10.1007/s10203-017-0185-9
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DOI: https://doi.org/10.1007/s10203-017-0185-9