Maximize: 3*v*m(adsbygoogle = window.adsbygoogle || []).push({});

subject to:

L - m - v >= 0

V - v >= 0

m - 6 >= 0

M - m >= 0

Where L, M, and V are positive integers.

Lagrangian (call it U):

U = 3vm + K1(L - m - v) + K2(V - v) + K3(m - 6) + K4(M - m)

Where K1-K4 are the slack variables/inequality Lagrange multipliers.

Which yield the KKT conditions:

dU/dv = 3m - K1 - K2 = 0

dU/dm = 3v - K1 + K3 - K4 = 0

K1(L - m - v) = 0

K2(V - v) = 0

K3(m - 6) = 0

K4(M - m) = 0

K1-K4 >= 0

Now, suppose we assume K1 = K2 = 0, K3 and K4 != 0.

This yields:

m - 6 = 0

M - m = 0

(so m = M)

3(M) - K1 - K2 = 0

but we assumed K1 = K2 = 0, and plugging into the second KKT condition yields:

3(M) - K1 - K2 = 0, 3M = 0, which is not true.

I do not understand if I have made an error, or if this result is to be interpreted in some fashion. Does this simply mean that the point is infeasible? It just seems strange to obtain the result that way; other times I can solve for all variables and clearly see that K1-K4 are not all positive, or that another constraint is being violated.

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# Finding and recognizing infeasible Lagrange multiplier points

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