Lecture 4 - Budget and space constraints Minimize s.t i =1...

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Budget and space constraints Minimize s.t. C Q c n i i i = 1 ( ) = + = n i i i i i i n Q D A Q h Q Q TAC 1 1 2 ,..., W Q w n i i Budget constraint Space constraint i = 1 Lagrange multipliers method: Minimize by solving necessary conditions: ( ) - Θ + - Θ + + = Θ Θ = = = n i i i n i i i n i i i i i i n W Q w B Q c Q D A Q h Q Q TAC 1 2 1 1 1 2 1 1 2 , , ,..., 2 , 1 ; ,..., 1 0 , 0 = = = Θ = j n i for G Q G j i
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Budget and space constraints Steps to Find Optimal Solution Double constraints: 1. Solve the unconstrained problem. If both constraints are satisfied, this solution is the optimal one. 2. Otherwise rewrite objective function using Lagrange multipliers by including one of the constraints, say budget, and solve one-constraint problem to find optimal solution. If the space constraint is satisfied, this solution is the optimal one. 3. Otherwise repeat the process for the only space constraint. 4. If both single-constraint solutions do not yield the optimal solution, then both constraints are active, and the Lagrange equation with both constraints must be solved. 5. Obtain optimal Q i * by solving (n+2) equations 2 , 1 ; ,..., 1 0 , 0 = = = Θ = j n i for G Q G j i ( ) - Θ + - Θ + + = Θ Θ = = = n i i i n i i i n i i i i i i
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