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

Pei Tang; Guoqi Li; Chen Ma; Ran Wang; Gaoxi Xiao; Luping Shi

doi:10.1016/j.isatra.2018.06.010

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

ISA Trans. 2018 Sep:80:232-243. doi: 10.1016/j.isatra.2018.06.010. Epub 2018 Jul 20.

Authors

Pei Tang¹, Guoqi Li², Chen Ma³, Ran Wang⁴, Gaoxi Xiao⁵, Luping Shi⁶

Affiliations

¹ Department of Precision Instrument, Center for Brain Inspired Computing Research, Tsinghua University, Beijing, 100084, China; Beijing Innovation Center for Future Chip, Tsinghua University, Beijing, China. Electronic address: tangp14@mails.tsinghua.edu.cn.
² Department of Precision Instrument, Center for Brain Inspired Computing Research, Tsinghua University, Beijing, 100084, China; Beijing Innovation Center for Future Chip, Tsinghua University, Beijing, China. Electronic address: liguoqi@mail.tsinghua.edu.cn.
³ Department of Precision Instrument, Center for Brain Inspired Computing Research, Tsinghua University, Beijing, 100084, China; Beijing Innovation Center for Future Chip, Tsinghua University, Beijing, China. Electronic address: macheng@mail.tsinghua.edu.cn.
⁴ College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing, China. Electronic address: wangran@nuaa.edu.cn.
⁵ School of EEE, Nanyang Technological University, Singapore. Electronic address: EGXXiao@ntu.edu.sg.
⁶ Department of Precision Instrument, Center for Brain Inspired Computing Research, Tsinghua University, Beijing, 100084, China; Beijing Innovation Center for Future Chip, Tsinghua University, Beijing, China. Electronic address: lpshi@mail.tsinghua.edu.cn.

PMID: 30037531
DOI: 10.1016/j.isatra.2018.06.010

Abstract

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.

Keywords: Matrix function optimization; Matrix variable; Network control; Weighted orthornormal constraint; Weighted trace constraint.