Skip to main content
SpringerLink
Log in

Computing Critical Angles Between Two Convex Cones

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

This paper addresses the numerical computation of critical angles between two convex cones in Euclidean spaces. We present a novel approach to computing these critical angles by reducing the problem to finding stationary points of a fractional programming problem. To efficiently compute these stationary points, we introduce a partial linearization-like algorithm that offers significant computational advantages. Solving a sequence of strictly convex subproblems with straightforward solutions in several settings gives the proposed algorithm high computational efficiency while delivering reliable results: our theoretical analysis demonstrates that the proposed algorithm asymptotically computes critical angles. Numerical experiments validate the efficiency of our approach, even when dealing with problems of relatively large dimensions: only a few seconds are necessary to compute critical angles between different types of cones in spaces of dimension 1000.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Algorithm 1
Fig. 1
Fig. 2

Similar content being viewed by others

Data Availability

All data generated or analyzed during this study are included in this article.

References

  1. Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program. 95, 3–51 (2003)

    Article  MathSciNet  Google Scholar 

  2. Baes, M.: Convexity and differentiability properties of spectral functions and spectral mappings on Euclidean Jordan algebras. Linear Algebra Appl. 422, 664–700 (2007)

    Article  MathSciNet  Google Scholar 

  3. Bauschke, H., Ouyang, H., Wang, X.: On angles between convex sets in Hilbert spaces. J. Math. Anal. Appl. 502, 125239 (2021)

    Article  MathSciNet  Google Scholar 

  4. Benson, H.P.: Fractional programming with convex quadratic forms and functions. Eur. J. Oper. Res. 173, 351–369 (2006)

    Article  MathSciNet  Google Scholar 

  5. Bot, R.I., Csetnek, E.R.: Proximal-gradient algorithms for fractional programming. Optimization 66, 1383–1396 (2017)

    Article  MathSciNet  Google Scholar 

  6. Bot, R.I., Dao, M.N., Li, G.: Extrapolated proximal subgradient algorithms for nonconvex and nonsmooth fractional programs. Math. Oper. Res. 47, 1707–2545 (2022)

    Article  MathSciNet  Google Scholar 

  7. Condat, L.: Fast projection onto the simplex and the \(l_1\) ball. Math. Program. 158, 575–585 (2016)

    Article  MathSciNet  Google Scholar 

  8. Currie, J., Wilson, D.I.: OPTI: Lowering the Barrier Between Open Source Optimizers and the Industrial MATLAB User. Foundations of Computer-Aided Process Operations, Georgia (2012)

    Google Scholar 

  9. Dinkelbach, W.: On nonlinear fractional programming. Manag. Sci. 13, 492–498 (1967)

    Article  MathSciNet  Google Scholar 

  10. Dolatnezhadsomarin, A., Khorram, E., Pourkarimi, L.: Efficient algorithms for solving nonlinear fractional programming problems. Filomat 33, 2149–2179 (2019)

    Article  MathSciNet  Google Scholar 

  11. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems I. Springer, New York (2011)

    Google Scholar 

  12. Faraut, J., Korányi, A.: Analysis on Symmetric Cones. Clarendon Press, Oxford (1994)

    Book  Google Scholar 

  13. Frenk, H., Schaible, S.: Fractional Programming. In: Floudas, C., Pardalos, P. (eds.) Encyclopedia of Optimization. Springer, Boston (2008)

    Google Scholar 

  14. Fukushima, M., Judice, J., de Oliveira, W., Sessa, V.: A sequential partial linearization algorithm for the symmetric eigenvalue complementarity problem. Comput. Optim. Appl. 77, 711–728 (2020)

    Article  MathSciNet  Google Scholar 

  15. Goldberg, F., Shaked-Monderer, N.: On the maximal angle between copositive matrices. Elec. J. Linear Algebra 27, 837–850 (2014)

    Article  MathSciNet  Google Scholar 

  16. Gorski, J., Pfeuffer, F., Klamroth, K.: Biconvex sets and optimization with biconvex functions: a survey and extensions. Math. Meth. Oper. Res. 66, 373–407 (2007)

    Article  MathSciNet  Google Scholar 

  17. Greer, R.: Chapter 2: A tutorial on polyhedral convex cones. In: Trees and Hills: Methodology for Maximizing Functions of Systems of Linear Relations. North-Holland Mathematics Studies vol. 96, 15–81 (1984)

  18. Hiriart-Urruty, J.-B., Seeger, A.: A variational approach to copositive matrices. SIAM Rev. 52, 593–629 (2010)

    Article  MathSciNet  Google Scholar 

  19. Iusem, A., Seeger, A.: On pairs of vectors achieving the maximal angle of a convex cone. Math. Program. 104, 501–523 (2005)

    Article  MathSciNet  Google Scholar 

  20. Iusem, A., Seeger, A.: Angular analysis of two classes of non-polyhedral convex cones: the point of view of optimization theory. Comput. Appl. Math. 26, 191–214 (2007)

    Article  MathSciNet  Google Scholar 

  21. Iusem, A., Seeger, A.: On convex cones with infinitely many critical angles. Optim. 56, 115–128 (2007)

    Article  MathSciNet  Google Scholar 

  22. Iusem, A., Seeger, A.: Searching for critical angles in a convex cone. Math. Program. 120, 3–25 (2009)

    Article  MathSciNet  Google Scholar 

  23. Miao, J., Ben-Israel, A.: On principal angles between subspaces in \({\mathbb{R} }^n\). Linear Algebra Appl. 171, 81–98 (1992)

    Article  MathSciNet  Google Scholar 

  24. Orlitzky, M.: When a maximal angle among cones is nonobtuse. Comput. Appl. Math. 39, 1–14 (2020)

    Article  MathSciNet  Google Scholar 

  25. Pataki, G.: On the closedness of the linear image of a closed convex cone. Math. Oper. Res. 32, 395–412 (2007)

    Article  MathSciNet  Google Scholar 

  26. Ramírez, H., Seeger, A., Sossa, D.: Commutation principle for variational problems on Euclidean Jordan algebras. SIAM J. Optim. 23, 687–694 (2013)

    Article  MathSciNet  Google Scholar 

  27. Ramírez, H., Sossa, D.: On the central paths in symmetric cone programming. J. Optim. Theory Appl. 172, 649–668 (2017)

    Article  MathSciNet  Google Scholar 

  28. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Heidelberg (1998)

    Book  Google Scholar 

  29. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Book  Google Scholar 

  30. Seeger, A., Sossa, D.: Complementarity problems with respect to Loewnerian cones. J. Glob. Optim. 62, 299–318 (2015)

    Article  MathSciNet  Google Scholar 

  31. Seeger, A., Sossa, D.: Critical angles between two convex cones I. General theory. TOP 24, 44–65 (2016)

    Article  MathSciNet  Google Scholar 

  32. Seeger, A., Sossa, D.: Critical angles between two convex cones II. Special cases. TOP 24, 66–87 (2016)

    Article  MathSciNet  Google Scholar 

  33. Seeger, A., Torki, M.: Ellipsoidal cones. J. Convex Anal. 27, 583–622 (2020)

    MathSciNet  Google Scholar 

  34. Schaible, S.: Fractional programming: applications and algorithms. Eur. J. Oper. Res. 7, 111–120 (1981)

    Article  MathSciNet  Google Scholar 

  35. Schaible, S.: Fractional programming. II, on Dinkelbach’s algorithm. Manag. Sci. 22, 868–873 (1976)

    Article  MathSciNet  Google Scholar 

  36. Sogi, N., Zhu, R., Xue, J.-H., Fukui, K.: Constrained mutual convex cone method for image set based recognition. Pattern Recognit. 121, 108190 (2022)

    Article  Google Scholar 

  37. Stancu-Minasian, I.M.: Fractional Programming, Theory, Methods and Applications. Kluwer Academic Publishers, London (1997)

    Book  Google Scholar 

  38. Vandenberghe, L., Boyd, S.: Semidefinite programming. SIAM Rev. 38, 49–95 (1996)

    Article  MathSciNet  Google Scholar 

  39. Wächter, A., Biegler, L.: On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math. Program. 106, 25–57 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors acknowledge the support of MATH-AMSUD 23-MATH-09 MORA-DataS project. The research of V. Sessa and W. de Oliveira benefited from the support of the Gaspard-Monge Program for Optimization and Operations Research (PGMO) project “SOLEM - Scalable Optimization for Learning and Energy Management”. D. Sossa is partially supported by FONDECYT (Chile) through grant 11220268.

Author information

Authors and Affiliations

  1. Centre de Mathématiques Appliquées (CMA), Mines Paris, Université PSL, Sophia Antipolis, Paris, France

    Welington de Oliveira & Valentina Sessa

  2. Instituto de Ciencias de la Ingeniería, Universidad de O’Higgins, Av. Libertador Bernardo O’Higgins 611, Rancagua, Chile

    David Sossa

Authors
  1. Welington de Oliveira
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Valentina Sessa
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. David Sossa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Sossa.

Additional information

Communicated by Radu Ioan Bot.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

de Oliveira, W., Sessa, V. & Sossa, D. Computing Critical Angles Between Two Convex Cones. J Optim Theory Appl (2024). https://doi.org/10.1007/s10957-024-02424-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10957-024-02424-3

Keywords

Mathematics Subject Classification

Search

Navigation