## A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization.(English)Zbl 1030.90077

Summary: In this paper, we present a nonlinear programming algorithm for solving semidefinite programs (SDPs) in standard form. The algorithm’s distinguishing feature is a change of variables that replaces the symmetric, positive semidefinite variable $$X$$ of the SDP with a rectangular variable $$R$$ according to the factorization $$X = RR^T$$. The rank of the factorization, i.e., the number of columns of $$R$$, is chosen minimally so as to enhance computational speed while maintaining equivalence with the SDP. Fundamental results concerning the convergence of the algorithm are derived, and encouraging computational results on some large-scale test problems are also presented.

### MSC:

 90C22 Semidefinite programming 90C30 Nonlinear programming 49M30 Other numerical methods in calculus of variations (MSC2010)

### Keywords:

augmented Lagrangian; limited memory BFGS

COL; SDPLR
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