Week 6 Flashcards
When is a matrix negative definite?
When it’s leading principle minors alternate in sign, with the first minor negative, the second positive, third negative etc.
When is a matrix negative semidefinite?
When it’s leading principle minors alternate in sign, with the first less than or equal to zero, the second positive or equal to zero, the third negative or equal to zero etc.
When is a matrix positive definite?
When all it’s principle leading minors are strictly positive.
When is a matrix positive semidefinite?
When all it’s principle minors are positive or equal to zero.
What is the condition needed to find critical point in UNCONSTRAINED optimisation problems?
Our function needs to be twice diferentiable in an open neighbourhood of x*
What is the first order necessary conditon in UNCONSTRAINED optimisation problems?
If x* is a local extremum, then f’(x*) = 0
What is the second order sufficient condition for UNCONSTRAINED optimisation problems?
if d2f(x*) / dx2 > 0 , then we have a local MINIMUM
if d2f(x*) / dx2 < 0 , then we have a local MAXIMUM
What is the second order sufficient condition using the hessian for UNCONSTRAINED optimisation?
If our hessian is positive definite, our critical point is a MINIMUM.
If our hessian is negative definite, our critical point is a MAXIMUM.
If our hessian is indefinite, we have a saddle point.
What is the condition for global extrema in UNCONSTRAINED optimisation problems?
If f is twice continuously differentiable and CONVEX (hessian is positive semidefinite), we have a global MINIMUM
If f is twice continuously differentiable and CONCAVE (hessian is negative semidefinite), we have a global MAXIMUM
