Optimization of the Shape of the Pareto Set in the Problems of Multi-criterial Programming

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.

I n mathematical modeling, it is often necessary to formalize preferences for states of the modeled object that generates several independent target functions. According to the historically established tradition, in this case, it is customary to talk about of multi-criterial optimization problems. A finite-dimensional multi-criterial model is a mathematical model with N objective functions: subject to maximization possessing at interior points of the set of elements n x E ∈ , and satisfying the following conditions: The incorrectness in the general case of such a statement is obvious, since the element x that is extremal for one of objective functions, in general, is not such for others.
However, useful information can be obtained by successively solving the following problems with a criterion for finding an extremum on the set (2) of each of the functions (1) separately for 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.
The second subset is called a Pareto-type set or, simply, a Pareto set. A general universal approach to the solution of multi-criterial optimization problems has not been proposed yet, but numerous approaches have been developed (see Fiacco & McCormick, 1968;Lotov & Pospelov, 2008), which limit the number of solutions.
For example, in the practical use of multi-criterial mathematical models, the set of independent objective functions is often replaced by a single one, thus passing to the standard problem of mathematical programming, allowing finding consistent or compromising solutions on the Pareto set in a certain sense.

Statement of the problem
In this article, the problem of finding an element on the set (2) that minimizes the gap between the objective functions will be considered as a compromise. In other words, this is a mathematical programming problem of the following form: and ρ is called "mismatch value".
The problem (4) is naturally called a two-level parametric problem since in its formulation it contains solutions of problems with single criterion (3), which we call first level problems. In this case, both in the problems of the first and the second level, it is assumed that the vector of the parameters u ∈Θ is fixed.
It is clear that the extreme value of the mismatch between the criteria in the general case is determined by the properties of the Pareto set and depends on the parameters vector u . Therefore, it is natural to indicate the third level optimization problem for the models (1)-(2) as follows: which solution will be the vector of parameters *** u ∈Θ and the number ( ) *** ** *** . u ρ = ρ In the present paper, possible solutions to problem (5) will be considered.

Solution method
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).
The specificity of this problem is based on the fact that the formulation of the problem (5) (the upper level or third level) includes the dependence It is proposed the use of the method of smooth penalty function to overcome this difficulty (see Umnov, 1975) and obtain a sufficiently smooth approximation dependences of ( ) It's assumed that the penalty function ( ) , P s τ , which penalizes the restriction 0 s ≤ , satisfies the following conditions: 1 0 ∀τ > and s ∀ , the function ( ) , P s τ has continuous derivatives with respect to all its arguments up to the second order 2 0 ∀τ > and s ∀ , 3 ( ) , 0 P s s τ > ∀ and 0 ∀τ > , and, When solving the third-level problem by an iterative method, for each step of the method, it is necessary preliminary to solve the problems of the second and first levels for a fixed vector of parameters u . Let us first consider a possible scheme for solving first-level problems. In fact, we will use an auxiliary function for the one-criterion problems (3), as follows: while a sufficiently smooth penalty function ( ) , P s τ satisfies conditions (6) and (7). As shown in Zhadan (2014) Since the condition of the second-level problem (4) includes the dependencies which are not differentiable functions for all their arguments, then for these dependencies it is also necessary to choose a smoothed approximation.
As an approximation, the auxiliary function calculated at a stationary point τ can be used, because (due to the properties of the penalty function method) its value for small positive τ is close to the optimal value of the objective function of the k -th problem (3).
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 ( ) k F u with respect to the components of the vector u of parameters. As , then according to the rule for differentiating a composite functionof several variables, we have: Note that the last simplification would be impossible if for Let us now look into the solution to the second-level problem. To make application of the penalty function method more convenient, the problem (4) is expressed as: x u . Let us define the auxiliary function for the problem (10) as follows: For the set of variables { } 1 2 , , , , n x x x ρ … , the conditions for the stationarity of the auxiliary function (12) will be: Let the solutions of system (13) From (13) Finally, we obtain formulas for the gradient components of ( ) E u in terms of the functions usedin the formulation of the multicriterial model (4)-(5) and the method of smooth penalty functions. From (12) it is obtained: can be found from (10).
Formulas (14) allow us to solve the third-level problem by applying any of the first-order methods, for example, conjugate directions. Note that second-order methods should also be considered here. However, this will be done at the end of the article, while now let us illustrate an example.

Proposed method in use
Let us consider multi-criterial mathematical model in which  is a vector of independent variables and  is a vector of parameters. The problem is to maximize for x and u ∈Θ the functions: , , a u u a u u a u u and b u u are given by the condition below.
A valid region of the model (with an allowable fixed u ) is a rectangular pyramid OABC. The Pareto set coincides with the face of ABC or is a part of it.
 . We assume that the set Θ in the parameter space is given by the condition that the sum of the lengths of the segments OA, OB and OC is constant and equal 3.

Optimization of the Shape of the Pareto Set in the Problems of Multi-criterial Programming
Applying the standard methods of analytic geometry, we find that for the compatibility of the system of model constraints, the existence of 0 r ≥ is necessary such that: The minimum value of the discrepancy between the criteria in this example depends on the form of the Pareto set, which is the triangle ABC, or a part of it. A graphic representation of the dependence of the error value of the objective functions on the parameters 1 u and 2 u is shown in pictures 2 and 3. Let us see with more details the properties of this dependence.
Clearly, the solutions of the first-level problems (3) for fixed 1 u and 2 u are: Consequently, the task of the second level (4) -minimizing the discrepancy of the criteria, will have the form: , u u ρ are primarily determined by how the set of constraints of a model of the «inequality» type is divided into active and inactive ones, that is, the first of which are satisfied as equalities, and the second -as strict inequalities.
This separation depends on the values of the parameters of the model and its optimal variant determines the solution of the second-level problem.
First, suppose that the values of the model parameters initiate a conflict of all three criteria simultaneously. In other words, the improvement of the value of any one of the objective functions of the model is possible only if the values of all the others deteriorate.
In this case, the last five constraints of the second-level problem must be active, and we obtain the following system of equations, which allows us to find the analytical form of the dependence ( ) ** 1 2 , u u ρ . − −       and 3 3 T   can be easily found, and for the first point the function has a local maximum with the value 2/3 according to the Sylvester criterion, not for the others because they don't satisfy the condition of non-negativity of the variables 1 2 3 , , x x x and r . The formula obtained is valid only in a certain area contained in Θ . An analysis of the isoline system shown in Picture 3, allows selecting five areas with different sets of active restrictions. Light lines determine the boundaries between the areas. The formula obtained above is valid only in area 4. In this area, the Pareto set of the model under consideration consists of the interior points of the triangle ABC.
Outside area 4, the formula for There are not stationary points for this dependency. For areas 2 and 3, the arguments and results are similar. The Pareto sets in areas 1, 2, and 3 are the sides of the triangle ABC: BC, AC, and AB, respectively. Finally, we note that in area 5 the system of conditions (2) is contradictory.
In this case, the exact solution to the problem of the upper (third) level has the form: Let us now describe the method for solving the third-level problem for the variant of the multicriterial model. We have: objective functions: maximize by Let us introduce the notations: and as a penalty function, we take ( ) Then, the auxiliary functions for the one-criterion problems (8) will be: The stationarity conditions of the auxiliary functions by the components of x give the following equations: , , 1,2,3 1,2 Let us now consider the problem (11) -optimization of the mismatch of the model criteria. In our case, this problem has the following form: minimize ρ according to the set of variables{ In addition, the following inequalities must be satisfied: This problem can also be solved by using the method of smooth penalty functions with the same ( ) , P s τ , for which it will be convenient to introduce (in addition to the previously defined) the notation 1, 2, 3 In this case, the auxiliary function (12) will be: The conditions of stationarity for the auxiliary function (13) in (18) take the form of a system of equations: The solution of the system (19) is denoted by ( ) u ρ and ( ) x u , and the function below is used as the smooth approximation of the dependency ( ) The derivatives of this function with respect to the components of the parameter vector u are found from formulas (14)  Finally, it must be mentioned that to calculate the derivatives (21) it is also necessary to know the values of the second derivatives of the functions ( ) k F u . These values can be found (similar to the one used above) by the method from formulas (10) and conditions (9) -the stationarity of the functions ( ) , , k A x u τ .