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

This paper is mainly concerned with solving the following two problems:
Problem I. Given $X$ $\in$ $ C^{n \times m} $, $\Lambda = {\rm diag}( \lambda_1, \lambda_2, \dots, \lambda_m) \in C^{m\times m}$. Find $ A \in ABSR^{n \times n} $ such that $$AX=X\Lambda$$where $ABSR^{n \times n}$ is the set of all real $n\times n$ anti-bisymmetric matrices.

Problem Ⅱ. Given $A^* \in R^{n \times n}$. Find $\hat{A} \in S_E $ such that $$||A^* - \hat{A}||_F=\underset{A\in S_E}{\min}||A^*- A ||_F,$$where $||\cdot||_F$ is Frobenius norm, and $S_E$ denotes the solution set of Problem Ⅰ.

The necessary and sufficient conditions for the solvability of Problem Ⅰ have been studied. The general form of $ S_E $ has been given. For Problem Ⅱ the expression of the solution has been provided.

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