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168 Dmitrii Lozovanuwhich determines the orthogonal projection Y of the set UY ⊆ R n+k on R k ; then we solvethe problem from section 2.2. The system (9) can be found by using method of elimination ofvariables u 1 , u 2 , . . . , u n from the system (7). The method of elimination of variables from thesystem of linear inequalities can be found in [5]. Note that in the final system (9) the numberof inequalities m ′ can be too big. Therefore such approach for solving our problem can beused only for a small class of problems.3. An algorithm for the establishment of the compatibility ofa parametrical problemWe propose an algorithm for the establishment of the compatibility of the system (6) for everyy satisfying (3). This algorithm works in the case when the sets of solutions of the consideredsystems are bounded. The case of the problem with unbounded sets of solutions can be easilyreduced to the bounded one.Step 1. Choose an arbitrary basic solution y 0 of the system (3). This solution correspondsto a solution of the system of linear equationsm∑b isjy j + b is0 = 0, s = 1, m. (10)j=1The matrix B = (b isj) of this system represents a submatrix of the matrix⎛⎞b 11 b 12 . . . b 1mb 21 b 22 . . . b 2m. . . . . . . . . . . .B ′ =b k1 b k2 . . . b km1 0 . . . 0⎜ 0 1 . . . 0⎟⎝ . . . . . . . . . . . . ⎠0 0 . . . 1and the vector b ′T = (b i1 , b i2 , . . . , b ik ) is a "subvector" of b ′T = (b 1 , b 2 , . . . , b k , 0, 0, . . . , 0). Thesystem of inequalitiesm∑b isjb j + b is0 ≤ 0, s = 1, m,j=1which corresponds to the system (10), determines in R m a cone originated in y 0 with thefollowing generating raysy s = y 0 + b s t, s = 1, m, t ≥ 0.Here b 1 , b 2 , . . . , b m represent directing vectors of respective rays originating in y 0 . Thesedirecting vectors correspond to columns of the matrix B −1 .Step 2.on subjectFor each s = 1, m, solve the following problem:⎧⎨⎩maximize tA 2 y ≤ b 2 ;y ≥ 0;y = y 0 + b s t, t ≥ 0