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

## TL;DR

This paper develops a comprehensive calculus for convex convex-composite functions using infimal convolution, enabling new applications in optimization and matrix analysis under relaxed conditions.

## Contribution

It provides a full conjugacy and subdifferential calculus for convex convex-composite functions with minimal assumptions, expanding theoretical tools for optimization.

## Key findings

- Derived conjugacy and subdifferential formulas under verifiable conditions
- Applied results to conic programming and matrix functions
- Extended calculus to spectral and variational Gram functions

## Abstract

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.