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A new relaxed variant of interior point method for low-rank semidefiniteprogramming problems is proposed in this paper. The method is a step outside ofthe usual interior point framework. In anticipation to converging to a low-rankprimal solution, a special nearly low-rank form of all primal iterates isimposed. To accommodate such a (restrictive) structure, the first orderoptimality conditions have to be relaxed and are therefore approximated bysolving an auxiliary least-squares problem. The relaxed interior pointframework opens numerous possibilities how primal and dual approximated Newtondirections can be computed. In particular, it admits the application of boththe first- and the second-order methods in this context. The convergence of themethod is established. A prototype implementation is discussed and encouragingpreliminary computational results are reported for solving theSDP-reformulation of matrix-completion problems.

A relaxed interior point method for low-rank semidefinite programming problems with applications to matrix completion