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Some LCPs solvable in strongly polynomial time with Lemke’s algorithm

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Abstract

We identify a class of Linear Complementarity Problems (LCPs) that are solvable in strongly polynomial time by Lemke’s Algorithm (Scheme 1) or by the Parametric Principal Pivoting Method (PPPM). This algorithmic feature for the class of problems under consideration here is attributable to the proper selection of the covering vector in Scheme 1 or the parametric direction vector in the PPPM which leads to solutions of limited and monotonically increasing support size; such solutions are sparse. These and other LCPs may very well have multiple solutions, many of which are unattainable by either algorithm and thus are said to be elusive. The initial conditions imposed on the new matrix class identified in Sect. 2 are subsequently relaxed in later sections.

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Notes

  1. Obviously, this property is of importance for (algorithms to solve) parametric linear complementarity problems.

  2. This acronym is due to Chu [3]. Under the present conditions, it is the same as the PPPM [7, Algorithm 4.5.2].

References

  1. Chandrasekaran, R.: A special case of the complementary pivot problem. Opsearch 7, 263–268 (1970)

    MathSciNet  Google Scholar 

  2. Chandrasekaran, R., Kabadi, S.N.: Strongly polynomial algorithm for a class of combinatorial LCPs. Op. Res. Lett. 6, 91–92 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chu, T.H.: A class of polynomially solvable linear complementarity problems. Math. Progr. 107, 461–470 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chung, S.J.: NP-completeness of the linear complementarity problem. J. Optim. Theory Appl. 60, 393–399 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cottle, R.W.: Observations on a class of nasty linear complementarity problems. Discret. Appl. Math. 2, 89–111 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cottle, R.W., Dantzig, G.B.: Complementary pivot theory of mathematical programming. Linear Algebra Appl. 1, 103–125 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, Boston, 1992. [See the revised edition in the Classics in Applied Mathematics series, Philadelphia: SIAM, 2009.]

  8. Cottle, R.W., Stone, R.E.: On the uniqueness of solutions to linear complementarity problems. Math. Progr. 27, 191–213 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  9. Eaves, B.C.: The linear complementarity problem. Manag. Sci. 17, 612–634 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fathi, Y.: Computational complexity of LCPs associated with positive definite symmetric matrices. Math. Progr. 17, 335–344 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gana, A.: Studies in the complementarity problem. Ph.D. thesis, Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor

  12. Kelly, L.M., Murty, K.G., Watson, L.T.: CP rays in simplicial cones. Math. Progr. 48, 387–414 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  13. Klee, V., Minty, G.J.: How good is the simplex algorithm? In: Shisha, O. (ed.) Inequalities—III, pp. 159–175. Academic Press, New York (1972)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Morris Jr., W.D., Lawrence, J.: Geometric properties of Hidden Minkowski matrices. SIAM J. Matrix Anal. Appl. 10, 229–232 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  16. Murty, K.G.: Computational complexity of complementary pivot methods. Math. Progr. Study 7, 61–73 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  17. Pang, J.S., Chandrasekaran, R.: Linear complementarity problems solvable by a polynomially bounded pivoting algorithm. Math. Progr. Study 25, 13–27 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  18. Saigal, R.: A note on a special linear complementarity problem. Opsearch 7, 175–183 (1970)

    MathSciNet  Google Scholar 

  19. Schiro, D.A., Pang, J.S., Shanbhag, U.V.: On the solution of affine generalized Nash equilibrium problems with shared constraints by Lemke’s method. Math. Progr., Ser. A 146, 1–46 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors are grateful to two referees who have made some constructive comments that have helped to improve the presentation of the paper.

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Authors and Affiliations

  1. Department of Industrial Engineering and Operations Research, University of California, Berkeley, CA, 94720, USA

    Ilan Adler

  2. Department of Management Science and Engineering, Stanford University, Stanford, CA, 94305, USA

    Richard W. Cottle

  3. Department of Industrial and Systems Engineering, University of Southern California, Los Angeles, CA, 90089, USA

    Jong-Shi Pang

Authors
  1. Ilan Adler
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  2. Richard W. Cottle
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  3. Jong-Shi Pang
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Correspondence to Jong-Shi Pang.

Additional information

J.-S. Pang: The work of this author is based on research supported by the National Science Foundation under grants CMMI 1402052 and 0969600.

Part of this work was done while the authors were visiting the Institute for Mathematical Sciences at the National University of Singapore.

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Adler, I., Cottle, R.W. & Pang, JS. Some LCPs solvable in strongly polynomial time with Lemke’s algorithm. Math. Program. 160, 477–493 (2016). https://doi.org/10.1007/s10107-016-0996-4

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  • DOI: https://doi.org/10.1007/s10107-016-0996-4

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