Optimizing a Linear Function with Constraint: A Tutorial on Lagrange Multipliers

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To minimize the function f(x_{1},x_{2}...x_{n})=Σ*x_{i}*a_{i} under the constraint Σ x_{i}*(2*i+1)=constant, Lagrange multipliers can be applied. This method involves setting up the Lagrangian function by incorporating the constraint with a multiplier. The necessary conditions for optimality are derived from taking partial derivatives and setting them to zero. Resources like Wikipedia provide detailed tutorials on the application of Lagrange multipliers for such optimization problems. This approach is essential for projects involving complex functions, such as those related to the Cosmic Microwave Background.
quantumfireball
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First let me clarify this is not a homework question.
This part has cropped up as part of a small project i am doing on Cosmic microwave background.

How would i go about minimizing the function

f(x_{1},x_{2}...x_{n})=\Sigma*x_{i}*a_{i}

subject to the constraint:
\Sigma x_{i}*(2*i+1)=constanta_{i} are constants
 
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Use Lagrange multipliers. See Wikipedia (or other Google reference) for tutorial.
 

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