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Hi, I want to implement a minimisation on a 2D spin Ising model with 30x30 grid. 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} \\ ss_{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} \\ ss_{i,j,2} \\ \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} \\ ss_{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] \\ & \qquad\qquad~~ + 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}\quad{\text{integral}} \\ & \\ & \forall i = 1...30\,j = 1...30 \\ & 0 \le s_{i,j} \le 1 \\ & s_{i,j}\quad{\text{integral}} \end{align} \]

asked 13 Jul '14, 23:47

Chivalry's gravatar image

Chivalry
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accept rate: 0%

edited 18 Jul '14, 06:36

fbahr's gravatar image

fbahr ♦
4.6k716

thank you sooo much for helping me to fix my format:D

(14 Jul '14, 11:44) Chivalry
1

Sometimes \(s\) has two subscripts, sometimes three.

(15 Jul '14, 15:54) Paul Rubin ♦♦

actually, those with two subscripts are spin, while those with 3 subscripts are spin products:)

(15 Jul '14, 16:18) Chivalry

I also edited the equation to make it more obvious...

(15 Jul '14, 16:32) Chivalry
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Asked: 13 Jul '14, 23:47

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Last updated: 18 Jul '14, 06:36

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