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I know that if the Hessian matrix of a multivariable function at a given stationary point has both positive and negative eigenvalues then that stationary point must be a saddle point. Does the same hold with multivariable optimization problems with only a single linear equality constraint where a reduced Hessian matrix has been calculated?

Ryan G
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    You have to use the bordered Hessian: see http://en.wikipedia.org/wiki/Hessian_matrix#Bordered_Hessian I think it's roughly the same idea, but check the details. I can refer you to more comprehensive articles if you would be interested in reading them. –  Apr 29 '14 at 23:38
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    Actually, if the constraint is linear, it might be easier to just solve for one of the variables, plug in, and then do a normal unconstrained problem and use the Hessian of the new function. –  Apr 29 '14 at 23:41
  • That's actually what I ended up doing, but I was curious because in the text I'm reading it says the necessary conditions for a local minimizer when dealing with linear equality constraints are 1). that the reduced gradient is zero at that point and 2). the reduced Hessian is positive semidefinite. The text doesn't elaborate on local maximizers or saddle points which is why I was unsure. – Ryan G Apr 29 '14 at 23:45
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    See Theorem 5 of this: http://www.math.northwestern.edu/~clark/285/2006-07/handouts/lagrange-2deriv.pdf It gives general conditions, and perhaps you will be able to see how it translates to the conditions you've been given. I don't actually what "reduced [thing]" refers to, so I'll leave you to the article :) –  Apr 29 '14 at 23:50
  • Thanks that's exactly what I was looking for! – Ryan G Apr 29 '14 at 23:54
  • No problem, cheers! –  Apr 29 '14 at 23:56

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