Sample Question #63 (mathematics – optimization)

How do you find the local maximum of a function subject to a linear equality constraint? What’s the dual of this maximization problem?

That is: max *f(x) x*

s.t. *g(x) = c *where *c *is a constant

(Here, *x *is a vector.)

Bonus question: what if the constraint *g(x)=c* is nonlinear?

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ANSWER (partial)

Graphically, you can just graph f(x) and find the highest point where it intercepts with g(x). Of course, this only works if x has dimension less than 3.

Mathematically, we need to consider a few cases. If f(x) is linear, you will get a corner solution if g(x) is also linear. If the first and second derivatives of f(x) exist, you can use the Lagrange method: use the Lagrange multiplier to find the solution, and use the 2nd derivatives to make sure if it’s really a local maxim. (But what if one of these derivatives doesn’t exist?)

The dual problem is to minimize g(x) subject to a constraint of f(x).