Skip to main content
SpringerLink
Log in

Isotonicity of the Metric Projection by Lorentz Cone and Variational Inequalities

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

Abstract

In this paper, we first discuss the geometric properties of the Lorentz cone and the extended Lorentz cone. The self-duality and orthogonality of the Lorentz cone are obtained in Hilbert spaces. These properties are fundamental for the isotonicity of the metric projection with respect to the order, induced by the Lorentz cone. According to the Lorentz cone, the quasi-sublattice and the extended Lorentz cone are defined. We also obtain the representation of the metric projection onto cones in Hilbert quasi-lattices. As an application, solutions of the classic variational inequality problem and the complementarity problem are found by the Picard iteration corresponding to the composition of the isotone metric projection onto the defining closed and convex set and the difference in the identity mapping and the defining mapping. Our results generalize and improve various recent results obtained by many others.

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

Similar content being viewed by others

References

  1. Isac, G., Németh, A.B.: Monotonicity of metric projections onto positive cones of ordered Euclidean spaces. Arch. Math. 46(6), 568–576 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  2. Isac, G., Németh, A.B.: Projection methods, isotone projection cones and the complementarity problem. J. Math. Anal. Appl. 153(1), 258–275 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. Németh, A.B., Németh, S.Z.: How to project onto an isotone projection cone. Linear Algebra Appl. 433(1), 41–51 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Deutsch, F., Ubhaya, V.: Characterizing best isotone approximations in \(L_{p}\) spaces, \(1\le p<\infty \). J. Approx. Theory 203, 1–27 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Isac, G.: On the Order Monotonicity of the Metric Projection Operator, Approximation Theory. In: Singh, S.P. (eds.) Wavelets and Applications (1995)

  6. Gajardo, P., Seeger, A.: Equilibrium problems involving the Lorentz cone. J. Glob. Optim. 58, 321–340 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Tang, J., He, G., Dong, L., Fang, L., Zhou, J.: A globally and quadratically convergent smoothing Newton method for solving second-order cone optimization. Appl. Math. Model. 39, 2180–2193 (2015)

    Article  MathSciNet  Google Scholar 

  8. Németh, S.Z.: Inequalities characterizing coisotone cones in Euclidean spaces. Positivity 11(3), 469–475 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Németh, A.B., Németh, S.Z.: A duality between the metric projection onto a convex cone and the metric projection onto its dual. J. Math. Anal. Appl. 392(2), 171–178 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Li, J., Yao, J.: The existence of maximum and minimum solutions to general variational inequalities in Hilbert lattices. J. Fixed Point Theory Appl. 17(1), 243–266 (2011)

    MathSciNet  Google Scholar 

  11. Németh, S.Z.: Characterisation of subdual latticial cones in Hilbert spaces by the isotonicity of the metric projection. Optimization 59(8), 1117–1121 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ferreira, O.P., Németh, S.Z.: Generalized isotone projection cones. Optimization 61(9), 1087–1098 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Abbas, M., Németh, S.Z.: Isotone Projection Cones and Nonlinear Complementarity Problems. In: Nonlinear Analysis. Springer India 323-347 (2014)

  14. Abbas, M., Németh, S.Z.: Finding solutions of implicit complementarity problems by isotonicity of the metric projection. Nolinear Anal. 75(4), 2349–2361 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Abbas, M., Németh, S.Z.: Solving nonlinear complementarity problems by isotonicity of the metric projection. J. Math. Anal. Appl. 386(2), 882–893 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Liu, L., Kong, D., Wu, Y.: The best approximation theorems and variational inequalities for discontinuous mappings in Banach spaces. Sci. China Math. 58(12), 2581–2592 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Wang, Y., Zhang, C.: Order-preservation of solution correspondence for parametric generalized variational inequalities on Banach lattices. Fixed Point Theory Appl. 108, (2015) doi:10.1186/s13663-015-0360-z

  18. Li, J., Ok, E.: Optimal solutions to variational inequalities on Banach lattices. J. Math. Anal. Appl. 388, 1157–1165 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kong, D., Liu, L., Wu, Y.: The best approximation theorems and fixed point theorems for discontinuous increasing mappings in Banach spaces. Abstr. Appl. Anal. 2015, 165053 (2015)

    Article  MathSciNet  Google Scholar 

  20. Adly, S., Rammal, H.: A new method for solving second-order cone eigenvalue complementarity problems. J. Optim. Theory Appl. 165, 563–585 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Németh, S.Z.: Iterative methods for nonlinear complementarity problems on isotone projection cones. J. Math. Anal. Appl. 350(1), 340–347 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kong, D., Liu, L., Wu, Y.: Best approximation and fixed-point theorems for discontinuous increasing maps in Banach lattices. Fixed Point Theory Appl. 1, 1–10 (2014)

    MathSciNet  MATH  Google Scholar 

  23. Capǎtǎ, A.: Optimality conditions for extended Ky Fan inequality with cone and affine constraints and their applications. J. Optim. Theory Appl. 152, 661–674 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Abbas, M., Khan, A.R., Németh, S.Z.: Complementary problems and common fixed points in vector lattices. Fixed Point Theory Appl. 60, 13 (2012)

    MATH  Google Scholar 

  25. Isac, G., Németh, S.Z.: Regular exceptional family of elements with respect to isotone projection cones in Hilbert spaces and complementarity problems. Optim. Lett. 2(4), 567–576 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Abbas, M., Németh, S.Z.: Implicit complementarity problems on isotone projection cones. Optimization 61(6), 765–778 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Németh, A.B., Németh, S.Z.: Lattice-like operations and isotone projection sets. Linear Algebra Appl. 439, 2815–2828 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Németh, S.Z.: A duality between the metric projection onto a convex cone and the metric projection onto its dual in Hilbert spaces. Nonlinear Anal. 97, 1–3 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Németh, A.B., Németh, S.Z.: Self-dual cones, generalized lattice operations and isotone projections. arXiv:1210.2324 (2012)

  30. Németh, S.Z., Zhang, G.: Extended Lorentz cone and variational inequalities on cylinders. J. Optim. Theory Appl. 168, 756–768 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Nishimura, H., Ok, E.: Solvability of variational inequalities on Hilbert lattices. Math. Oper. Res. 37(4), 608–625 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Messerschmidt, M.: Normality of spaces of operators and quasi-lattices. Positivity 19, 695–724 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kong, D., Liu, L., Wu, Y.: Isotonicity of the metric projection with applications to variational inequalities and fixed point theory in Banach spaces. J. Fixed Point Theory Appl. (2016) doi:10.1007/s11784-016-0337-5

  34. Németh, S.Z., Zhang, G.: Extended Lorentz cones and mixed complementarity problems. J. Glob. Optim. 62, 443–457 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referee for his/her very important comments that improved the results and the quality of the paper. The authors were supported financially by the National Natural Science Foundation of China (11371221, 11571296).

Author information

Authors and Affiliations

  1. College of Information Science and Engineering, Shandong Agricultural University, Taian, Shandong, China

    Dezhou Kong

  2. School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong, China

    Lishan Liu

  3. Department of Mathematics and Statistics, Curtin University, Perth, Australia

    Lishan Liu & Yonghong Wu

Authors
  1. Dezhou Kong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lishan Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yonghong Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lishan Liu.

Additional information

Antonino Maugeri.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kong, D., Liu, L. & Wu, Y. Isotonicity of the Metric Projection by Lorentz Cone and Variational Inequalities. J Optim Theory Appl 173, 117–130 (2017). https://doi.org/10.1007/s10957-017-1084-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-017-1084-5

Keywords

Mathematics Subject Classification

Search

Navigation