# Linear function synthesis algorithm and it’s applications.

We consider the ordinal approach to the completion of a linear function on the basis of consistent initial information, substantiate the functions synthesis algorithm and discuss the applications of this algorithm. In a number of decision-making problems, allowing the formalization in the form of linear models is certainly known that the objective function is linear, but the vector of its coeﬃcients is not installed. Partial information $I(D)$ of the function is deﬁned on a subset of feasible variables $D$ by the binary relation $\tilde x \succ \tilde y ⇔ f(\tilde x) > f(\tilde y),

\tilde x, \tilde y \subset D$ which is established by expert evaluation. This information is consistent if and only if this relation is a strict order.

Partially speciﬁed function is called linear allowing an approach is, if there is a vector $\tilde c ∈ \mathbb{R}^n$ such that $f(\tilde x) > f(\tilde y) ⇔ (\tilde c, \tilde x) > (\tilde c, \tilde y)$. A necessary and suﬃcient condition for a linear completion of the function, partly given by the order relation is formulated. Provides a way completions linear function based on linear correction procedures and function synthesis algorithm based on the ordinal initial information is formulated. The ambiguity of reconstruction and examples of software algorithms аre discuss.

Next, we consider a number of problems with the search for optimal solutions using the algorithm of synthesis of a linear function. One such problem is a weakly-deﬁned linear programming problem, which is completely determined by the set of feasible solutions, and the objective function is deﬁned in part by the order relation. The examples discuss ambiguity of reconstruction and additional information narrows the uncertainty.

Another area of application of the synthesis algorithm associated with the proposed approach to solving multiobjective optimization problem by convolution of criteria. If such a problem, there is a subset of feasible solutions, which can be the same order by all criteria, the vector of coeﬃcients of the convolution can be restored by a linear correction procedures.

Next, we propose an approach to the reconstruction of continuous nonlinear functions satisfying a monotonicity and convexity and partially deﬁned by the order relation on a ﬁnite subset of the domain of deﬁnition.

Finally, the problem of reconstruction of a linear function of collective utility in the problem of collective choice-making. If a subset of feasible solutions all individuals equally set our preferences, the coeﬃcient vector of collective utility function can be restored by the linear correction procedures.