An interior trust-region-based algorithm for linearly constrained minimization problems is proposed and analyzed. This algorithm is similar to trust region algorithms forunconstrained minimization: a trust region subproblem on a subspace is solved in eachiteration. We establish that the proposed algorithm has convergence properties analogousto those of the trust region algorithms for unconstrained minimization. Namely, every limitpoint of the generated sequence satisfies the Krush-Kuhn-Tucker (KKT) conditions andat least one limit point satisfies second order necessary optimality conditions. In adidition,if one limit point is a strong local minimizer and the Hessian is Lipschitz continuous in aneighborhood of that point, then the generated sequence converges globally to that pointin the rate of at least 2-step quadratic. We are mainly conceed with the theoretical properties of the algorithm in this paper. Implementation issues and adaptation to large-scaleproblems will be addressed in a future report.