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Parametrical approach for studying and solving bilinear programming problem 167The following Theorem holds [4].Theorem 3. The system (7) is compatible with respect to u 1 , u 2 , . . . , u n for every y 1 , y 2 , . . . , y k satisfying(8) if and only if the following system⎧r∑m∑− g is v i ≤ b is z i , s = 0, q;⎪⎨ i=1 i=1r∑m∑− g is v i = b is z i , s = q + 1, k;i=1 i=1⎪⎩v i ≥ 0, i = 1, ris compatible with respect to v 1 , v 2 , . . . , v r for every z 1 , z 2 , . . . , z m satisfying the following system⎧m∑− a ij z i ≤ 0, j = 1, p;⎪⎨ i=1m∑− a ij z i = 0, j = p + 1, n;i=1 ⎪⎩z i ≥ 0, i = 1, m.2.2 Two special cases of the parametrical problemNote that if r = 0 and q = k in the system (8) then we obtain the problem of determiningcompatibility of the system (7) for every nonnegative values of parameters y 1 , y 2 , . . . , y k . It iseasy to observe that in this case the system (7) is compatible for every nonnegative values ofparameters y 1 , y 2 , . . . , y k if and only if each of the following systems⎧ n∑ ⎪⎨a ij u j ≤ b is , s = 0, k;j=1 ⎪⎩u j ≥ 0, j = 1, pis compatible.An another special case of the problem is the one when n = 0. This case can be reduced tothe previous one using the duality problem for it.In such a way, our problem can be solved in polynomial time for the mentioned above cases.2.3 General approach for determining the compatibility property forparametrical systemsIt is easy to observe that the compatibility property of the system (7) for all admissible valuesof parameters y 1 , y 2 , . . . , y k satisfying (8) can be verified by checking compatibility of thesystem (7) for every basic solution of the system (8). This fact follows from the geometricalinterpretation of the problem. The set Y ⊆ R k of vectors y = (y 1 , y 2 , . . . , y k ), for which thesystem (7) is compatible, corresponds to an orthogonal projection on R k of the set UY ⊆ R n+kof solutions of the system (7) with respect to variables u 1 , u 2 , . . . , u n , y 1 , y 2 , . . . , y k .Another general approach which can be argued on the basis of the mentioned above geometricalinterpretation is the following.We find the system of linear inequalitiesr∑b ′ ij y j + b ′ i0 ≤ 0, i = 1, m ′ , (9)j=1