Decomposed structured subsets for semidefinite and sum-of-squares optimization

Semidefinite programs (SDPs) are standard convex problems that frequently found in control and optimization applications. Interior-point methods can solve SDPs polynomial time up to arbitrary accuracy, but scale poorly as the size of matrix variables number constraints increases. To improve scalability, be approximated with lower upper bounds through use structured subsets (e.g., diagonally-dominant scaled-diagonally dominant matrices). Meanwhile, any underlying sparsity or symmetry structure may leveraged form an equivalent SDP smaller positive semidefinite constraints. In this paper, we present a notion decomposed approximate after conversion. The lower/upper by approximation conversion become tighter than obtained approximating original directly. We apply sum-of-squares examples H ∞ norm estimation constrained optimization. An existing basis pursuit method is adapted into framework iteratively refine bounds.