Commit e9d1e9cd authored by Michael Keller's avatar Michael Keller
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parent aeb6b0a5
......@@ -213,8 +213,18 @@ statement with a condition on $R$ might be helpful:
&+ R(c_\beta, c_\alpha) \cdot ((u_\beta + \lambda) \cdot (v_\alpha + \lambda) - u_\beta \cdot v_{\alpha})\\
&+ R(c_\beta, c_\beta) \cdot ((u_\beta + \lambda) \cdot (v_\beta - \lambda) - u_\beta \cdot v_{\beta})\\
&= R(c_\alpha, c_\alpha) \cdot (u_\alpha \cdot \lambda - v_\alpha \cdot \lambda - \lambda^2)\\
&+ R(c_\alpha, c_\beta) \cdot (- u_\alpha \cdot \lambda - v_\beta \cdot \lambda + \lambda^2)\\
&- R(c_\alpha, c_\beta) \cdot (u_\alpha \cdot \lambda + v_\beta \cdot \lambda - \lambda^2)\\
&+ R(c_\beta, c_\alpha) \cdot (u_\beta \cdot \lambda + v_\alpha \cdot \lambda + \lambda^2)\\
&+ R(c_\beta, c_\beta) \cdot (- u_\beta \cdot \lambda + v_\beta \cdot \lambda - \lambda^2)\\
&- R(c_\beta, c_\beta) \cdot (u_\beta \cdot \lambda - v_\beta \cdot \lambda + \lambda^2)\\
&= \lambda^2 \cdot \left ( R(c_\alpha, c_\beta) + R(c_\beta, c_\alpha) - R(c_\alpha, c_\alpha) - R(c_\beta, c_\beta) \right )\\
&+ \lambda \cdot (R(c_\alpha, c_\alpha) - R(c_\alpha, c_\beta)) \cdot u_\alpha\\
&+ \lambda \cdot (R(c_\beta, c_\alpha) - R(c_\beta, c_\beta)) \cdot u_\beta\\
&+ \lambda \cdot (R(c_\beta, c_\alpha) - R(c_\alpha, c_\alpha)) \cdot v_\alpha\\
&+ \lambda \cdot (R(c_\beta, c_\beta) - R(c_\alpha, c_\beta)) \cdot v_\beta
\end{align*}
Which, if every crop likes all other crops
more than itself, can be made positive with the
appropriate $\lambda$. Then the quadratic
term in the formula above will always be positive,
and we can focus only on all of the linear constraints.
\end{proof}
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