<p>If it possible to have a local maximum when the Hessian is only negative semi-definite (i.e., there is one zero eigenvalue and all other eigenvalues are negative).</p>

<p>If not, what it the ultimate recourse to prove that a point is a local maximum is such cases?</p>

CC BY-SA 3.0
2016-08-23T12:54:45.470Z
If it's NSD then it's inconclusive :http://calculus.subwiki.org/wiki/Second_derivative_test_for_a_function_of_two_variables  You could try to visually look at it.
VCG
CC BY-SA 3.0
2016-08-23T16:43:37.880Z
Useful pointer. Thanks. Looks like I can rule out it being a local minimum, but what is the next course of action? Suppose that I am dealing with a firm's profit maximization problem, it seems like the eigen-vector associated with the zero eigen-value can be interpreted as a direction along which the firm would be indifferent as marginal step in that direction does not impact the value of the objective. Does this intuition generalize? Has some mathematician formalized it?
davidrpugh
CC BY-SA 3.0
2016-08-23T17:47:35.220Z
Can you give me such a profit function? Looking at it may help me get an idea.
VCG
<optimization>
Local maximum when Hessian is negative semi-definite?
davidrpugh
# Local maximum when Hessian is negative semi-definite?

(Asked by: username_0 on 2016-08-23)

If it possible to have a local maximum when the Hessian is only negative semi-definite (i.e., there is one zero eigenvalue and all other eigenvalues are negative).

If not, what it the ultimate recourse to prove that a point is a local maximum is such cases?

**Tags:** optimization

> **Comments:**
> **username_1**: If it's NSD then it's inconclusive :http://calculus.subwiki.org/wiki/Second_derivative_test_for_a_function_of_two_variables  You could try to visually look at it.  
> **username_0**: Useful pointer. Thanks. Looks like I can rule out it being a local minimum, but what is the next course of action? Suppose that I am dealing with a firm's profit maximization problem, it seems like the eigen-vector associated with the zero eigen-value can be interpreted as a direction along which the firm would be indifferent as marginal step in that direction does not impact the value of the objective. Does this intuition generalize? Has some mathematician formalized it?  
> **username_1**: Can you give me such a profit function? Looking at it may help me get an idea.  

---

## 0 answers

