Let $A = (a_{ij})_{1 \le i, j \le n}$ be a square matrix with all $a_{ij} \ge 0$. Moreover, assume $x \in \mathbb{R}^n$ is constrained to
$$\Delta = \{x\in \mathbb{R}^n \mid x_1 + \cdots + x_n = 1, x_j \ge 0 \text{ for all }j \}$$
and denote $y := Ax$. Is there any explicit way of obtaining the following?
$$\max_{x \in \Delta } \left (\min _{1\le j \le n}(y_j) \right)$$
How about introducing an optimization variable $t \in \mathbb R$ and solving the following linear program in $\mathrm x \in \mathbb R^n$ and $t \in \mathbb R$?
$$\boxed{\begin{array}{ll} \text{maximize} & t\\ \text{subject to} & \mathrm e_1^\top\mathrm A \mathrm x \geq t\\ & \quad\vdots\\ & \mathrm e_n^\top\mathrm A \mathrm x \geq t\\ & \mathrm x \in \Delta\end{array}}$$