Optimization on matrix manifold based on gradient information and its applications in network control

Vector function optimization problems, in which one or more variables are multidimensional vectors or infinite-dimensional vectors, have been extensively studied and demonstrated in existing schemes. In various real life applications, a cost function to be optimized usually involves matrix variables subjected to certain constraints. Locating its minimum can be modeled as an optimization problem on matrix manifold which is investigated in this paper. We first present an index-notation-arrangement based chain rule (I-Chain rule) to obtain the gradient information of the cost function. Two iterative algorithms, namely, trace-constraint-based projected gradient method (TPGM) and orthonormal-constraint-based projected gradient method (OPGM) are proposed and their convergence properties are established. We find that the network control problems can be effectively solved by both TPGM and OPGM. Two important phenomena are observed. For controlling directed networks with selectable inputs, both TPGM and OPGM tend to locate the nodes that divides the elementary stem/circle/dilation equally for consuming less energy, with OPGM having a slightly higher chance than TPGM. For controlling directed networks by only evolving the connection strengths on a fixed network structure, we find that after a network adaptively changes its topology in such a way that many similar sub-networks are gradually evolved, the control cost attains its minimum. Our work takes a further step from understanding optimization problems on matrix manifold to extending their applications in science and engineering.