A polynomial matrix inequality (PMI) is a formula asserting that a polynomial matrix is positive semidefinite. Polynomial matrix optimization (PMO) concerns minimizing the smallest eigenvalue of a symmetric polynomial matrix subject to a tuple of PMIs. This work explores the use of sparsity methods in reducing the complexity of sum of squares--based methods in verifying PMIs or solving PMO. In the unconstrained setting, Newton polytopes can be employed to sparsify the monomial basis, resulting in smaller semidefinite programs. In the general setting, we show how to exploit different types of sparsity (term sparsity, correlative sparsity, matrix sparsity) encoded in polynomial matrices to derive sparse semidefinite programming relaxations for PMO. For term sparsity, we show that the block structures of the term sparsity iterations with maximal chordal extensions converge to the one determined by PMI sign symmetries. For correlative sparsity, unlike the scalar case, we provide a counterexample showing that asymptotic convergence does not hold under the Archimedean condition and the running intersection property. By employing the theory of matrix-valued measures, we establish several results on detecting global optimality and retrieving optimal solutions under correlative sparsity. The effectiveness of sparsity methods on reducing computational complexity is demonstrated on various examples of PMO.

Publication:

SIAM JOURNAL ON OPTIMIZATION

http://dx.doi.org/10.1137/24M1719761

Author:

JARED MILLER

Automatic Control Laboratory (IfA), Department of Information Technology and Electrical En-gineering (D-ITET), ETH Z\"urich, 8092, Z\"urich, Switzerland jarmiller@control.ee.ethz.ch

JIE WANG

State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science,Chinese Academy of Sciences, Beijing, China

wangjie212@amss.ac.cn

FENG GUO

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, LiaoningProvince, China

fguo@dlut.edu.cn