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This question is for sure a duplicate, but different users seem to give different answers. The question is: suppose you find that the Hessian matrix for a function $f(\textbf{x})$ is semidefinite positive on the whole domain. Are all stationary points also minima?

Here it seems like the answer is positive, while here it seems like it is negative.

There is a nice figure in Chiang, Fundamental Methods of Mathematical Economics (I don't know whether I can post a scanned image of it) in which you can read something like this: 1) if the Hessian is everywhere positive semidefinite, the function is convex; 2) if the function is convex, a stationary point is a global minimum. Moreover: 1) if the Hessian is everywhere positive definite, the function is strictly convex; 2) if the function is strictly convex, a stationary point is a unique global minimum.

Is that always right for $C^2$ functions?

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Luigi
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  • Both of your links point to the same question. But it does not apply in your case because that question is talking about the Hessian at a single stationary point, not the whole domain like you are looking at. – Michael Grant Apr 10 '15 at 14:50

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I had a similar question, including your question as a special case: See Local optimality of a KKT point..

The answer to your question depends on what is the whole domain of $x$. I.e. if the domain is a non-convex set (defined by non-convex constraints), then, a stationary point can be everything but is not necessarily a local or global minimum.

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The Pheromone Kid
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