Convex Optimization. Mikhail Moklyachuk

higher order) If the function f(x) has a derivative of order n at point and

      for some m ≥ 1, 2m ≤ 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. □

1.3. Optimization problems with objective functions of several variables

      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 ∈ .

      THEOREM 1.8.– A function is lower semicontinuous on ℝⁿ if and only if for all a ∈ ℝ the set f^–1((a, +∞]) is open (or the complementary set f^–1 ((–∞, a]) is closed).

      PROOF.– Let f be a lower semicontinuous on ℝⁿ function, let a ∈ ℝ and let ∈ f^–1((a, +∞]). Then there exists a δ-neighborhood of the point such that for all points x ∈ the inequality f(x) > a holds true. This means that . Consequently, the set f^–1((a, +∞]) is open.

      Vice versa, if the set f^–1((a, +∞]) is open for any a ∈ ℝ and ∈ ℝⁿ, then and the function f is lower semicontinuous at point by agreement, or and ∈ f^–1((a, +∞]), when . Since the set f^–1((a, +∞]) is open, then there exists a δ-neighborhood of the point such that and f(x) > a for all x ∈ . Consequently, the function f is lower semicontinuous at point . □

      THEOREM 1.9.– (Weierstrass theorem) A lower (upper) semicontinuous on a non-empty bounded closed subset X of the space ℝⁿ function is bounded from below (from above) on X and attains the smallest (largest) value.

      THEOREM 1.10.– (Weierstrass theorem) If a function f(x) is lower semicontinuous and for some number a the set {x: f(x) ≤ a} is non-empty and bounded, then the function f(x) attains its absolute minimum.

      COROLLARY 1.1.– If a function f(x) is lower (upper) semicontinuous on ℝⁿ and

      then the function f(x) attains its minimum (maximum) on each closed subset of the space ℝⁿ.

      THEOREM 1.11.– (Necessary conditions of the first order) If is a point of local extremum of the differentiable at the point function f(x), then all partial derivatives of the function f(x) are equal to zero at the point :

      THEOREM 1.12.– (Necessary conditions of the second order) If is a point of local minimum of the function f(x)>, which has the second-order partial derivatives at the point , then the following condition holds true:

      This condition means that the matrix

      is non-negative definite.

      THEOREM 1.13.– (Sufficient conditions of the second order) Let a function have the second-order partial derivatives at a point and let the following conditions hold true:

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