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.
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