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Set containment characterization with strict and weak quasiconvex inequalities

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Abstract

Dual characterizations of the containment of a convex set, defined by infinite quasiconvex constraints, in an evenly convex set, and in a reverse convex set, defined by infinite quasiconvex constraints, are provided. Notions of quasiconjugate for quasiconvex functions, λ-quasiconjugate and λ-semiconjugate, play important roles to derive the characterizations of the set containments.

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References

  1. Goberna M.A., Rodríguez M.M.L.: Analyzing linear systems containing strict inequalities via evenly convex hulls. Eur. J. Oper. Res. 169, 1079–1095 (2006)

    Article  Google Scholar 

  2. Goberna M.A., Jeyakumar V., Dinh N.: Dual characterizations of set containments with strict convex inequalities. J. Global Optim. 34, 33–54 (2006)

    Article  Google Scholar 

  3. Greenberg H.J., Pierskalla W.P.: Quasi-conjugate functions and surrogate duality. Cah. Cent. Étud. Rech. Opér. 15, 437–448 (1973)

    Google Scholar 

  4. Jeyakumar V.: Characterizing set containments involving infinite convex constraints and reverse-convex constraints. SIAM J. Optim. 13, 947–959 (2003)

    Article  Google Scholar 

  5. Mangasarian O.L.: Set containment characterization. J. Global Optim. 24, 473–480 (2002)

    Article  Google Scholar 

  6. Penot J.P., Volle M.: On quasi-convex duality. Math. Oper. Res. 15, 597–625 (1990)

    Article  Google Scholar 

  7. Singer I.: The lower semi-continuous quasi-convex hull as a normalized second conjugate. Nonlinear Anal. Theory Methods Appl. 7, 1115–1121 (1983)

    Article  Google Scholar 

  8. Singer I.: Conjugate functionals and level sets. Nonlinear Anal. Theory Methods Appl. 8, 313–320 (1984)

    Article  Google Scholar 

  9. Suzuki, S., Kuroiwa, D.: Set containment characterization for quasiconvex programming. J. Global Optim. (to appear)

  10. Thach P.T.: Quasiconjugates of functions, duality relationship between quasiconvex minimization under a reverse convex constraint and quasiconvex maximization under a convex constraint, and applications. J. Math. Anal. Appl. 159, 299–322 (1991)

    Article  Google Scholar 

  11. Thach P.T.: Diewert-Crouzeix conjugation for general quasiconvex duality and applications. J. Optim. Theory Appl. 86, 719–743 (1995)

    Article  Google Scholar 

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  1. Interdisciplinary Graduate School of Science and Engineering, Shimane University, 1060 Nishikawatsu-cho, Matsue, Shimane, Japan

    Satoshi Suzuki

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  1. Satoshi Suzuki
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Correspondence to Satoshi Suzuki.

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Suzuki, S. Set containment characterization with strict and weak quasiconvex inequalities. J Glob Optim 47, 273–285 (2010). https://doi.org/10.1007/s10898-009-9473-4

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  • DOI: https://doi.org/10.1007/s10898-009-9473-4

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