On the Rank of Extreme Matrices in Semidefinite Programs and the Multiplicity of Optimal Eigenvalues

Abstract

We derive some basic results on the geometry of semidefinite programming (SDP) and eigenvalue-optimization, i.e., the minimization of the sum of the k largest eigenvalues of a smooth matrix-valued function.We provide upper bounds on the rank of extreme matrices in SDPs, and the first theoretically solid explanation of a phenomenon of intrinsic interest in eigenvalue-optimization. In the spectrum of an optimal matrix, the kth and (k / 1)st largest eigenvalues tend to be equal and frequently have multiplicity greater than two. This clustering is intuitively plausible and has been observed as early as 1975.When the matrix-valued function is affine, we prove that clustering must occur at extreme points of the set of optimal solutions, if the number of variables is sufficiently large. We also give a lower bound on the multiplicity of the critical eigenvalue. These results generalize to the case of a general matrix-valued function under appropriate conditions.

A Unified Theorem on SDP Rank Reduction

“…, m; X 0, symmetric (1) where P • Q = Tr(P T Q) is the Frobenius inner product of the two matrices P and Q. It is well-known (Barvinok [2]; see also Barvinok [3], Pataki [13]) that if (1) is feasible, then there exists a solution X 0 of rank no more than √ 2m. However, in many applications, such as graph realization (So and Ye [14]) and dimensionality reduction (Matoušek [10], Weinberger and Saul [15]), it is desirable to have a low-rank solution, say, a solution of rank at most d, where d ≥ 1 is fixed.…”

On the Low Rank Solutions for Linear Matrix Inequalities

“…The following result was shown by Barvinok [2] and Pataki [14] for the real case, and Huang and Zhang [11] for the complex case.…”

“…The same bound was independently rediscovered by Pataki in [14]. Later it was shown by Barvinok in [3] that the bound is essentially tight.…”

“…Moreover, Huang and Zhang [11] also proposed a polynomial-time procedure, in the complex case, for finding a low rank solution with a bound on the rank similar to that in Barvinok [2] and Pataki [14]. In the case where the number of constraints in the SDP is small, the rank-1 decomposition method of Sturm and Zhang [19] (and Huang and Zhang [11] for the complex SDP) can be considered as an alternative polynomial-time procedure to find the low rank matrix solutions (in this case rank-1), other than the procedure suggested in Barvinok [2] and Pataki [14].…”

“…Essentially, Au-Yeung and Poon [1] showed that there is a chance to further reduce the rank by one. For instance, for a bounded standard real SDP with 3 constraints, Barvinok [2] and Pataki [14]'s bound can only conclude that there exists a rank-2 feasible solution. However, Au-Yeung and Poon [1]'s result concludes that there is in fact a rank-1 feasible solution!…”