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

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. Special attention is given to support and indicator functions. Particular instances yield all weighted Ky Fan norms and squared gauges on $\mathbb R^{n\times m}$, 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, which opens the door to a range of unexplored optimization methods

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

In this note we provide a full conjugacy and subdifferential calculus for convex convex-composite functions in finite-dimensional space. Our approach, based on infimal convolution and cone-convexity, is straightforward and yields the desired results under a verifiable Slater-type condition, with relaxed monotonicity and without lower semicontinuity assumptions on the functions in play. The versatility of our findings is illustrated by a series of applications in optimization and matrix analysis, including conic programming, matrix-fractional, variational Gram, and spectral functions.Comment: 30 page

A note on the K-epigraph

We study the question as to when the closed convex hull of a K-convex map equals its K-epigraph. In particular, we shed light onto the smallest cone K such that a given map has convex and closed K-epigraph, respectively. We apply our findings to several examples in matrix space as well as to convex composite functions