Skip to main content
SpringerLink
Log in

An active index algorithm for the nearest point problem in a polyhedral cone

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

We consider the problem of finding the nearest point in a polyhedral cone C={x∈R n:D x≤0} to a given point b∈R n, where D∈R m×n. This problem can be formulated as a convex quadratic programming problem with special structure. We study the structure of this problem and its relationship with the nearest point problem in a pos cone through the concept of polar cones. We then use this relationship to design an efficient algorithm for solving the problem, and carry out computational experiments to evaluate its effectiveness. Our computational results show that our proposed algorithm is more efficient than other existing algorithms for solving this problem.

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. Al-Sultan, K.S., Murty, K.G.: Nearest points in nonsimplicial cones and LCPs with PSD symmetric matrices. In: Recent Developments in Mathematical Programming. Gordon and Breach Science Publishers, Philadelphia (1991). Edited by Santosh Kumar on behalf of the Australian Society for Operations Research

    Google Scholar 

  2. Anstreicher, K.M., Hertog, D.D., Ross, C., Terlaky, T.: A long-step Barrier method for convex quadratic programming. Algorithmica 10, 365–382 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Fang, S.C.: An iterative method for generalized complementarity problems. IEEE Trans. Automat. Contr. AC 25, 1225–1227 (1980)

    Article  MATH  Google Scholar 

  4. Goldfarb, D., Idnani, A.: A numerically stable dual method for solving strictly convex quadratic programs. Math. Program. 27, 1–33 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  5. Goldfarb, D., Liu, S.: An O(n 3 L) primal interior point algorithm for convex quadratic programming. Math. Program. 49, 325–340 (1990)

    Article  MathSciNet  Google Scholar 

  6. Han, S.P.: A successive projection method. Math. Program. 40, 1–14 (1988)

    Article  MATH  Google Scholar 

  7. Kostina, E., Kostyukova, O.: A primal-dual active-set method for convex quadratic programming. http://www.iwr.uni-heidelberg.de/organization/sfb359/PP/Preprint2003-05.pdf

  8. Lemke, C.E.: Bimatrix equilibrium points and mathematical programming. Manag. Sci. 11, 681–689 (1965)

    Article  MathSciNet  Google Scholar 

  9. Liu, Z.: The nearest point problem in a polyhedral cone and its extensions. Ph.D. Dissertation, North Carolina State University (2009)

  10. Monteiro, R.D.C., Adler, I.: Interior path following primal-dual algorithms, part II: convex quadratic programming. Math. Program. 44, 43–66 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  11. Murty, K.G., Fathi, Y.: A critical index algorithm for nearest point problems on simplicial cones. Math. Program. 23, 206–215 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  12. Murty, K.G.: A new practically efficient interior point method for LP. Algorithmic Oper. Res. 1, 3–19 (2006)

    MathSciNet  MATH  Google Scholar 

  13. Ruggiero, V., Zanni, L.: A modified projection algorithm for large strictly convex quadratic programs. J. Optim. Theory Appl. 104, 281–299 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Stoer, J., Witzgall, C.: Convexity and Optimization in Finite Dimensions. Springer, Berlin (1970)

    MATH  Google Scholar 

  15. Wilhelmsen, D.R.: A nearest point algorithm for convex polyhedral cones and applications to positive linear approximation. Math. Comput. 30, 48–57 (1976)

    MathSciNet  MATH  Google Scholar 

  16. Wolfe, P.: Finding the nearest point in a polytope. Math. Program. 11, 128–149 (1976)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate Program in Operations Research, North Carolina State University, Raleigh, NC, 27695, USA

    Zhe Liu

  2. Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC, 27695, USA

    Yahya Fathi

Authors
  1. Zhe Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yahya Fathi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhe Liu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, Z., Fathi, Y. An active index algorithm for the nearest point problem in a polyhedral cone. Comput Optim Appl 49, 435–456 (2011). https://doi.org/10.1007/s10589-009-9303-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-009-9303-0

Search

Navigation