Mikhail Moklyachuk - Convex Optimization

then is the point of local minimum (maximum) of the function f ( x ).

The necessary and sufficient conditions of the higher order of existence of an extremum of the function f ( x ) are given in the following theorems.

THEOREM 1.6.– ( Necessary conditions of higher order ) If is a point of local minimum (maximum) of the function f ( x ), which has at this point the n th order derivative, then

or

for some m ≥ 1, 2 m ≤ n .

PROOF.– According to Taylor’s theorem, for a function which is n times differentiable at the point we have

If n = 1, then the assertion of the theorem is true as a result of Fermat’s theorem. Let n > 1, then

or

Let l be an odd number. Then the function , u ∈ R can be expanded in a series by Taylor’s theorem

The function g ( u ) has a derivative at point u = 0. Since ∈ locmin f , then 0 ∈ locming g . According to Fermat’s theorem, . Hence . This contradicts the condition . Therefore, the number l is even, l = 2 m . According to Taylor’s theorem

Since , then if ∈ locmin f and if ∈ locmax f . □

THEOREM 1.7.– ( Sufficient conditions of higher order ) If the function f ( x ) has a derivative of order n at point and

for some m ≥ 1, 2 m ≤ n , then the function f ( x ) attains a local minimum (maximum) at the point .

PROOF.– Since , then according to Taylor’s theorem

If , then for sufficiently small x , i.e. ∈ locmin f . If , then for sufficiently small x , i.e. ∈ locmax f . □

Let be a function of n real variables.

DEFINITION 1.3.– A function f is said to be lower semicontinuous (upper semicontinuous) at a point if there exists a δ-neighborhood

of the point such that the inequality

holds true for all x ∈ .