I am working on the following problem.
Consider the following function $\textit{f}: \mathbb{R^n}$ × $\mathbb{R^n}$ → $\mathbb{R}$.
$$f(\vec{z},\vec{d}) := \min_{t \in \mathbb{R},\vec{v} \in \mathbb{R}^n} t \\ \text{subject to:}\,z_i+t\ge\vec{v}\,'(\vec{a}_{i}+\vec{d}) \\ \sum_{i=1}^n v_i=1 \\ v_i \ge 0.$$
Demonstrate whether $f(\vec{z},\vec{d})$ is always convex, concave, both, or neither in terms of $\vec{z}$. Demonstrate whether $f(\vec{z},\vec{d})$ is always convex, concave, both, or neither in terms of $\vec{d}$.
Recognizing that this is the epigraph form of the convex optimization problem, I began by converting this problem into standard form as shown below:
$$f(\vec{z},\vec{d}) := \min_{t \in \mathbb{R},\vec{v} \in \mathbb{R}^n} \vec{v}\,'(\vec{a}_{i}+\vec{d})-z_i \\ \text{subject to:}\,\sum_{i=1}^n v_i=1 \\ v_i \ge 0.$$
I don't think that I can make use of the second order condition (Hessian) given this form of the objective function, but I am stumped on alternatives for proving concavity. Any suggestions would be greatly appreciated. Perhaps there is a textbook I can use as a reference?