linear programming, solution of a mathematical problem concerning maximum and minimum values of a first-degree (linear) algebraic expression, with variables subject to certain stated conditions (restraints). For example, the problem might be to find the minimum value of the expression x + y subject to the restraints x ≥0, y ≥0, 2 x + y ≥12, 5 x +8 y ≥74, and x +6 y ≥24. The solution was set forth by the Russian mathematician L. V. Kantorovich in 1939 and was developed independently by the American George B. Dantzig, whose first work on the subject appeared in 1947. A faster, but more complex technique, that is suitable for problems with hundreds or thousands of variables, was developed by Bell Laboratories mathematician Naranda Karmarkar in 1983. Linear programming is particularly important in military and industrial planning.