Convex and Concave Functions

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 4 0 0 0 6 0 0 0 8 H2= = 24 4 0 0 0 6 0 0 0 8 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 2 1 2 1 -2 0 2 0 2 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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