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A Review of Quasi-Convex Functions

Author

Listed:
  • Harvey J. Greenberg

    (Southern Methodist University, Dallas, Texas)

  • William P. Pierskalla

    (The Technological Institute, Northwestern University, Evanston, Illinois)

Abstract

Many theorems involving convex functions have appeared in the literature since the pioneering work of Jensen. Recently some results have been obtained for a larger class of functions: quasi-convex. This review summarizes in condensed form results known to date, providing some refinements to gain further generality. An additional objective of this review is to clarify the structure underlying quasi-convex functions by presenting analogues to properties of convex functions, and by illustrating where analogues do not exist.

Suggested Citation

  • Harvey J. Greenberg & William P. Pierskalla, 1971. "A Review of Quasi-Convex Functions," Operations Research, INFORMS, vol. 19(7), pages 1553-1570, December.
    • Handle: RePEc:inm:oropre:v:19:y:1971:i:7:p:1553-1570
    DOI: 10.1287/opre.19.7.1553
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    Cited by:

    1. Cassottana, Beatrice & Shen, Lijuan & Tang, Loon Ching, 2019. "Modeling the recovery process: A key dimension of resilience," Reliability Engineering and System Safety, Elsevier, vol. 190(C), pages 1-1.
    2. Carlo Alberto Bernardi, 2020. "On the Extension of Continuous Quasiconvex Functions," Journal of Optimization Theory and Applications, Springer, vol. 187(2), pages 421-430, November.
    3. Javier Cabello S'anchez & Juan Antonio Fern'andez Torvisco & Mariano R. Arias, 2024. "TAC Method for Fitting Exponential Autoregressive Models and Others: Applications in Economy and Finance," Papers 2402.04138, arXiv.org.
    4. Ricci, Roberto Ghiselli, 2015. "Penalty functions based upon a general class of restricted dissimilarity functions," European Journal of Operational Research, Elsevier, vol. 241(3), pages 806-814.
    5. Tamás L. Balogh & Christian Ewerhart, 2015. "On the origin of r-concavity and related concepts," ECON - Working Papers 187, Department of Economics - University of Zurich.
    6. Larry Epstein & Massimo Marinacci, 2005. "Coarse Contingencies," RCER Working Papers 515, University of Rochester - Center for Economic Research (RCER).
    7. Kieran Donaghy, 2011. "Models of travel demand with endogenous preference change and heterogeneous agents," Journal of Geographical Systems, Springer, vol. 13(1), pages 17-30, March.
    8. Hamid Nazerzadeh & Georgia Perakis, 2016. "Technical Note—Nonlinear Pricing Competition with Private Capacity Information," Operations Research, INFORMS, vol. 64(2), pages 329-340, April.
    9. Jean-Paul Penot, 2015. "Projective dualities for quasiconvex problems," Journal of Global Optimization, Springer, vol. 62(3), pages 411-430, July.
    10. Elisa Mastrogiacomo & Emanuela Rosazza Gianin, 2019. "Time-consistency of risk measures: how strong is such a property?," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(1), pages 287-317, June.
    11. M.J. Penttinen, 2000. "Timber Harvesting with Variable Prices and Costs," Working Papers ir00045, International Institute for Applied Systems Analysis.
    12. Groetzner, Patrick & Werner, Ralf, 2022. "Multiobjective optimization under uncertainty: A multiobjective robust (relative) regret approach," European Journal of Operational Research, Elsevier, vol. 296(1), pages 101-115.
    13. Ramsharan Rangarajan, 2017. "On the resolution of certain discrete univariate max–min problems," Computational Optimization and Applications, Springer, vol. 68(1), pages 163-192, September.
    14. Javier Cabello Sánchez & Juan Antonio Fernández Torvisco & Mariano R. Arias, 2021. "TAC Method for Fitting Exponential Autoregressive Models and Others: Applications in Economy and Finance," Mathematics, MDPI, vol. 9(8), pages 1-20, April.
    15. Gerchak, Yigal & Hassini, Elkafi & Ray, Saibal, 2002. "Capacity selection under uncertainty with ratio objectives," European Journal of Operational Research, Elsevier, vol. 143(1), pages 138-147, November.
    16. Michael Jong Kim, 2016. "Robust Control of Partially Observable Failing Systems," Operations Research, INFORMS, vol. 64(4), pages 999-1014, August.

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