BIZNIQUE / MATHEMATICAL OPTIMISATION / 58
Use a multiplier to express a constraint
Use a multiplier to express a constraint
Connect the slope of an objective with the normal to an equality constraint.
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THE IDEA
Read the example.
Read the example.
At the constrained optimum, stationarity gives x = 3−λ/2 and y = 2−λ/2. Combining these with x+y=b gives λ=5−b. The multiplier depends on the sign convention used in the Lagrangian.
L = (x−3)² + (y−2)² + λ(x+y−b)
Use it well
Use it well
An equality multiplier may have either sign. Inequality constraints add sign and complementary-slackness conditions; first-order conditions alone do not always certify a global optimum.
TRY IT YOURSELF
A quick check.
A quick check.
For the equality x+y = 7, find x, y and λ.
Check your reasoning
Check your reasoning
x = 4, y = 3 and λ = −2. The constraint forces the solution past the unconstrained target.
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