Optimisation Flashcards
(8 cards)
Define a maximum, minimum and a point of inflection, in terms of second derivatives of f(x).
At point (a, f(a)), the point can be defined as:
A maxima - if f’‘(a) < 0.
A minima - if f’‘(a) > 0.
A point of inflection - if f’‘(a) = 0 and the sign changes.
What is the difference between a critical points and points of inflection?
Critical points are any type of stationary point on a graph, i.e. where f’(x) = 0.
A point of inflection is any point which f’(x) = 0 and the sign of f’‘(x) changes.
What expansion can be used to explore the nature of a point of inflection (a, f(a))?
A Taylor series expansion.
What a saddle point on a 3D graph?
One in which the point is a maximum in two axes but a minimum on the third.
What type of point P can a hessian matrix be used for?
A hessian matrix acts upon a position vector P and its determinant gives the nature of point P, i.e. its type of critical point.
What does a hessian matrix of function F look like?
[Fxx Fxy]
[Fxy Fyy]
if detH > 0, what does this mean about position (a, b) of function F?
Similarly for detH < 0 and detH = 0.
if detH > 0, (a, b) is a local minimum or maximum of F. if Fxx > 0 or Fyy > 0, it is a minimum, and if Fxx < 0 or Fyy < 0, it is a maximum.
if detH < 0 then point (a, b) is a saddle point of F.
if detH = 0 further investigation is required.
What does it mean to consider the lagrangian of function F(x, y)
The lagrangian is a function that must be used if F is subject to the constraints of some g(x, y). it looks like this:
F(x, y, λ) = f(x, y) + λg(x, y)
Consider it like a constant of integration which we must find when g(x, y) = 0 for a maxima or minima.