1. Convex and Concave Functions
• How to determine if a function f(x1, x2,… xn)is convex or concave
• Use the Hessian matrix and determine principal minors
Note: H will always be a square symmetric matrix

2. Convex and Concave Functions
Example:
[ 4 ] = 4
H1 =
4 0
0 6
H2 =
= 24
H =
H3 =
= 192

3. Convex and Concave Functions
Example: f(x) = 2x12 + 3x22 +4x32 - 8x1 - 12x2 - 24x3 +110
H1 = 4
H2 = 24
H3 = 192
• If all leading principal minors of H are nonnegative f(x) is convex
• If all leading principal minors of H are positive f(x) is strictly convex
•• Therefore, f(x) is a strictly convex function
(i.e. a local minimum will be a global minimum)
Example: f(x) = x12 - x22 + x32 + 2x1x3 + x1x2
Is f(x) convex or concave?
or neither?

4. Convex and Concave Functions
Example: f(x) = x12 - x22 + x32 + 2x1x3 + x1x2
H1 = 2
H2 = - 5
H3 = - 2
H =
• If the kth leading principal minors of H has the same sign as (-1)k ,,
then f(x) is strictly concave (in which case a local maximum
will be a global maximum).
• If the kth leading principal minors of H has the same sign as (-1)k ,,
or zero, f(x) is concave.
•• Therefore, f(x) is neither convex nor concave

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