Matrix Function Optimization Problems Under Orthonormal Constraint
研究了矩阵变量在正交约束下的函数优化问题，提出一种基于索引符号链式法则的投影梯度方法，放宽了收敛条件，并通过仿真验证有效性。
We investigate the matrix function optimization under the orthonormal constraint on the matrix variable. By introducing an index-notation-arrangement-based chain rule (I-Chain rule), we obtain the gradient of the cost function and propose a revisited orthonormal-constraint-based projected gradient method to locate a minimum of an objective/cost function of matrix variables iteratively subject to orthonormal constraint. To guarantee the convergence the proposed method, existing schemes require the gradient can be represented by the multiplication of a symmetrical matrix and the matrix variable itself. This condition has been relaxed in this paper. New techniques are proposed to establish the convergence property of the iterative algorithm. Simulation results show the effectiveness of our framework. This paper allows more extensive applications of matrix function optimization problems in science and engineering.