My Math Forum Lagrangian Multiplier (discrete version)

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 September 30th, 2013, 03:45 AM #1 Newbie   Joined: Sep 2013 Posts: 5 Thanks: 0 Lagrangian Multiplier (discrete version) Hi everyone, I have transformed my previously posted problem into discrete time (viewtopic.php?f=46&t=43103) : A school is trying to maximize its profits, which consist of the price charged for students of type i, p_i, times the number of students of type i, x_i, minus the Costs of having k customers, C(k). k is simply the sum of all customers x_i. Theta is the average ability of student all students. $\max_{\theta,k,p_i,x_i} \pi=\sum_{i=0}^n p_i x_i-C(k)\\ s.t.\\ k=\sum_{i=0}^n x_i\\ \theta=\frac{1}{k}\sum_{i=0}^n b_i x_i\\$ According to the solution, we can solve this by assuming we have an optimal $p_i^*$ and then forming the Lagrangian taking into account both constraints,i.e. the solution would be: $p_i^*= V#39;(k)+\eta (\theta_i-b_i)$ where $\eta$ is the Lagrangian multiplier of the second constraint: $\eta= \frac{1}{k} \sum_{i=0}^n \frac{\partial p_i^*}{\partial \theta} x_i$

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