8.2.4.二元函数局部最优条件 A function H(1,02)has a local maximum at (01,62),the following three conditions hold (1)The first-order partial derivatives are 0, H0,2la=6-i=0 001 H0,2o=成-4=0 002 (2)At least one second-order partial derivative is negative, 气0%=或6= 02 or a,l=6>0 15/56
8.2.4. 二元函数局部最优条件 A function H(θ1, θ2) has a local maximum at ( ˆθ1, ˆθ2), the following three conditions hold (1) The first-order partial derivatives are 0, ∂ ∂θ1 H(θ1, θ2)| θ1=θˆ 1,θ2=θˆ 2 = 0 ∂ ∂θ2 H(θ1, θ2)| θ1=θˆ 1,θ2=θˆ 2 = 0 (2) At least one second-order partial derivative is negative, ∂ 2 ∂θ2 1 H(θ1, θ2)| θ1=θˆ 1,θ2=θˆ 2 < 0 or ∂ 2 ∂θ2 2 H(θ1, θ2)| θ1=θˆ 1,θ2=θˆ 2 > 0 15 / 56
(3)The determinant of Hessian matrix is positive at the interesting points, H0,2) mH0,6） 02 a00mH01,02) Hl6,a） 01=A,02=2 a高ra-(e. 2 >0 01=01,02=02 This means thatH)and)must have the same sign to be larger than a nonnegative number. local max(min)for negative(positive)definite. 16/56
(3) The determinant of Hessian matrix is positive at the interesting points, ∂ 2 ∂θ2H(θ1, θ2) ∂ 2 ∂θ1∂θ2 H(θ1, θ2) ∂ 2 ∂θ1∂θ2 H(θ1, θ2) ∂ 2 ∂θ2 2 H(θ1, θ2) θ1=θˆ 1,θ2=θˆ 2 = ∂ 2 ∂θ2 1 H(θ1, θ2) ∂ 2 ∂θ2 2 H(θ1, θ2)− ∂ 2 ∂θ1∂θ2 H(θ1, θ2) 2 θ1=θˆ 1,θ2=θˆ 2 > 0 This means that ∂ 2 ∂θ2 1 H(θ1, θ2) and ∂ 2 ∂θ2 2 H(θ1, θ2) must have the same sign to be larger than a nonnegative number. local max (min) for negative (positive) definite. 16 / 56
For a function f of more than two variables and has first order partial derivative equals zero at(a,b,...): If the Hessian is positive definite (equivalently,has all eigenvalues positive)at (a,b,...)then fattains a local minimum at (a,6,...). If the Hessian is negative definite (equivalently,has all eigenvalues negative)at (a,b,...),then fattains a local maximum at (a,b,...). If the Hessian has both positive and negative eigenvalues then (a,b,...)is a saddle point forf. 17/56
For a function f of more than two variables and has first order partial derivative equals zero at (a, b, ...): 1 If the Hessian is positive definite (equivalently, has all eigenvalues positive) at (a, b, ...), then f attains a local minimum at (a, b, ...). 2 If the Hessian is negative definite (equivalently, has all eigenvalues negative) at (a, b, ...), then f attains a local maximum at (a, b, ...). 3 If the Hessian has both positive and negative eigenvalues then (a, b, ...) is a saddle point for f. 17 / 56
Determinant of Hessian matrix of function fis defined as: i12 ff …2m IHI= f fe ..Jmn For function fof two variables The second order condition for a local maximum is then that fi<0 and which is just the condition that H is negative definite. 18/56
Determinant of Hessian matrix of function f is defined as: |H| = f11 f12 ... f1n f21 f22 ... f2n . . . . . . . . . . . . fn1 fn2 ... fnn For function f of two variables |H| = f11 f12 f21 f22 The second order condition for a local maximum is then that f11 < 0 and f11 f12 f21 f22 > 0 which is just the condition that H is negative definite. 18 / 56