Let a linear program \begin{aligned} &\max_{\textbf{x}\in\mathbb{R}^n}& \textbf{c}^{\text{T}}\textbf{x} \\ &\text{ s.t.}& \textbf{A}\textbf{x}&\leq \textbf{b},\\ &&\textbf{A}_{eq}\textbf{x}&= \textbf{b}_{eq}. \end{aligned} Let $\textbf{A},\textbf{A}_{eq},\textbf{b},\textbf{b}_{eq}$ be such that there is some trivial feasible solution $\textbf{x}_0$ that satisfies the constraints, i.e, $\exists \mathbf{x}_0\in\mathbb{R}^n\text{ such that }\textbf{A}\textbf{x}_0\leq \textbf{b},\textbf{A}_{eq}\textbf{x}_0=\textbf{b}_{eq}$.
What is the best (read: fastest) method to find out, if there exists some other solution $\textbf{x}\neq \textbf{x}_0$ that satisfies the constraints? (Actually solving another feasible $\textbf{x}$ is unnecessary for my application.)
Ideally, this method would have implementations in some common optimization solver, such as Gurobi or Optimization Toolbox of Matlab.