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Hölderian Error Bounds and Kurdyka-Łojasiewicz Inequality for the Trust Region Subproblem

Author

Listed:
  • Rujun Jiang

    (School of Data Science, Fudan University, Shanghai 200433, China)

  • Xudong Li

    (School of Data Science, Fudan University, Shanghai 200433, China)

Abstract

In this paper, we study the local variational geometry of the optimal solution set of the trust region subproblem (TRS), which minimizes a general, possibly nonconvex, quadratic function over the unit ball. Specifically, we demonstrate that a Hölderian error bound holds globally for the TRS with modulus 1/4, and the Kurdyka-Łojasiewicz (KL) inequality holds locally for the TRS with a KL exponent 3/4 at any optimal solution. We further prove that, unless in a special case, the Hölderian error bound modulus and the KL exponent is 1/2. Finally, as a byproduct, we further apply the obtained KL property to show that projected gradient methods studied elsewhere for solving the TRS achieve a local sublinear or even linear rate of convergence with probability 1 by choosing a proper initial point.

Suggested Citation

  • Rujun Jiang & Xudong Li, 2022. "Hölderian Error Bounds and Kurdyka-Łojasiewicz Inequality for the Trust Region Subproblem," Mathematics of Operations Research, INFORMS, vol. 47(4), pages 3025-3050, November.
    • Handle: RePEc:inm:ormoor:v:47:y:2022:i:4:p:3025-3050
    DOI: 10.1287/moor.2021.1243
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