Abstract
A new relaxed variant of interior point method for low-rank semidefinite programming problems is proposed in this paper. The method is a step outside of the usual interior point framework. In anticipation to converging to a low-rank primal solution, a special nearly low-rank form of all primal iterates is imposed. To accommodate such a (restrictive) structure, the first order optimality conditions have to be relaxed and are therefore approximated by solving an auxiliary least-squares problem. The relaxed interior point framework opens numerous possibilities how primal and dual approximated Newton directions can be computed. In particular, it admits the application of both the first- and the second-order methods in this context. The convergence of the method is established. A prototype implementation is discussed and encouraging preliminary computational results are reported for solving the SDP-reformulation of matrix-completion problems.
Highlights
We are concerned with an application of an interior point method (IPM) for solving large, sparse and specially structured positive semidefinite programming problems (SDPs)
Applications include problems arising in engineering, finance, optimal control, power flow, various SDP relaxations of combinatorial optimization problems, matrix completion or other applications originating from modern computational statistics and machine learning
We focus on the linear algebra phase and present a matrix-free implementation well suited for structured constraint matrices as those arising in the SDP reformulation of matrix completion problems [13]
Summary
We are concerned with an application of an interior point method (IPM) for solving large, sparse and specially structured positive semidefinite programming problems (SDPs). When positive semidefinite programs are solved using interior-point algorithms, because of the nature of logarithmic barrier function promoting the presence of nonzero eigenvalues, the primal variable X typically converges to a maximum-rank solution [24,34]. Special low-rank structure of X may be imposed directly in problem (1.1), but this excludes the use of an interior point algorithm (which requires all iterates X to be strictly positive definite). We would like to preserve as many of the advantageous properties of interior point methods as possible and expect to achieve it by (i) working with the original problem (1.1) and (ii) exploiting the low-rank structure of X. Knowing that at optimality X is low-rank we impose a special form of the primal variable throughout the interior point algorithm.
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