alt="image"/> ∈ absmin, Smin = 3.
EXAMPLE 1.5.– An example of the irregular problem. Consider the constrained optimization problem
Figure 1.1. Example 1.5
Solution. Figure 1.1 depicts the admissible set of the problem and the level line of the objective function. The solution of the problem is the point
The vector
at the point
Gradients
Answer.
EXAMPLE 1.6.– Solve the convex constrained optimization problem
Solution. Slater’s condition is fulfilled. Therefore, we write the regular Lagrange function:
The system for finding stationary points in this case (s = 0, n = 2, k = m = 3) can be written in the form
At point
System solution
Answer.
EXAMPLE 1.7.– Let the numbers a > 0, b > 0, and let a < b. Find points of the local minimum and the local maximum of the function
on the set of solutions of the system
Solution. We denote this set by X. Let us write the Lagrange function
Figure 1.2. Example 1.6
The system for determining stationary points has the form
[1.4]
[1.5]
[1.6]
[1.7]
[1.8]
Let xi = 0. Then it follows from the system that x2 ≥ 1,
1 1) x1 = 0, x2 = 1, bλ0 + 3λ1 − 2λ2 = 0, λ1 ≥ 0, λ2 ≥ 0, (λ1, λ2) ≠ 0;
2 2) x1 = 0, x2 = −1, bλ0 − 2λ2 = 0, λ1 = 0, λ2 > 0.
Similarly, assuming that x2 = 0, we obtain two other groups of solutions:
1 3) x1 = 1, x2 = 0, aλ0 + 3λ1 − 2λ2 = 0, λ1 ≥ 0, λ2 ≥ 0, (λ1, λ2) ≠ 0;
2 4) x1 = −1, x2 = 0, aλ0 − 2λ2 = 0, λ1 = 0, λ2 > 0.
Assume that x1 ≠ 0, x2 ≠ 0. Then equations of the system can be presented in the form
