The trust region subproblem with non-intersecting linear constraints
- 501 Downloads
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.
KeywordsTrust-region subproblem Second-order cone programming Semidefinite programming Nonconvex quadratic programming
Mathematics Subject Classification90C20 90C22 90C25 90C26 90C30
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 \(m\).
- 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.Google Scholar
- 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)Google Scholar
- 4.Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust-Region Methods. MPS/SIAM Series on Optimization. SIAM, Philadelphia, PA (2000)Google Scholar
- 9.Pong, T., Wolkowicz, H.: The generalized trust region subproblem. University of Waterloo, Waterloo, Ontario (2012)Google Scholar
- 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).Google Scholar
- 11.Sherali, H.D., Adams, W.P.: A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Kluwer, Dordrecht (1997)Google Scholar
- 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