A penalty method for rank minimization problems in symmetric matrices

Abstract

The problem of minimizing the rank of a symmetric positive semidefinite matrix subject to constraints can be cast equivalently as a semidefinite program with complementarity constraints (SDCMPCC). The formulation requires two positive semidefinite matrices to be complementary. This is a continuous and nonconvex reformulation of the rank minimization problem. We investigate calmness of locally optimal solutions to the SDCMPCC formulation and hence show that any locally optimal solution is a KKT point.We develop a penalty formulation of the problem. We present calmness results for locally optimal solutions to the penalty formulation. We also develop a proximal alternating linearized minimization (PALM) scheme for the penalty formulation, and investigate the incorporation of a momentum term into the algorithm. Computational results are presented.

Two Relaxation Methods for Rank Minimization Problems

“…This variational formulation was due to Li and Qi [38], with important work on optimality conditions derived by Ding et al [36]. Other recent work on this formulation includes [7,39,40]. We begin with a special case of (1), in which the matrix variable X ∈ R n×n is restricted to be symmetric and positive semidefinite.…”

“…Proposition 3.1 [7] Assume φ(X ) ≡ 0. Each (X , U ) with X feasible and U given by ( 6) is a local optimal solution in Problem (5).…”

“…where λ, μ and Y are Lagrangian multipliers corresponding to the constraints A(X ) = b, X , U = 0 and I − U 0 respectively. Proposition 3.2 [7] The KKT conditions hold at local optima of Problem (5).…”

“…Proposition 3.3 [7] Assume φ(X ) ≡ 0. Any feasible pair (X , U ) with U given by ( 6) is a KKT stationary point of problem (5).…”

“…In general, a local minimizer of an SDCMPCC problem may not satisfy first order optimality conditions because of the complementarity constraints. We have previously shown [7] that the first order optimality conditions do indeed hold at local minimizers of the SDCMPCC lifting of the rank minimization. In the current paper, we show in Sect.…”