Abstract
Dual characterizations of containment of a convex set, defined by quasiconvex constraints, in a convex set, and in a reverse convex set, defined by a quasiconvex constraint, are provided. Notions of quasiconjugate for quasiconvex functions, H-quasiconjugate and R-quasiconjugate, play important roles to derive characterizations of the set containments.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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)
Goberna M.A., Jeyakumar V., Dinh N.: Dual characterizations of set containments with strict convex inequalities. J. Global Optim. 34, 33–54 (2006)
Greenberg H.J., Pierskalla W.P.: Quasi-conjugate functions and surrogate duality. Cah. Cent. Étud. Rech. Opér. 15, 437–448 (1973)
Jeyakumar V.: Characterizing set containments involving infinite convex constraints and reverse-convex constraints. SIAM J. Optim. 13, 947–959 (2003)
Mangasarian O.L.: Set containment characterization. J. Global Optim. 24, 473–480 (2002)
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)
Thach P.T.: Diewert-Crouzeix conjugation for general quasiconvex duality and applications. J. Optim. Theory Appl. 86, 719–743 (1995)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Suzuki, S., Kuroiwa, D. Set containment characterization for quasiconvex programming. J Glob Optim 45, 551–563 (2009). https://doi.org/10.1007/s10898-008-9389-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-008-9389-4