Convex and Concave Functions How to determine if

Convex and Concave Functions • How to determine if a function f(x 1, x 2, … xn)is convex or concave • Use the Hessian matrix and determine principal minors Note: H will always be a square symmetric matrix

Convex and Concave Functions Example: H= 4 0 0 0 6 0 0 0 8 H 1 = H 3 = [4 ] = 4 H 2 = 4 0 0 0 6 0 = 192 0 0 8 4 0 0 6 = 24

Convex and Concave Functions Example: H 1 = 4 f(x) = 2 x 12 + 3 x 22 +4 x 32 - 8 x 1 - 12 x 2 - 24 x 3 +110 H 2 = 24 H 3 = 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) = x 12 - x 22 + x 32 + 2 x 1 x 3 + x 1 x 2 Is f(x) convex or concave? or neither?

Convex and Concave Functions Example: H= f(x) = x 12 - x 22 + x 32 + 2 x 1 x 3 + x 1 x 2 2 1 -2 0 2 H 1 = 2 H 2 = - 5 H 3 = - 2 • 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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