Infimal-convolution dual subgradients ($\partial (h\square k)(z)=\arg {\max }_p\{⟨p,z⟩-h^{\ast }(p)-k^{\ast }(p)\}$)

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.