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Solving second-order conic systems with variable precision

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Abstract

We describe and analyze an interior-point method to decide feasibility problems of second-order conic systems. A main feature of our algorithm is that arithmetic operations are performed with finite precision. Bounds for both the number of arithmetic operations and the finest precision required are exhibited.

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Notes

  1. http://www.mosek.com/.

  2. http://www.ilog.com/products/cplex/.

References

  1. Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program. 95, 3–51 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bartels, R.H.: A stabilization of the simplex method. Numer. Math. 16, 414–434 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  3. Björck, A.: Numerical Methods for Least Squares Problems. SIAM, Philadelphia (1996)

  4. Cheung, D., Cucker, F.: Solving linear programs with finite precision: II. Algorithms. J. Complex. 22, 305–335 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Clasen, R.J.: Techniques for automatic tolerance control in linear programming. Commun. ACM 9, 802–803 (1966)

    Article  MATH  Google Scholar 

  6. Cucker, F., Peña, J.: A primal-dual algorithm for solving polyhedral conic systems with a finite-precision machine. SIAM J. Optim. 12, 522–554 (2002)

    Article  Google Scholar 

  7. Higham, N.: Accuracy and Stability of Numerical Algorithms. SIAM, Philadelphia (1996)

  8. Lawson, C.L., Hanson, R.J.: Solving Least Squares Problems. Prentice-Hall, Englewood Cliffs (1974)

  9. Lewis, A.: Ill-conditioned convex processes and linear inequalities. Math. Oper. Res. 24, 829–834 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  10. Lewis, A.: Ill-conditioned inclusions. Set-Valued Anal. 9, 375–381 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ogryczak, W.: The simplex method is not always well behaved. Linear Algebra Appl. 109, 41–57 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  12. Peña, J.: Understanding the geometry of infeasible perturbations of a conic linear system. SIAM J. Optim. 10, 534–550 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  13. Peña, J., Renegar, J.: Computing approximate solutions for conic systems of constraints. Math. Program. 87, 351–383 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  14. Renegar, J.: A Mathematical View of Interior-Point Methods in Convex Optimization. SIAM, Philadelphia (2000)

  15. Renegar, J.: Is it possible to know a problem instance is ill-posed? J. Complex. 10, 1–56 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  16. Renegar, J.: Linear programming, complexity theory and elementary functional analysis. Math. Program. 70, 279–351 (1995)

    MATH  MathSciNet  Google Scholar 

  17. Robinson, S.M.: A characterization of stability in linear programming. Oper. Res. 25, 435–447 (1977)

    Article  MATH  Google Scholar 

  18. Sousa Lobo, M., Vandenberghe, L., Boyd, S., Lebret, H.: Applications of second-order cone programming. Linear Algebra Appl. 284, 193–228 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  19. Storoy, S.: Error control in the simplex-technique. BIT 7, 216–225 (1967)

    Article  MATH  Google Scholar 

  20. Tsuchiya, T.: A convergence analysis of the scaling-invariant primal-dual path-following algorithms for second-order cone programming. Optim. Methods Softw. 11, 141–182 (1999)

    Article  MathSciNet  Google Scholar 

  21. Vera, J.R.: On the complexity of linear programming under finite precision arithmetic. Math. Program. 80, 91–123 (1998)

    MATH  MathSciNet  Google Scholar 

  22. Vera, J.C., Rivera, J.C., Pe na, J., Hui, Y.: A primal-dual symmetric relaxation for homogeneous conic systems. J. Complex. 23, 245–261 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  23. Wolfe, P.: Error in the solution of linear programming problems. In: Ball, L.R. (ed.) Error in Digital Computation, pp. 271–284. Wiley, London (1965)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Jiulong, Hong Kong

    Felipe Cucker

  2. Tepper School of Business, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA, 15213-3890, USA

    Javier Peña

  3. Collaborative Research Network, Federation University Australia, Mount Helen Campus, CRN F Building, PO Box 663, Ballarat, VIC, 3353, Australia

    Vera Roshchina

Authors
  1. Felipe Cucker
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  2. Javier Peña
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  3. Vera Roshchina
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Corresponding author

Correspondence to Javier Peña.

Additional information

Partially supported by GRF grant CityU 100810.

Supported by NSF grant CCF-0830533.

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Cucker, F., Peña, J. & Roshchina, V. Solving second-order conic systems with variable precision. Math. Program. 150, 217–250 (2015). https://doi.org/10.1007/s10107-014-0767-z

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  • DOI: https://doi.org/10.1007/s10107-014-0767-z

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