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The generalized trust region subproblem

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  • Ting Pong
  • Henry Wolkowicz

Abstract

The interval bounded generalized trust region subproblem (GTRS) consists in minimizing a general quadratic objective, q 0 (x)→min, subject to an upper and lower bounded general quadratic constraint, ℓ≤q 1 (x)≤u. This means that there are no definiteness assumptions on either quadratic function. We first study characterizations of optimality for this implicitly convex problem under a constraint qualification and show that it can be assumed without loss of generality. We next classify the GTRS into easy case and hard case instances, and demonstrate that the upper and lower bounded general problem can be reduced to an equivalent equality constrained problem after identifying suitable generalized eigenvalues and possibly solving a sparse system. We then discuss how the Rendl-Wolkowicz algorithm proposed in Fortin and Wolkowicz (Optim. Methods Softw. 19(1):41–67, 2004 ) and Rendl and Wolkowicz (Math. Program. 77(2, Ser. B):273–299, 1997 ) can be extended to solve the resulting equality constrained problem, highlighting the connection between the GTRS and the problem of finding minimum generalized eigenvalues of a parameterized matrix pencil. Finally, we present numerical results to illustrate this algorithm at the end of the paper. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Ting Pong & Henry Wolkowicz, 2014. "The generalized trust region subproblem," Computational Optimization and Applications, Springer, vol. 58(2), pages 273-322, June.
    • Handle: RePEc:spr:coopap:v:58:y:2014:i:2:p:273-322
    DOI: 10.1007/s10589-013-9635-7
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    References listed on IDEAS

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    1. Yichuan Ding & Dongdong Ge & Henry Wolkowicz, 2011. "On Equivalence of Semidefinite Relaxations for Quadratic Matrix Programming," Mathematics of Operations Research, INFORMS, vol. 36(1), pages 88-104, February.
    2. Flippo, Olaf E. & Jansen, Benjamin, 1996. "Duality and sensitivity in nonconvex quadratic optimization over an ellipsoid," European Journal of Operational Research, Elsevier, vol. 94(1), pages 167-178, October.
    3. Gábor Pataki, 1998. "On the Rank of Extreme Matrices in Semidefinite Programs and the Multiplicity of Optimal Eigenvalues," Mathematics of Operations Research, INFORMS, vol. 23(2), pages 339-358, May.
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    4. A. Taati & M. Salahi, 2019. "A conjugate gradient-based algorithm for large-scale quadratic programming problem with one quadratic constraint," Computational Optimization and Applications, Springer, vol. 74(1), pages 195-223, September.
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    7. Van-Bong Nguyen & Thi Ngan Nguyen & Ruey-Lin Sheu, 2020. "Strong duality in minimizing a quadratic form subject to two homogeneous quadratic inequalities over the unit sphere," Journal of Global Optimization, Springer, vol. 76(1), pages 121-135, January.
    8. Amaioua, Nadir & Audet, Charles & Conn, Andrew R. & Le Digabel, Sébastien, 2018. "Efficient solution of quadratically constrained quadratic subproblems within the mesh adaptive direct search algorithm," European Journal of Operational Research, Elsevier, vol. 268(1), pages 13-24.

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