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An O(n) Algorithm for Projecting a Vector on the Intersection of a Hyperplane and a Box in Rn

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Abstract

We consider the problem of projecting a vector on the intersection of a hyperplane and a box in Rn. This paper extends a previous result of Maculan, Minoux, and Plateau (Ref. 1) concerning the projection of a vector on the intersection of a hyperplane and Rn +. We present an O(n) time algorithm based on the linear-time median-finding algorithm. This algorithm determines the median of the components of the vector to be projected. Computational results are also presented in order to evaluate the algorithm and its time complexity. We consider two sets of instances which are randomly generated for any given n. The algorithm was successful in solving all the instances in a reasonable time.

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References

  1. Maculan, N., Minoux, M., and Plateau, G., An O(n) Algorithm for Projecting a Vector on the Intersection of a Hyperplane and Rn+, RAIRO—Operations Research, Vol. 3, pp. 17–16, 1997.

    Google Scholar 

  2. Held, M., Wolfe, P., and Crowder, H. P., Validation of the Subgradient Optimization, Mathematical Programming, Vol. 6, pp. 62–88, 1974.

    Google Scholar 

  3. De Paula, G. G., Jr., Um Algoritmo de Decomposição Primal para um Problema Dinâmico de Localização na Produçãode Carvão em Plantação de Eucaliptus, Systems Engineering and Computer Science, Graduate Programs in Engineering (COPPE), Federal University of Rio de Janeiro, PhD Thesis, 1986.

  4. Wynants, C., Network Synthesis Problems, Université Libre de Bruxelles, Instírationnelle, PhD Thesis, 1999.

  5. Maculan, N., and De Paula, G.G., Jr.., A Linear-Time Median-Finding Algorithm for Projecting a Vector on the Simplex of Rn, Operations Research Letters, Vol. 8, pp. 219–222, 1989.

    Google Scholar 

  6. Brucker, P., An O(n) Algorithm for Quadratic Knapsack Problems, Operations Research Letters, Vol. 3, pp. 163–166, 1984.

    Google Scholar 

  7. Michelot, C., A Finite Algorithm for Finding the Projection of a Point onto the Canonical Simplex of Rn, Journal of Optimization Theory and Applications, Vol. 50, pp. 195–200, 1986.

    Google Scholar 

  8. Pardalos, P. M., and Kovoor, N., An Algorithm for a Singly Constrained Class of Quadratic Programs Subject to Upper and Lower Bounds, Mathematical Programming, Vol. 46, pp. 321–328, 1990.

    Google Scholar 

  9. Cormen, T. H., Leiserson, C. E., and Rivest, R. L., Introduction to Algorithms, MIT Press, Cambridge, Massachusetts, 1990.

    Google Scholar 

  10. Minoux, M., Mathematical Programming, John Wiley and Sons, New York, NY, 1986.

    Google Scholar 

  11. Kenninghon, J., and Shalaby, M., An Effective Subgradient Procedure for Minimal Cost Multicommodity Flow Problems, Management Science, Vol. 23, pp. 994–1004, 1977.

    Google Scholar 

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Authors and Affiliations

  1. COPPE-PESC, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brasil

    N. Maculan (Professor), C.P. Santiago (MS Fellow, COPPE-PESC, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brasil) & E.M. Macambira (PhD Fellow, COPPE-PESC, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brasil)

  2. IM-DCC, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brasil

    M.H.C. Jardim (Professor)

Authors
  1. N. Maculan
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  2. C.P. Santiago
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  3. E.M. Macambira
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  4. M.H.C. Jardim
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Maculan, N., Santiago, C., Macambira, E. et al. An O(n) Algorithm for Projecting a Vector on the Intersection of a Hyperplane and a Box in Rn . Journal of Optimization Theory and Applications 117, 553–574 (2003). https://doi.org/10.1023/A:1023997605430

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  • DOI: https://doi.org/10.1023/A:1023997605430

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