In the trust-region framework for optimizing a general nonlinear function subject to nonlinear inequality constraints, sequential quadratic programming (SQP) techniques generate subproblems in which a quadratic function must be minimized over a spherical region subject to linear inequality constraints. An interior-point algorithm proposed by Kearsley approximately solves these subproblems when the objective functions are large-scale and convex. Kearsley's algorithm handles the inequality constraints with a classical log-barrier function, minimizing quadratic models of the log-barrier function for fixed values of the barrier parameter subject to the trust-region constraint. Kearsley recommends the LSTRS algorithm of Rojas et al. for minimizing these models. For the convex case, we prove convergence of Kearsley's algorithm and suggest alternatives to the LSTRS algorithm. These alternatives include the new annulus algorithm of Griffin et al., which blends the conjugate gradient and sequential subspace minimization methods to yield promising numerical results. For the nonconvex case, we propose and test a new interior-point algorithm that incorporates the annulus algorithm into an SQP framework with trust regions.