On Three Open Problems Related to Quasi Relative Interior

Constantin Zălinescu

doi:10.68381/jca22032

Abstract

We give answers to two questions formulated by J. Borwein and R. Goebel [Notions of relative interior in Banach spaces, J. Math. Sci. (N. Y.) 115(4) (2003) 2542–2553] and to a conjecture formulated by S. M. Grad and E. L. Pop [Vector duality for convex vector optimization problems by means of the quasi-interior of the ordering cone, Optimization 63(1) (2014) 21–37] related to calculus rules for quasi (relative) interior.

Author information

Contact details are reproduced from the original publication and may be historical.

Constantin Zălinescu

Faculty of Mathematics, University Alexandru Ioan Cuza, Iaşi, Romania
and: Octav Mayer Institute of Mathematics (Romanian Academy), Iaşi, Romania

zalinesc@uaic.ro

Keywords

Quasi interior
quasi relative interior
open problem

References

J. Borwein, R. Goebel: Notions of relative interior in Banach spaces, J. Math. Sci., New York 115(4) (2003) 2542–2553.https://doi.org/10.1023/A:1022988116044 Google Scholar
J. M. Borwein, A. S. Lewis: Partially finite convex programming. I. Quasi relative interiors and duality theory, Math. Program., Ser. B 57(1) (1992) 15–48.https://doi.org/10.1007/BF01581072 Google Scholar
R. I. Boţ, E. R. Csetnek: Regularity conditions via generalized interiority notions in convex optimization: new achievements and their relation to some classical statements, Optimization 61(1) (2012) 35–65.https://doi.org/10.1080/02331934.2010.505649 Google Scholar
R. I. Boţ, E. R. Csetnek, G. Wanka: Regularity conditions via quasi-relative interior in convex programming, SIAM J. Optim. 19(1) (2008) 217–233.https://doi.org/10.1137/07068432X Google Scholar
S. M. Grad, E. L. Pop: Vector duality for convex vector optimization problems by means of the quasi-interior of the ordering cone, Optimization 63(1) (2014) 21–37.https://doi.org/10.1080/02331934.2013.775283 Google Scholar
C. Zălinescu: Convex Analysis in General Vector Spaces, World Scientific, River Edge (2002).https://doi.org/10.1142/5021 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2015. Rights now held by Banach Press LLC.

[1] J. Borwein, R. Goebel: Notions of relative interior in Banach spaces, J. Math. Sci., New York 115(4) (2003) 2542–2553.https://doi.org/10.1023/A:1022988116044 Google Scholar

[2] J. M. Borwein, A. S. Lewis: Partially finite convex programming. I. Quasi relative interiors and duality theory, Math. Program., Ser. B 57(1) (1992) 15–48.https://doi.org/10.1007/BF01581072 Google Scholar

[3] R. I. Boţ, E. R. Csetnek: Regularity conditions via generalized interiority notions in convex optimization: new achievements and their relation to some classical statements, Optimization 61(1) (2012) 35–65.https://doi.org/10.1080/02331934.2010.505649 Google Scholar

[4] R. I. Boţ, E. R. Csetnek, G. Wanka: Regularity conditions via quasi-relative interior in convex programming, SIAM J. Optim. 19(1) (2008) 217–233.https://doi.org/10.1137/07068432X Google Scholar

[5] S. M. Grad, E. L. Pop: Vector duality for convex vector optimization problems by means of the quasi-interior of the ordering cone, Optimization 63(1) (2014) 21–37.https://doi.org/10.1080/02331934.2013.775283 Google Scholar

[6] C. Zălinescu: Convex Analysis in General Vector Spaces, World Scientific, River Edge (2002).https://doi.org/10.1142/5021 Google Scholar