Penalty and barrier methods
Compare the constrained problem with a quadratic penalty and a logarithmic barrier. The slice is taken along x = y = t, the line containing all three maximizers.
Surface plot
constraint: x² + y² = 1
Diagonal slice
x = y = t
Displayed maximizer
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Constraint value x² + y²
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Actual constrained optimum
(1/√2, 1/√2), value √2