# Generating Random Polytopes for Nonconvex Optimization

I am working on comparing different approaches for solving the following nonconvex optimization problem:

\begin{align*} \min_{x} \quad & g(x) \\ \text{s.t.} \quad & Ax = b, \\ & x \geq 0. \end{align*}

Specifically, I'm interested in generating random standard bounded polytopes defined by:

\begin{align*} S=\{x:~Ax = b, ~ x \geq 0\}, \end{align*}

where $$A$$ is an $$m \times n$$ matrix.

I've observed the following:

-When $$m > n$$ (more constraints than variables), global solvers tend to find optimal solutions easily and quickly. \

-When $$m < n$$ (more variables than constraints), global solvers often struggle to find optimal solutions. \end{itemize}

To ensure a fair comparison of different approaches, I'm wondering whether it's better to have $$m > n$$ or $$n > m$$ for these polytopes.

TLDR: for problems stated in standard form, it is customary to assume that $$m < n$$ (more variables than constraints) and that $$A$$ has full row rank.
The following only applies if your polyhedra are defined in standard form $$S = \{ x \in \mathbb{R}^{n} | Ax = b, x \geq 0\}$$ where $$A \in \mathbb{R}^{m \times n}$$.
Assuming $$A$$ has full row rank, the set defined by $$Ax = b$$ is an affine subspace of dimension $$n - m$$: this is because each linear equality constraint eliminates one degree of freedom. If you have more than $$n$$ constraints, i.e., if $$m \geq n$$, then $$Ax = b$$ may have no solution, in which case your problem will be infeasible. If your problem is feasible despite having more constraints than variables, then 1) $$A$$ does not have full row rank and 2) the feasible space is likely a low-dimensional affine subspace, which may explain why solvers have an easier time.