Conditions for the Preservation of Motzkin Decomposability and the Pareto Bordered Property Under Addition

Bogdan Daniel Moldovan; Cornel Pintea

doi:10.68381/jca32061

Abstract

We provide some sufficient conditions on pairs of Motzkin decompasable sets and Pareto bordered sets in order to get the Minkowski sum of their components Motzkin decomposable and Pareto bordered respectively. We also prove that minimal faces of a closed convex set are also extreme faces of the set and vice-versa. This result allows us to define the generalized Minkowski sets by using the extreme faces. A Minkowski type theorem is proved with extreme faces playing the role of the extreme points in the classical Minkowski Theorem. The special class of Pareto bordered sets, which is a subclass of that of generalized Minkowski sets, is also taken into account. Indeed, as mentioned above, we show that the Minkowski sum of some Pareto bordered sets with the same lineality remains Pareto bordered. Note that the class of generalized Minkowski sets is not closed with respect to the Minkowski sum. It is however worth to mention that the class of closed convex sets which are both Motzkin decomposable and generalized Minkowski (or shortly, MdgM sets) is closed both with respect to Minkowski sum and Cartesian product [see J. E. Martínez-Legaz and C. Pintea: Closed convex sets that are both Motzkin and generalized Minkowski sets, J. Nonlinear Var. Analysis 8/4 (2024) 571–579].

Author information

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

Bogdan Daniel Moldovan

Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania

bogdan.moldovan1@ubbcluj.ro

Cornel Pintea

Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania

cornel.pintea@ubbcluj.ro

Keywords

Motzkin decomposable sets
Pareto bordered sets
generalized Minkowski sets

Mathematics Subject Classification

52A20, 53A07

References

E. Beutner: On the closedness of the sum of closed convex cones in reflexive Banach spaces, J. Convex Analysis 14/1 (2007) 99–102.https://doi.org/10.68381/jca14007 Google Scholar
M. A. Goberna, J. E. Martínez-Legaz, M. I. Todorov: On Motzkin decomposable sets and functions, J. Math. Analysis Appl. 372/2 (2010) 525–537.https://doi.org/10.1016/j.jmaa.2010.07.007 Google Scholar
J. E. Martínez-Legaz, C. Pintea: Closed convex sets of Minkowski type, J. Math. Analysis Appl. 444/2 (2016) 1195–1202.https://doi.org/10.1016/j.jmaa.2016.06.073 Google Scholar
J. E. Martínez-Legaz, C. Pintea: Closed convex sets with an open or closed Gauss range, Math. Program. B 189/1-2 (2021) 433–454.https://doi.org/10.1007/s10107-020-01561-7 Google Scholar
J. E. Martínez-Legaz, C. Pintea: Closed convex sets of Motzkin, generalized Minkowski, and Pareto bordered types, Commun. Optim. Theory 2024 (2024), art.no. 2, 17p.Google Scholar
J. E. Martínez-Legaz, C. Pintea, Closed convex sets that are both Motzkin and generalized Minkowski sets, J. Nonlinear Var. Analysis 8/4 (2024) 571–579.https://doi.org/10.23952/jnva.8.2024.4.06 Google Scholar
B. S. Mordukhovich, N. M. Nam, An Easy Path to Convex Analysis and Applications, Synthesis Lectures on Mathematics and Statistics 14, Morgan & Claypool, San Rafael (2014).https://doi.org/10.1007/978-3-031-02406-1 Google Scholar
C. Pintea: Pareto bordered sets, Optimization 73/11 (2024) 3351–3367.https://doi.org/10.1080/02331934.2023.2280038 Google Scholar
R. T. Rockafellar: Convex Analysis, Princeton Mathematics Series 28, Princeton University Press, Princeton (1970).Google Scholar
V. Soltan: Lectures on Convex Sets, World Scientific, Hackensack (2015).https://doi.org/10.1142/9508 Google Scholar
R. Webster: Convexity, Oxford University Press, Oxford (1994).https://doi.org/10.1093/oso/9780198531470.001.0001 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2025. Rights now held by Banach Press LLC.

[1] E. Beutner: On the closedness of the sum of closed convex cones in reflexive Banach spaces, J. Convex Analysis 14/1 (2007) 99–102.https://doi.org/10.68381/jca14007 Google Scholar

[2] M. A. Goberna, J. E. Martínez-Legaz, M. I. Todorov: On Motzkin decomposable sets and functions, J. Math. Analysis Appl. 372/2 (2010) 525–537.https://doi.org/10.1016/j.jmaa.2010.07.007 Google Scholar

[3] J. E. Martínez-Legaz, C. Pintea: Closed convex sets of Minkowski type, J. Math. Analysis Appl. 444/2 (2016) 1195–1202.https://doi.org/10.1016/j.jmaa.2016.06.073 Google Scholar

[4] J. E. Martínez-Legaz, C. Pintea: Closed convex sets with an open or closed Gauss range, Math. Program. B 189/1-2 (2021) 433–454.https://doi.org/10.1007/s10107-020-01561-7 Google Scholar

[5] J. E. Martínez-Legaz, C. Pintea: Closed convex sets of Motzkin, generalized Minkowski, and Pareto bordered types, Commun. Optim. Theory 2024 (2024), art.no. 2, 17p.Google Scholar

[6] J. E. Martínez-Legaz, C. Pintea, Closed convex sets that are both Motzkin and generalized Minkowski sets, J. Nonlinear Var. Analysis 8/4 (2024) 571–579.https://doi.org/10.23952/jnva.8.2024.4.06 Google Scholar

[7] B. S. Mordukhovich, N. M. Nam, An Easy Path to Convex Analysis and Applications, Synthesis Lectures on Mathematics and Statistics 14, Morgan & Claypool, San Rafael (2014).https://doi.org/10.1007/978-3-031-02406-1 Google Scholar

[8] C. Pintea: Pareto bordered sets, Optimization 73/11 (2024) 3351–3367.https://doi.org/10.1080/02331934.2023.2280038 Google Scholar

[9] R. T. Rockafellar: Convex Analysis, Princeton Mathematics Series 28, Princeton University Press, Princeton (1970).Google Scholar

[10] V. Soltan: Lectures on Convex Sets, World Scientific, Hackensack (2015).https://doi.org/10.1142/9508 Google Scholar

[11] R. Webster: Convexity, Oxford University Press, Oxford (1994).https://doi.org/10.1093/oso/9780198531470.001.0001 Google Scholar