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Computational Optimization and Applications

, Volume 71, Issue 2, pp 435–456 | Cite as

Duality of nonconvex optimization with positively homogeneous functions

  • Shota YamanakaEmail author
  • Nobuo Yamashita
Article
  • 167 Downloads

Abstract

We consider an optimization problem with positively homogeneous functions in its objective and constraint functions. Examples of such positively homogeneous functions include the absolute value function and the p-norm function, where p is a positive real number. The problem, which is not necessarily convex, extends the absolute value optimization proposed in Mangasarian (Comput Optim Appl 36:43–53, 2007). In this work, we propose a dual formulation that, differently from the Lagrangian dual approach, has a closed-form and some interesting properties. In particular, we discuss the relation between the Lagrangian duality and the one proposed here, and give some sufficient conditions under which these dual problems coincide. Finally, we show that some well-known problems, e.g., sum of norms optimization and the group Lasso-type optimization problems, can be reformulated as positively homogeneous optimization problems.

Keywords

Positively homogeneous functions Duality Nonconvex optimization 

Notes

Acknowledgements

The authors are grateful to Prof. Ellen. H. Fukuda for helpful comments and suggestions. This work was supported in part by a Grant-in-Aid for Scientific Research (C) (17K00032) from Japan Society for the Promotion of Science.

References

  1. 1.
    Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program. 95(1), 3–51 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aravkin, A.Y., Burke, J.V., Drusvyatskiy, D., Friedlander, M.P., MacPhee, K.: Foundations of gauge and perspective duality. arXiv preprint arXiv:1702.08649 (2017)
  3. 3.
    Caccetta, L., Qu, B., Zhou, G.: A globally and quadratically convergent method for absolute value equations. Comput. Optim. Appl. 48(1), 45–58 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chartrand, R.: Exact reconstruction of sparse signals via nonconvex minimization. IEEE Signal Process. Lett. 14(10), 707–710 (2007)CrossRefGoogle Scholar
  5. 5.
    Chartrand, R., Yin, W.: Iteratively reweighted algorithms for compressive sensing. In: IEEE International Conference on Acoustics, Speech and Signal Processing, 2008. ICASSP 2008, IEEE, pp. 3869–3872 (2008)Google Scholar
  6. 6.
    Eldar, Y.C., Mishali, M.: Robust recovery of signals from a structured union of subspaces. IEEE Trans. Inf. Theory 55(11), 5302–5316 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Freund, R.M.: Dual gauge programs, with applications to quadratic programming and the minimum-norm problem. Math. Program. 38(1), 47–67 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Friedlander, M.P., Macedo, I., Pong, T.K.: Gauge optimization and duality. SIAM J. Optim. 24(4), 1999–2022 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hu, S., Huang, Z.: A note on absolute value equations. Optim. Lett. 4(3), 417–424 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hu, S., Huang, Z., Zhang, Q.: A generalized Newton method for absolute value equations associated with second order cones. J. Comput. Appl. Math. 235(5), 1490–1501 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kirmaci, U.S., Bakula, M.K., Özdemir, M.E., Pecaric, J.E.: On some inequalities for \(p-\)norms. J. Inequal. Pure Appl. Math. 9(1), 1–8 (2008)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)CrossRefzbMATHGoogle Scholar
  13. 13.
    Mangasarian, O.L.: Absolute value equation solution via concave minimization. Optim. Lett. 1(1), 3–8 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mangasarian, O.L.: Absolute value programming. Comput. Optim. Appl. 36(1), 43–53 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mangasarian, O.L.: A generalized Newton method for absolute value equations. Optim. Lett. 3(1), 101–108 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mangasarian, O.L., Meyer, R.R.: Absolute value equations. Linear Algebra Appl. 419(2), 359–367 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Meier, L., van de Geer, S., Bühlmann, P.: The group Lasso for logistic regression. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 70(1), 53–71 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Miao, X., Yang, J., Hu, S.: A generalized Newton method for absolute value equations associated with circular cones. Appl. Math. Comput. 269, 155–168 (2015)MathSciNetGoogle Scholar
  19. 19.
    Mourad, N., Reilly, J.P.: Minimizing nonconvex functions for sparse vector reconstruction. IEEE Trans. Signal Process. 58(7), 3485–3496 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Prokopyev, O.: On equivalent reformulations for absolute value equations. Comput. Optim. Appl. 44(3), 363–372 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rohn, J.: A theorem of the alternatives for the equation \(ax+ b|x|= b\). Linear Multilinear Algebra 52(6), 421–426 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Rohn, J.: An algorithm for solving the absolute value equation. Electron. J. Linear Algebra 18(5), 589–599 (2009)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Stojnic, M., Parvaresh, F., Hassibi, B.: On the reconstruction of block-sparse signals with an optimal number of measurements. IEEE Trans. Signal Process. 57(8), 3075–3085 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Wolf, G.W.: Facility Location: Concepts, Models, Algorithms and Case Studies. Taylor & Francis, Routledge (2011)Google Scholar
  25. 25.
    Xue, G., Ye, Y.: An efficient algorithm for minimizing a sum of \(p-\)norms. SIAM J. Optim. 10(2), 551–579 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Yamanaka, S., Fukushima, M.: A branch-and-bound method for absolute value programs. Optimization 63(2), 305–319 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Yuan, M., Lin, Y.: Model selection and estimation in regression with grouped variables. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 68(1), 49–67 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Zhang, C., Wei, Q.J.: Global and finite convergence of a generalized Newton method for absolute value equations. J. Optim. Theory Appl. 143(2), 391–403 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Physics, Graduate School of InformaticsKyoto UniversityKyotoJapan

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