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.

中文翻译：

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正交约束下的矩阵函数优化问题

我们研究了矩阵变量正交约束下的矩阵函数优化。通过引入基于索引符号排列的链式规则（I-Chain 规则），我们获得了成本函数的梯度，并提出了一种重新审视的基于正交约束的投影梯度方法来定位目标/成本函数的最小值矩阵变量迭代地受正交约束。为了保证所提出的方法的收敛性，现有方案要求梯度可以由对称矩阵和矩阵变量本身的乘积来表示。该条件在本文中有所放宽。提出了新的技术来建立迭代算法的收敛性。仿真结果表明了我们框架的有效性。