Finite-valued infimal convolution

If two nonnegative finite-valued convex functions are defined on all of a finite-dimensional space, their infimal convolution is finite, convex and continuous everywhere. Convex perturbation duality then gives strong duality and dual attainment at every argument. It does not require or imply attainment of the primal decomposition infimum.

Under the finite-valued assumptions, subgradients of an infimal convolution are obtained as its attained dual optimizers. Equivalently $z\in \partial (h^{\ast }+k^{\ast })(p)$. Splitting the latter subdifferential into two separate ones needs a subdifferential sum rule qualification; solving the aggregate dual objective avoids that extra assumption.