Lagrange's method
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Lagrange's method solves constrained optimization problems by forming an augmented function that combines the objective function and constraints, using Lagrange multipliers (λ) as weighting factors. The method finds extrema by taking partial derivatives of the augmented function with respect to the objective variables and λ, setting the results equal to zero. This produces a system of equations that can be solved simultaneously to identify values that satisfy the constraint and optimize the original objective function.
Lagrange's method
- 2. Lagrange Multipliers The constrained optima problem can be stated as find ing the extreme value of subject to . So Lagrange (a mathematician) formed the augmente d function. denotes augmented function will behave like the function if the constraint is followed.
- 3. Given the augmented function, the first order conditi on for optimization (where the independent variable s are , and λ) is as follows:
- 4. Using the previous example: note: to be on the b udget line
- 5. Lagrange Multipliers Solving these 3 equations simultaneously:
- 6. Solving these 3 equations simultaneously (cont’d):
- 7. Solving these 3 equations simultaneously (cont’d):
- 8. If , then This solution yields the same answer as the substit ution method, i.e., and .