Matrix function optimization under weighted boundary constraints and its applications in network control

The matrix function optimization under weighted boundary constraints on the matrix variables is investigated in this work. An “index-notation-arrangement based chain rule” (I-Chain rule) is introduced to obtain the gradient of a matrix function. By doing this, we propose the weighted trace-constraint-based projected gradient method (WTPGM) and weighted orthornormal-constraint-based projected gradient method (WOPGM) to locate a point of minimum of an objective/cost function of matrix variables iteratively subject to weighted trace constraint and weighted orthonormal constraint, respectively. New techniques are implemented to establish the convergence property of both algorithms. In addition, compared with the existing scheme termed “orthornormal-constraint-based projected gradient method” (OPGM) that requires the gradient has to be represented by the multiplication of a symmetrical matrix and the matrix variable itself, such a condition has been relaxed in WOPGM. Simulation results show the effectiveness of our methods not only in network control but also in other learning problems. We believe that the results reveal interesting physical insights in the field of network control and allow extensive applications of matrix function optimization problems in science and engineering.