Please, be aware that test cases don't necessarily use all the same penalty parameters. For instance, I have used 1e6 for cases 2,3,4 & 6 and 1e4 for cases 1 & 5. It all depends on which method one uses for the exterior penalty (and approximations may fail because of it).
In a sense, it is like trying to find the secret formula instead of finding the minimum value of a function.
High school cafeteria
First non-zero element in each column
Function composition - harder
Gauss Eliminate 2-by-2 example
Augmented Lagrange Multiplier (ALM) Method
Sequential Unconstrained Minimization (SUMT) using Interior Penalty
Free Fall analytical solution (Chapra 2012 textbook Example 1.1)
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