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In this paper, we study the extended trust region subproblem (eTRS) in whichthe trust region intersects the unit ball with a single linear inequalityconstraint. By reformulating the Lagrangian dual of eTRS as a two-parameterlinear eigenvalue problem, we state a necessary and sufficient condition forits strong duality in terms of an optimal solution of a linearly constrainedbivariate concave maximization problem. This results in an efficient algorithmfor solving eTRS of large size whenever the strong duality is detected.Finally, some numerical experiments are given to show the effectiveness of theproposed method.

A Fast Eigenvalue Approach for Solving the Trust Region Subproblem with an Additional Linear Inequality