简化版的优化问题

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Published on: 2010/07/02

提问:一个优化问题一文中提出了一个5种属性点的分配问题。如果n=2,并且假定没有主属性和副属性相同的技能,也就是只有两种属性,那么问题可以简化到很容易求出解析解。

问题描述:

\mathrm{min} \frac{A_1}{2p_1+p_2}+\frac{A_2}{p_1+2p_2}

\mathrm{s.t.}p_1+p_2=C, p_1, p_2 >= 0

定义F(p_1, p_2, \lambda) = \frac{A_1}{2p_1+p_2}+\frac{A_2}{p_1+2p_2}-\lambda(p_1+p_2)

\frac{\partial F}{\partial p_1}=-\frac{2A_1}{(2p_1+p_2)^2}-\frac{A_2}{(p_1+2p_2)^2}+\lambda=0\frac{\partial F}{\partial p_2} = -\frac{A_1 }{(2p_1+p_2)^2}-\frac{2A_2}{(p_1+2p_2)^2}+\lambda=0

那么有:\frac{A_1}{(2p_1 +p_2)^2}=\frac{A_2}{(p_1+2p_2)^2}

k= \sqrt{\frac{A_1}{A_2}},则:

k= \frac{2p_1+p_2}{p_1+2p_2}

So, \frac{p_1}{p_2}=\frac{2-k}{2k-1}

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