Abstract

We study the problem of when the linear image of a fixed closed convex subset X of Rn\mathbb{R}^n is closed. Specifically, we improve results of J. M. Borwein and W. B. Moors [Stability of closedness of convex cones under linear mappings I, J. Convex Analysis 16 (2009) 699–705; Stability of closedness of convex cones under linear mappings II, J. Nonlinear Analysis Optim. 1 (2010) 1–7] by showing that for almost all linear mappings T from Rn\mathbb{R}^n into Rm\mathbb{R}^m, not only T(X) is closed, but there is also an open neighborhood of T whose members also preserve the closedness of X.

Author information

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

Si Tiep Dinh

Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam

dstiep@math.ac.vn

Tien-Son Pham

Department of Mathematics, Dalat University, Dalat, Vietnam

sonpt@dlu.edu.vn

Keywords

  • Asymptotic cone
  • closedness
  • convex cone
  • convex set
  • linear mapping
  • stability
  • sigma-porosity

Mathematics Subject Classification

47N10; 90C25, 90C22

References

  1. A. Auslender, M. Teboulle: Asymptotic Cones and Functions in Optimization and Variational Inequalities, Springer Monographs in Mathematics, Springer, New York (2003).Google Scholar
  2. R. Benedetti, J.-J. Risler: Real Algebraic and Semi-Algebraic Sets, Actualités Mathématiques, Hermann, Paris (1990).Google Scholar
  3. J. Bochnak, M. Coste, M.-F. Roy: Real Algebraic Geometry, Ergebnisse der Mathematik, 3. Folge, Band 36, Springer, Berlin (1998).https://doi.org/10.1007/978-3-662-03718-8Google Scholar
  4. J. M. Borwein, W. B. Moors: Stability of closedness of convex cones under linear mappings, J. Convex Analysis 16/3-4 (2009) 699–705.https://doi.org/10.68381/jca16041Google Scholar
  5. J. M. Borwein, W. B. Moors: Stability of closedness of convex cones under linear mappings II, J. Nonlinear Analysis Optim. 1/1 (2010) 1–7.Google Scholar
  6. J.-P. Dedieu: Cône asymptote d’un ensemble non convexe. Application à l’optimisation, C. R. Acad. Sci. Paris 285/7 (1977) 501–503.Google Scholar
  7. J.-P. Dedieu: Critères de fermeture pour l’image d’un fermé non convexe par une multiapplication, C. R. Acad. Sci. Paris 287/14 (1978) 941–943.Google Scholar
  8. J. Dieudonné: Sur la séparation des ensembles convexes, Math. Annalen 163 (1966) 1–3.https://doi.org/10.1007/BF02052480Google Scholar
  9. M. Dür, B. Jargalsaikhan, G. Still: Genericity results in linear conic programming – a tour d’horizon, Math. Oper. Res. 42/1 (2017) 77–94.https://doi.org/10.1287/moor.2016.0793Google Scholar
  10. H. V. Hà, T. S. Pham: Genericity in Polynomial Optimization, Series on Optimization and its Applications 3, World Scientific, Singapore (2017).https://doi.org/10.1142/q0066Google Scholar
  11. M. Liu, G. Pataki: Exact duals and short certificates of infeasibility and weak infeasibility in conic linear programming, Math. Program. Ser. A 167/2 (2018) 435–480.https://doi.org/10.1007/s10107-017-1136-5Google Scholar
  12. G. Pataki: On the closedness of the linear image of a closed convex cone, Math. Oper. Res. 32 (2007) 395–412.https://doi.org/10.1287/moor.1060.0242Google Scholar
  13. G. Pataki, L. Tunçel: On the generic properties of convex optimization problems in conic form, Math. Program. Ser. A 89/3 (2001) 449–457.https://doi.org/10.1007/PL00011408Google Scholar
  14. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  15. R. T. Rockafellar, R. Wets: Variational Analysis, Grundlehren der Mathematischen Wissenschaften 317, Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3Google Scholar
  16. E. Steinitz: Bedingt konvergente Reihen und konvexe Systeme I, II, III, J. Reine Angew. Math. 143 (1913) 128–175; 144 (1914) 1–40; 146 (1916) 1–52.https://doi.org/10.1515/crll.1914.144.1Google Scholar
  17. L. Zajíček: On σ-porous sets in abstract spaces, Abstract Appl. Analysis 5 (2005) 509–534.https://doi.org/10.1155/AAA.2005.509Google Scholar

Suggested citation

S. T. Dinh, T.-S. Pham. “Stability of Closedness of Closed Convex Sets under Linear Mappings.” Journal of Convex Analysis 28 (2021), No. 4, 1281–1291. https://doi.org/10.68381/jca28075

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Published by Heldermann Verlag, 2021. Rights now held by Banach Press.