A Study of Convex Convex-Composite Functions via Infimal Convolution with Applications

In this paper, 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. The results are established 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.