Variational Properties of Matrix Functions via the Generalized Matrix-Fractional Function

Abstract

We show that many important convex matrix functions can be represented as the partial infimal projection of the generalized matrix fractional (GMF) and a relatively simple convex function. This representation provides conditions under which such functions are closed and proper as well as formulas for the ready computation of both their conjugates and subdifferentials. Particular instances yield all weighted Ky Fan norms and squared gauges on R n×m , and as an example we show that all variational Gram functions are representable as squares of gauges. Other instances yield weighted sums of the Frobenius and nuclear norms. The scope of applications is large and the range of variational properties and insight is fascinating and fundamental. An important byproduct of these representations is that they lay the foundation for a smoothing approach to many matrix functions on the interior of the domain of the GMF function.

A note on the K-epigraph

“…We now show that F is K-convex: To this end, first note that u, F is convex for all u ∈ ri (−K • ), by (9). Any u ∈ −K • is a limit {u k ∈ ri (−K • )} → u, and therefore u, F is the pointwise limit of convex functions u k , F , and hence convex (Lemma 2).…”

“…We now prove that F is also K-closed: To this end, let {(x k , y k ) ∈ K-epi F } → (x, y). In particular, there exists {v k ∈ K} such that F (x k ) + v k = y k for all k ∈ N. Moreover, as K ⊂ KF (see above) and KF -epi F is closed (by definition), we have (x, y) ∈ KF -epi F , and consequently, x ∈ D. Thus we can use the fact that, by (9), u, F is D-closed for all u ∈ ri (−K • ), and hence…”

“…It plays a central role in study of the matrix-fractional [7][8][9] and variational Gram functions [10,13].…”

A study of convex convex-composite functions via infimal convolution with applications

“…These go beyond what was considered before, and include connections to conic programming in Section 5.1, and to matrix analysis and modern matrix optimization, see Sections 5.3-5.5. More concretely, Section 5.3 contains a new extension of the matrix-fractional function [15,16,18] to the complex domain. Section 5.4 provides new, short proofs for the conjugate and subdifferential of variational Gram functions [16,36], and Section 5.5 gives a new proof of Lewis' well-known result on spectral functions [40,41] for the convex case.…”

“…More concretely, Section 5.3 contains a new extension of the matrix-fractional function [15,16,18] to the complex domain. Section 5.4 provides new, short proofs for the conjugate and subdifferential of variational Gram functions [16,36], and Section 5.5 gives a new proof of Lewis' well-known result on spectral functions [40,41] for the convex case. Section 5.6 extends a Farkas-type result due to Bot et al [4].…”