\begin{equation} Ax \leq 0, c^{T}x < 0 \end{equation}
and
\begin{equation} A^{T}y + c = 0, y \geq 0 \end{equation}
are strong alteratives. This is through duality. investment arbitrage invest x_{j} in each of n assets 1 … n with prices p_1 … p_{n} suppose V_{ij} is the payoff of asset j and outcome i risk-free (cash): p_{1} = 1, V_{i1} = 1 for all i as our first investment Arbitrage means there exists some x with p^{T}x < 0, V x \succeq 0 (first thing is you borrow money?, and then second thing is you get more money back). By Farkas’ Lemma, there being no arbicharnge implies there exists some y such that V^{T}y = p. Recall our first column is 1 (from risk-free cash) making 1^{T}y = 1.