Characterizations via linear scalarization of minimal and properly minimal elements

1. Aliprantis, C.D., Florenzano, M., Martins-da-Rocha, V.F., Tourky, R.: Equilibrium analysis in financial markets with countably many securities. J. Math. Econ. 40, 683–699 (2004)

   Article  MathSciNet  MATH  Google Scholar 

2. Borwein, J.M.: Proper efficient points for maximizations with respect to cones. SIAM J. Control Optim. 15, 57–63 (1977)

   Article  MathSciNet  MATH  Google Scholar 

3. Boţ, R.I.: Conjugate Duality in Convex Optimization, Lecture Notes in Economics and Mathematical Systems, vol. 637. Springer, Berlin (2010)

   Google Scholar 

4. Boţ, R.I., Csetnek, E.R.: Regularity conditions via generalized interiority notions in convex optimization: new achievements and their relation to some classical statements. Optimization 61, 35–65 (2012)

   Article  MathSciNet  MATH  Google Scholar 

5. Boţ, R.I., Csetnek, E.R., Moldovan, A.: Revisiting some duality theorems via the quasirelative interior in convex optimization. J. Optim. Theory Appl. 139, 67–84 (2008)

   Article  MathSciNet  MATH  Google Scholar 

6. Boţ, R.I., Csetnek, E.R., Wanka, G.: Regularity conditions via quasi-relative interior in convex programming. SIAM J. Optim. 19, 217–233 (2008)

   Article  MathSciNet  MATH  Google Scholar 

7. Boţ, R.I., Grad, S.-M., Wanka, G.: Classical linear vector optimization duality revisited. Optim. Lett. 6, 199–210 (2012)

   Article  MathSciNet  MATH  Google Scholar 

8. Boţ, R.I., Grad, S.-M., Wanka, G.: Duality in Vector Optimization. Springer, Berlin (2009)

   MATH  Google Scholar 

9. Boţ, R.I., Wanka, G.: An alternative formulation for a new closed cone constraint qualification. Nonlinear Anal. Theory Methods Appl 64, 1367–1381 (2006)

   Article  MathSciNet  MATH  Google Scholar 

10. Breckner, W.W., Kassay, G.: A systematization of convexity concepts for sets and functions. J. Convex Anal. 1, 109–127 (2015)

    MathSciNet  MATH  Google Scholar 

11. Gong, X.H.: Optimality conditions for Henig and globally proper efficient solutions with ordering cone has empty interior. J. Math. Anal. Appl. 307, 12–31 (2005)

    Article  MathSciNet  MATH  Google Scholar 

12. Grad, S.-M.: On gauge functions for convex cones with possibly empty interiors. J. Convex Anal. 24, 519–524 (2017)

13. Grad, S.-M.: Vector Optimization and Monotone Operators via Convex Duality. Springer, Cham (2015)

    Book  MATH  Google Scholar 

14. Grad, S.-M., Pop, E.L.: Vector duality for convex vector optimization problems by means of the quasi interior of the ordering cone. Optimization 63, 21–37 (2014)

    Article  MathSciNet  MATH  Google Scholar 

15. Isermann, H.: Proper efficiency and the linear vector maximum problem. Oper. Res. 22, 189–191 (1974)

    Article  MathSciNet  MATH  Google Scholar 

16. Jahn, J.: Vector optimization—theory, applications, and extensions. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

17. Lehmann, R., Oettli, W.: The theorem of the alternative, the key-theorem, and the vector-maximum problem. Math. Program. 8, 332–344 (1975)

    Article  MathSciNet  MATH  Google Scholar 

18. Vogel, W.: Vektoroptimierung in Produkträumen. Verlag Anton Hain, Meisenheim am Glan (1977)

    MATH  Google Scholar 

19. Tanaka, T., Kuroiwa, D.: The convexity of \(A\) and \(B\) assures \({{\rm int}} A +B={{\rm int}} (A+B)\). Appl. Math. Lett. 6, 83–86 (1993)

    Article  MathSciNet  MATH  Google Scholar 

20. Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)

    Book  MATH  Google Scholar 

21. Zălinescu, C.: On the use of the quasi-relative interior in optimization. Optimization 64, 1795–1823 (2015)

    Article  MathSciNet  MATH  Google Scholar 

22. Zălinescu, C.: On three open problems related to quasi relative interior. J. Convex Anal. 22, 641–645 (2015)

    MathSciNet  MATH  Google Scholar 

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