Convex Optimization. Mikhail Moklyachuk
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Название: Convex Optimization

Автор: Mikhail Moklyachuk

Издательство: John Wiley & Sons Limited

Жанр: Математика

Серия:

isbn: 9781119804086

isbn:

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      The formalized problem consists of the following elements:

       – objective function ;

       – domain X of the definition of the objective functional f;

       – constraint: C ⊂ X.

      Here,

is an extended real line, that is, the set of all real numbers, supplemented by the values +∞ and –∞, C is a subset of the domain of definition of the objective functional f. So to formalize an optimization problem means to clearly define and describe elements f, C and X. The formalized problem is written in the form

      The maximization problem can always be reduced to the minimization problem by replacing the functional f with the functional g = –f. And, on the contrary, the minimization problem in the same way can be reduced to the maximization problem. If the necessary conditions for the extremum in the minimization problem and maximization problem are different, then we write these conditions only for the minimization problem. If it is necessary to investigate both problems, then we write down

      An admissible point image is a point of absolute or global minimum (maximum) of the extremum problem if for any x. ∈ C the inequality holds true

image

      Then we write image ∈ absmin (absmax). The point of the absolute minimum (maximum) is called a solution of the problem. The value image, where image is a solution of the problem, is called a numerical value of the problem. This value is denoted by Smin (Smax).

      In addition to global extremum problems, local extremum problems are also studied. Let X be a normed space. A local minimum (maximum) of the problem is reached at a point image that is image ∈ locmin (locmax), if image ∈ C and there exists a number δ > 0 such that for any admissible point x ∈ C that satisfies condition image, the inequality holds true

image

      In other words, if image ∈ locmin (locmax), then there exists a neighborhood image of the point image such that

image

      in the problem

image

      The theory of extremum problems gives general rules for solving extremum problems. The theory of necessary conditions of the extremum is more developed. The necessary conditions of the extremum make it possible to allocate a set of points among which solutions of the problem are situated. Such a set is called a critical set, and the points themselves are called critical points. As a rule, a critical set does not contain many points and a solution of the problem can be found by one or another method.

      Let f: ℝ → ℝ be a function of one real variable.

      DEFINITION 1.1.– A function f is said to be lower semicontinuous (upper semicontinuous) at a point if for every ε > 0, there exists a δ > 0 such that the inequality

image

      holds true for all x ∈ (image, image).

      DEFINITION 1.2.– (Equivalent) A function f is said to be lower semicontinuous (upper semicontinuous) at a point image if for every a ∈ ℝ, image there exists δ > 0, such that the inequality

image

      holds true for all x ∈ (image, image).

      If the function takes values in image, then definitions 1.1 and 1.2 make sense when image. In the case where image, the function f is considered to be lower semicontinuous (upper semicontinuous) by agreement.

      Here are examples of semicontinuous functions:

      1 1) the function y = [x] (integer part of x) is upper semicontinuous at the points of discontinuity;

      2 2) the function y = {x} (fractional part of x) is lower semicontinuous at the points of discontinuity;

      3 3) the Dirichlet function, which is equal to 0 at rational points and equal to 1 at irrational points, is lower semicontinuous at each rational point and upper semicontinuous at each irrational point;

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