Moment-SOS hierarchies for arrow-type polynomial matrix inequalities with applications to structural optimization

Abstract: The Arrow Decomposition (AD) technique, initially introduced in [Mathematical Programming 190(1-2) (2021), pp 105-134], demonstrated superior scalability over the classical chordal decomposition in the context of Linear Matrix Inequalities (LMIs) if the matrix in question satisfied suitable assumptions. The primary objective of this paper is to extend the AD method to address Polynomial Optimization Problems (POPs) involving large-scale Polynomial Matrix Inequalities (PMIs), with the solution framework relying on moment-sum of square (mSOS) hierarchies. As a first step, we revisit the LMI case and weaken the conditions necessary for the key AD theorem presented in [Mathematical Programming 190(1-2) (2021), pp 105-134]. This modification allows the method to be applied to a broader range of problems. Next, we propose a practical procedure that reduces the number of additional variables, drawing on physical interpretations often found in structural optimization applications. For the PMI case, we explore two distinct approaches to combine the AD technique with mSOS hierarchies. One approach involves applying AD to the original POP before implementing the mSOS relaxation. The other approach applies AD directly to the mSOS relaxations of the POP. We establish convergence guarantees for both approaches and prove that theoretical properties extend to the polynomial case. Finally, we illustrate the significant computational advantages offered by the application of AD, particularly in the context of structural optimization problems.