Skip to main content
SpringerLink
Log in

Projection onto polyhedra in outer representation

  • Published:
Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

Abstract

The projection of the origin onto an n-dimensional polyhedron defined by a system of m inequalities is reduced to a sequence of projection problems onto a one-parameter family of shifts of a polyhedron with at most m + 1 vertices in n + 1 dimensions. The problem under study is transformed into the projection onto a convex polyhedral cone with m extreme rays, which considerably simplifies the solution to an equivalent problem and reduces it to a single projection operation. Numerical results obtained for random polyhedra of high dimensions are presented.

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. E. A. Nurminskii, “Method of Successive Projections for Solving the Least Distance Problem for Simplexes,” Elektron. Zh. “Issledovano v Rossii” 160, 1732–1739 (2004); http://zhurnal.ape.relarn.ru/articles/2004/160.pdf.

    Google Scholar 

  2. E. A. Nurminskii, “Convergence of the Suitable Affine Subspace Method for Finding the Least Distance to a Simplex,” Zh. Vychisl. Mat. Mat. Fiz. 45, 1996–2004 (2005) [Comput. Math. Math. Phys. 45, 1915–1922 (2005)].

    MathSciNet  Google Scholar 

  3. E. A. Nurminskii, “Acceleration of Iterative Methods of Protection onto a Polyhedron,” Dal’nevost. Mat. Sb., 1, 51–62 (1995).

    Google Scholar 

  4. C. Lemarechal and J.-B. Hiriart-Urruty, Convex Analysis and Minimization Algorithms II (Springer-Verlag, Berlin, 1993).

    MATH  Google Scholar 

  5. V. F. Dem’yanov and A. M. Rubinov, Fundamentals of Nonsmooth Analysis and Quasi-Differential Calculus (Nauka, Moscow, 1990) [in Russian].

    Google Scholar 

  6. M. A. Kozlov, S. P. Tarasov, and L. G. Khachiyan, “Polynomial Solvability of Convex Quadratic Programming,” Zh. Vychisl. Mat. Mat. Fiz. 20, 1319–1323 (1980).

    MATH  MathSciNet  Google Scholar 

  7. Octave, http://www.octave.org.

  8. ILOG CPLEX, http://www.ilog.com/products/cplex/.

Download references

Author information

Authors and Affiliations

  1. Institute for Automation and Control Processes, Far East Division, Russian Academy of Sciences, ul. Radio 5, Vladivostok, 690041, Russia

    E. A. Nurminski

Authors
  1. E. A. Nurminski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. A. Nurminski.

Additional information

Original Russian Text © E.A. Nurminski, 2008, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2008, Vol. 48, No. 3, pp. 387–396.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nurminski, E.A. Projection onto polyhedra in outer representation. Comput. Math. and Math. Phys. 48, 367–375 (2008). https://doi.org/10.1134/S0965542508030044

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542508030044

Keywords

Search

Navigation