Abstract
In this paper, a scheme for using the method of smooth penalty functions for the dependence of solutions of multi-criterial optimization problems on parameters is being considered. In particular, algorithms based on the method of smooth penalty functions are given to solve problems of optimization by the parameters of the level of consistency of the objective functions and to find the corresponding shape of the Pareto’s set.
Highlights
Subject to maximization possessing at interior points of the set of elements x ∈ E n, and satisfying the following conditions: yi ( x,u) ≤ 0 i = [1,m]
The concept of improving the multi-criterial objective function allows the feasible points to be divided into two subsets: for the first, all feasible points improve all objective functions and for the second, there are points for which the improvement of one function causes the deterioration of at least one other function
A general universal approach to the solution of multi-criterial optimization problems has not been proposed yet, but numerous approaches have been developed, which limit the number of solutions
Summary
Let us consider the problem of finding in the parameter space a standard method (for example, gradient) of finding the extremum of the mismatch value of the objective functions of the multicriterial model (3)–(4)–(5). Standard optimization methods used for lower-level tasks, based on the use of continuous gradients or other differential characteristics, suggest that in addition to the solving system (9), these characteristics themselves can be found Let us demonstrate this using the example of calculating the derivatives of the function Fk (u) with respect to the components of the vector u of parameters. Let the solutions of system (13) be ρ(u) and x (u) , as a smoothed approximation of the ( ) dependency ρ** (u) , we can use the function E (u) = −E ρ,ρ(u), x (u),u The derivatives of this function by the components of the vector u ofparameters and the rule for differentiating a composite function of several variables give us:. This will be done at the end of the article, while let us illustrate an example
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