Low-Rank Semidefinite Programming : Theory and Applications

Finding low-rank solutions of semidefinite programs is important in many applications.

For example, semidefinite programs that arise as relaxations of polynomial optimization problems are exact relaxations when the semidefinite program has a rank-1 solution.

Unfortunately, computing a minimum-rank solution of a semidefinite program is an NP-hard problem. This monograph reviews the theory of low-rank semidefinite programming, presenting theorems that guarantee the existence of a low-rank solution, heuristics for computing low-rank solutions, and algorithms for finding low-rank approximate solutions.

It then presents applications of the theory to trust-region problems and signal processing.