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The trust region subproblem with non-intersecting linear constraints

Abstract

This paper studies an extended trust region subproblem (eTRS) in which the trust region intersects the unit ball with \(m\) linear inequality constraints. When \(m=0,\,m = 1\), or \(m = 2\) and the linear constraints are parallel, it is known that the eTRS optimal value equals the optimal value of a particular convex relaxation, which is solvable in polynomial time. However, it is also known that, when \(m \ge 2\) and at least two of the linear constraints intersect within the ball, i.e., some feasible point of the eTRS satisfies both linear constraints at equality, then the same convex relaxation may admit a gap with eTRS. This paper shows that the convex relaxation has no gap for arbitrary \(m\) as long as the linear constraints are non-intersecting.

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References

  1. 1.

    Barvinok, A.: Problems of distance geometry and convex properties of quadratic maps. Discrete Comput. Geom. 13, 189–202 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  2. 2.

    Burer, S., Anstreicher, K.: Second-order cone constraints for extended trust-region subproblems. University of Iowa (March 2011). Revised July 2012 and Nov 2012. To appear in SIAM J. Optim.

  3. 3.

    Celis, M.R., Dennis, J.E., Tapia, R.A.: A trust region strategy for nonlinear equality constrained optimization. In: Numerical Optimization, vol. 1984, pp. 71–82 (Boulder, Colo., 1984). SIAM, Philadelphia, PA (1985)

  4. 4.

    Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust-Region Methods. MPS/SIAM Series on Optimization. SIAM, Philadelphia, PA (2000)

  5. 5.

    Fu, M., Luo, Z.-Q., Ye, Y.: Approximation algorithms for quadratic programming. J. Comb. Optim. 2, 29–50 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  6. 6.

    Gould, N.I.M., Lucidi, S., Roma, M., Toint, P.L.: Solving the trust-region subproblem using the Lanczos method. SIAM J. Optim. 9(2), 504–525 (1999). (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  7. 7.

    Moré, J.J., Sorensen, D.C.: Computing a trust region step. SIAM J. Sci. Stat. Comput. 4(3), 553–572 (1983)

    Article  MATH  Google Scholar 

  8. 8.

    Pataki, G.: On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues. Math. Oper. Res. 23, 339–358 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  9. 9.

    Pong, T., Wolkowicz, H.: The generalized trust region subproblem. University of Waterloo, Waterloo, Ontario (2012)

    Google Scholar 

  10. 10.

    Rendl, F., Wolkowicz, H.: A semidefinite framework for trust region subproblems with applications to large scale minimization. Math. Program. 77(2, Ser. B):273–299 (1997).

  11. 11.

    Sherali, H.D., Adams, W.P.: A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Kluwer, Dordrecht (1997)

    Google Scholar 

  12. 12.

    Sturm, J.F., Zhang, S.: On cones of nonnegative quadratic functions. Math. Oper. Res. 28(2), 246–267 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. 13.

    Ye, Y.: A new complexity result on minimization of a quadratic function with a sphere constraint. In: Floudas, C., Pardalos, P. (eds.) Recent Advances in Global Optimization. Princeton University Press, Princeton, NJ (1992)

    Google Scholar 

  14. 14.

    Ye, Y., Zhang, S.: New results on quadratic minimization. SIAM J. Optim. 14(1), 245–267 (2003). (electronic)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

The authors are in debt to two anonymous referees who provided detailed suggestions that have improved the paper immensely. The authors would also like to thank Kurt Anstreicher for numerous discussions and for suggesting first that the case \(m=2\) might be extensible to arbitrary \(m\).

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Correspondence to Samuel Burer.

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Burer, S., Yang, B. The trust region subproblem with non-intersecting linear constraints. Math. Program. 149, 253–264 (2015). https://doi.org/10.1007/s10107-014-0749-1

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Keywords

  • Trust-region subproblem
  • Second-order cone programming
  • Semidefinite programming
  • Nonconvex quadratic programming

Mathematics Subject Classification

  • 90C20
  • 90C22
  • 90C25
  • 90C26
  • 90C30