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

Rujun Jiang () and Xudong Li ()
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Rujun Jiang: School of Data Science, Fudan University, Shanghai 200433, China
Xudong Li: School of Data Science, Fudan University, Shanghai 200433, China

Mathematics of Operations Research, 2022, vol. 47, issue 4, 3025-3050

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.

Keywords: Primary: 90C20; 90C26; 49J52; secondary: 90C30; trust region subproblem; error bounds; KL inequality; convergence rate (search for similar items in EconPapers)
Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:47:y:2022:i:4:p:3025-3050

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