Linear Programming (for the Encyclopedia of Microcomputers)

Linear programming is one of the most successful disciplines within the eld of operations research. In its standard form, the linear programming problem calls for nding nonnegative x 1 ;. . .; x n so as to maximize a linear function P n j=1 c j x j subject to a system of linear equations: a 11 x 1 + + a 1n x n = b 1. .. a m1 x 1 + + a mn x n = b m : This problem can be stated in vector notation as Maximize c T x subject to Ax = b x 0 where A 2 R mn is assumed to have linearly independent rows, and b 2 R m and c;x 2 R n. In fact, any problem of maximizing or minimizing a linear function subject to linear equations and inequalities can be easily reduced to the standard form. The dual problem of the linear programming problem in standard form is Minimize b T y subject to A T y c : The former problem is then referred to as the primal. The duality theorem asserts that (i) for any x that satisses the constraints of the primal and for any y that satisses the conditions of the dual, c T x b T y, and (ii) if there exist such x and y, then the maximum of the primal equals the minimum of the dual. The duality theorem plays a central role in the theory of linear programming.