Decomposed Structured Subsets for Semidefinite Optimization

Abstract Semidefinite programs (SDPs) are important computational tools in controls, optimization, and operations research. Standard interior-point methods scale poorly for solving large-scale SDPs. With certain compromise of solution quality, one method for scalability is to use the notion of structured subsets (e.g. diagonally-dominant (DD) and scaled-diagonally dominant (SDD) matrices), to derive inner/outer approximations for SDPs. For sparse SDPs, chordal decomposition techniques have been widely used to derive equivalent SDP reformations with smaller PSD constraints. In this paper, we investigate a notion of decomposed structured subsets by combining chordal decomposition with DD/SDD approximations. This notion takes advantage of any underlying sparsity via chordal decomposition, while embracing the scalability of DD/SDD approximations. We discuss the applications of decomposed structured subsets to semidefinite optimization. Basis pursuit for refining DD/SDD approximations are also incorporated into the decomposed structured subset framework, and numerical performance is improved as compared to standard DD/SDD approximations. These results are demonstrated on H∞ norm estimation problems for networked systems.