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As I understand it, the Lagrange multiplier is always positive. Can it be negative?

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    $\begingroup$ The Lagrange multipliers for enforcing inequality constraints ($\le$) are non-negative. The Lagrange multipliers for equality constraints ($=$) can be positive or negative depending on the problem and the conventions used. $\endgroup$
    – Nick Alger
    Commented Nov 25, 2016 at 9:40

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If you have a purely geometric problem of maximizing or minimizing some function $f$ under a constraint $g=0$ then some people set up the auxiliary function $\Phi:=f-\lambda g$, and other people define $\Phi:=f+\lambda g$. Both will of course obtain the same conditionally stationary points of $f$, but the sign of $\lambda$ in these points will be opposite for the two.

If however the problem at stake comes from a physical or economical context then it is sometimes possible to give the $\lambda$ a meaning in this context, in so far as there is a clear cut "positive side" of $g$. People then speak about "virtual forces" or "shadow prices".

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