I want to implement a minimisation on a 2D spin Ising model with 30x30 grid. The spin variables is 0,1 and the objective is to minimize the sum of products of spins. For simplicity, I only include NN pair interaction and a "triangular" interaction term. The formulation I give is as followed. And I wish to ask for suggestions whether there would be some stronger formulation that could make computation faster. Thank you :D

the original objective is:

$$\sum\limits_{i = 1}^{29} {\sum\limits_{j = 1}^{30} {{J_1}{s_{i,j}}{s_{i + 1,j}}} } + \sum\limits_{i = 1}^{30} {\sum\limits_{j = 1}^{29} {{J_1}{s_{i,j}}{s_{i,j + 1}}} } + \sum\limits_{i = 1}^{29} {\sum\limits_{j = 1}^{29} {{J_3}{s_{i,j}}{s_{i + 1,j}}{s_{i,j + 1}}} } $$

$${s_{i,j}} \in \{ 0,1\} $$

we linearize it so that:

[\begin{align} & \min : \\ & \sum\limits_{i=1}^{29}{\sum\limits_{j=1}^{30}{{{J}_{1}}{{s}_{i,j,1}}}}+\sum\limits_{i=1}^{30}{\sum\limits_{j=1}^{29}{{{J}_{1}}{{s}_{i,j,2}}}}+\sum\limits_{i=1}^{29}{\sum\limits_{j=1}^{29}{{{J}_{3}}{{s}_{i,j,3}}}} \\ & \text{subject to:} \\ & \forall i=1...29\,j=1...30\,\ k=1...4 \\ & \left[ \begin{matrix} {{s}_{i,j}} \\ {{s}_{i+1,j}} \\ {{s}_{i,j,1}} \\ \end{matrix} \right]={{y}_{i,j,1,1}}\left[ \begin{matrix} 0 \\ 0 \\ 0 \\ \end{matrix} \right]+{{y}_{i,j,1,2}}\left[ \begin{matrix} 1 \\ 0 \\ 0 \\ \end{matrix} \right]+{{y}_{i,j,1,3}}\left[ \begin{matrix} 0 \\ 1 \\ 0 \\ \end{matrix} \right]+{{y}_{i,j,1,4}}\left[ \begin{matrix} 1 \\ 1 \\ 1 \\ \end{matrix} \right] \\ & \sum\limits_{k=1}^{4}{{{y}_{i,j,1,k}}=1} \\ & 0\le y\le 1 \\ & {{y}_{i,j,1,k}}\text{ is integral} \\ & \\ & \forall i=1...30\,j=1...29\,\,k=1...4 \\ & \left[ \begin{matrix} {{s}_{i,j}} \\ {{s}_{i,j+1}} \\ {{s}_{i,j,1}} \\ \end{matrix} \right]={{y}_{i,j,2,1}}\left[ \begin{matrix} 0 \\ 0 \\ 0 \\ \end{matrix} \right]+{{y}_{i,j,2,2}}\left[ \begin{matrix} 1 \\ 0 \\ 0 \\ \end{matrix} \right]+{{y}_{i,j,2,3}}\left[ \begin{matrix} 0 \\ 1 \\ 0 \\ \end{matrix} \right]+{{y}_{i,j,2,4}}\left[ \begin{matrix} 1 \\ 1 \\ 1 \\ \end{matrix} \right] \\ & \sum\limits_{k=1}^{4}{{{y}_{i,j,2,k}}=1} \\ & 0\le {{y}_{i,j,2,k}}\le 1 \\ & {{y}_{i,j,2,k}}\text{ is integral} \\ & \\ & \forall i=1...29\,j=1...29\,\,k=1...8 \\ & \left[ \begin{matrix} {{s}_{i,j}} \\ {{s}_{i+1,j}} \\ {{s}_{i,j+1}} \\ {{s}_{i,j,3}} \\ \end{matrix} \right]={{y}_{i,j,3,1}}\left[ \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ \end{matrix} \right]+{{y}_{i,j,3,2}}\left[ \begin{matrix} 1 \\ 0 \\ 0 \\ 0 \\ \end{matrix} \right]+{{y}_{i,j,3,3}}\left[ \begin{matrix} 0 \\ 1 \\ 0 \\ 0 \\ \end{matrix} \right]+{{y}_{i,j,3,4}}\left[ \begin{matrix} 1 \\ 1 \\ 0 \\ 0 \\ \end{matrix} \right]+{{y}_{i,j,3,5}}\left[ \begin{matrix} 0 \\ 0 \\ 1 \\ 0 \\ \end{matrix} \right]+{{y}_{i,j,3,6}}\left[ \begin{matrix} 1 \\ 0 \\ 1 \\ 0 \\ \end{matrix} \right]+{{y}_{i,j,3,7}}\left[ \begin{matrix} 0 \\ 1 \\ 1 \\ 0 \\ \end{matrix} \right]+{{y}_{i,j,3,8}}\left[ \begin{matrix} 1 \\ 1 \\ 1 \\ 1 \\ \end{matrix} \right] \\ & \sum\limits_{k=1}^{8}{{{y}_{i,j,3,k}}=1} \\ & 0\le {{y}_{i,j,3,k}}\le 1 \\ & {{y}_{i,j,3,k}}\text{ is integral} \\ & \\ & \forall i=1...30\,j=1...30 \\ & 0\le {{s}_{i,j}}\le 1 \\ & {{s}_{i,j}}\quad \text{integral} \\ \end{align}]