Sufficient conditions for the uniqueness of the maxima of the optimization problem in the framework of a stochastic model with priorities depending on one random variable

. An objective function arising in the optimization problem of a quasilinear complex system with dependent priorities is considered. In the case of three priorities based on the results of one experiment, sufficient conditions are obtained for all stationary points of the objective function under consideration to be local maximum points.


Introduction
In papers [1][2][3] the problem of optimization of interaction within the framework of a unified system of a number of institutions and an "optimizer" interested in the successful functioning of the system and acting on the basis of expert assessments was considered.Expert assessments should be implemented in the choice of priorities (generally random and taking values between zero and one) with different distributions.Under certain natural conditions, a theorem on the existence and uniqueness of the local (and global) maximum of the objective function of a certain complex system consisting of functions of a quasilinear type and independent random priorities was proved.These maxima are functions of a finite number of the positive parameters.In the present work, the problem of minimizing these maxima in the domain of the parameters` definition is solved.In particular, an exact solution of the minimax problem for constant priorities is given.The theorem was proved that if the function of maximum arising in the optimization problem of a quasilinear complex system with independent priorities has a stationary point with respect to natural parameters, then this point is a local minimum point.This point is unique.It implements minimax of the objective function of the arbiter, whose task is to optimize the relationship of a number of the institutions with help of the expert assessments.In practice, expert judgments should enable a design of the priority's distribution functions.And the arbiter sets up the mathematical model of the described complex system using the components of the obtained minimax point [4][5][6][7][8][9][10].In this work, we began to investigate the described model, when the priorities are dependent random variables.This task turned out to be much more difficult.In the case of three priorities based on the results of one experiment, sufficient conditions are obtained for all stationary points of the objective function under consideration to be local maximum points..

Formulation of the problem
Let   ,, F P  be a probability space.The mentioned problem of optimizing the relationship between the three institutions and the "arbiter" is based on the investigation of the maximum points of the objective function where 1 2 3 ,,    are random variables (priorities) taking values on the interval (0,1) ; E is the expectation with respect to the probability P ; is continuous on D , continuously differentiable and strictly positive on 0 D , and is equal to zero on the frontier D  of this domain.It follows from the well-known theorem of analysis that this function has at least one local maximum point, which is simultaneously a global maximum point.This work is devoted to the study of stationary points of function ( 2), provided that the priorities 1 2 3 ,,    are determined by one experiment associated with some uniformly distributed random variable  .So, we suppose that 11 ()   where 1 2 3 ,, f f f are Borel functions (0,1) , taking values on the same interval (0,1) .By the change of variable theorem under the Lebesgue integral, we obtain: In this paper, a number of conditions on functions 1 2 3 ,, f f f are obtained under which all stationary points of function ( 3) are points of local maximum.

Main results and their proofs
Introduce the notation: Introduce also the following notations: .
is a stationary point of a function F if and only if the following equalities hold: Proof.The assertion of Lemma 1 follows immediately from the equalities:

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Let us calculate the partial derivatives of the second order of the function F .We have: be a stationary point of function F . Then equalities (4) hold.Substituting them in (5), we obtain: Let us calculate at the stationary point the determinant After a series of transformations, we get: .In all further statements, the fulfillment of the inequality 0  will be verified, from which it follows that any stationary point of the function F is a local maximum point.In this case, it will be taken into account that in formula (6) all terms in the first three rows are strictly positive, and in the fourth row are strictly negative.Proposition 1.Let us consider numbers Proof.We will ensure the inequality It is clear that for this it is sufficient to provide the inequalities 0,1 0,9 , 1, 2,3. 1, Obviously, this is true if we take 0,05min i a   .From (7) and the inequalities for i f it follows that the system of inequalities is satisfied and hence the system The required result follows from (6).
Proposition 4. Let one of the functions , 1,

  
The conclusion of this proposition follows from the last inequality and formula (6). Corollary By the Bunyakovsky-Schwartz inequality, we have The inequalities (1 ) (1 ) ( 1) Applying formula (6) we obtain the required inequality.

Conclusion
The authors are confident that the inequality 0  can be proved for arbitrary

Fig. 1 .
Fig. 1.All possible points of the local maximum of function (1) lie in the interior of the closed triangle.

1
positive parameters.Thus, all possible points of the local maximum of function (1) lie in the interior of the closed triangle shown in Figure implies the assertion of the lemma.
this has not been possible.The next step in the investigation should be the solution of the minimax problem, as it was done in[2][3] with independent priorities.
(8)of.Both points are proved by checking the fulfillment of inequalities(8)and further reasoning, as in Proposition 1.  2. Let one of the functions , 1,2,3