Characterizations of the set containment with star-shaped constraints

Document Type : Research Paper


1 Department of Mathematics, Islamic Azad University, Kerman Branch, Kerman, Iran

2 Department of Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran


In this paper, we first give a separation theorem for a closed star-shaped set at the origin and a point outside it in terms of separation by an upper semi-continuous and super-linear function, and also, we introduce a $\nu$-star-shaped-conjugation. By using this facts, we present characterizations of the set containment with infinite star-shaped constraints defined by weak inequalities. Next, we give characterizations of the set containment with infinite evenly radiant constraints defined by strict or weak inequalities. Finally, we give a characterization of the set containment with an upper semi-continuous and radiant constraint, in a reverse star-shaped set, defined by a co-star-shaped constraint. These results have many applications in Mathematical Economics, in particular, in Utility Theory.


[1] T. Bartoszynski, M. Dzamonja, L. Halbeisen, E. Murtinova and A. Plichko, On bases in Banach spaces, Studia Math. 170(2) (2005) 147–171.
[2] A.R. Doagooei and H. Mohebi, Dual characterizations of the set containments with strict cone-convex inequalities in Banach spaces, J. Global Optim. 43(4) (2009) 577–591.
[3] M.A. Goberna, V. Jeyakumar and N. Dinh, Dual characterizations of set containments with strict convex inequalities, J. Global Optim. 34 (2006) 33–54.
[4] M.A. Goberna and M.M. L. Rodrıguez, Analyzing linear systems containing strict inequalities via evenly convex hulls, European J. Oper. Res. 169 (2006) 1079–1095.
[5] V. Jeyakumar, Characterizing set containments involving infinite convex constraints and reverse-convex constraints, SIAM J. Optim., 13 (2003) 947959.
[6] O.L. Mangasarian, Set containment characterization, J. Global Optim. 24 (2002) 473–480.
[7] R.A. Megginson, An introduction to Banach space theory, Springer-Verlag, New York, 1998.
[8] H. Mohebi and E. Naraghirad, Cone-separation and star-shaped separability with applications, Nonlinear Anal.Theory Meth. Appl. 69 (2008) 2412–2421.
[9] J.-P. Penot, Duality for radiant and shady programs, Acta Math. Viet. 22 (1997) 541–566.
[10] A.M. Rubinov, Abstract convexity and global optimization, Kluwer Academic Publisher, Boston, Dordrecht, London, 2000.
[11] A.M. Rubinov and E.V. Sharikov, Star-shaped separability with application, J. Convex Anal. 13(3-4) (2006)849–860.
[12] I. Singer, The lower semi-continuous quasi-convex hull as a normalized second conjugate, Nonlinear Anal. Theory Meth. Appl. 7 (1983) 1115–1121.
[13] I. Singer, Conjugate functionals and level sets, Nonlinear Anal. Theory Meth. Appl. 8 (1984) 313–320.
[14] S. Suzuki and D. Kuroiwa, Set containment characterization for quasiconvex programming, J. Global Optim., to appear.
[15] S. Suzuki, Set containment characterization with strict and weak quasiconvex inequalities, J. Global Optim. 47 (2009) 577–591.
[16] A. Shveidel, Separability of star-shaped sets and its application to an optimization problem, Optim. 40 (1997) 207–227.
[17] P.T. Thach, 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 (1991) 299–322.
[18] P.T. Thach, Diewert-Crouzeix conjugation for general quasiconvex duality and applications, J. Optim. Theory Appl. 86 (1995) 719–743.
[19] A. Zaffaroni, Is every radiant function the sum of quasiconvex functions?, Math. Meth. Oper. Res. 59 (2004), 221–233.
[20] A. Zaffaroni, Superlinear separation of radiant and coradiant sets, Optim. 56 (2007) 267–285.
[21] A. Zaffaroni, Representing nonhomotetic preferences, to appear.
Volume 12, Issue 1
May 2021
Pages 790-811
  • Receive Date: 03 February 2018
  • Revise Date: 19 November 2018
  • Accept Date: 25 November 2018