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Unformatted text preview: 5 The tableau Solving simple linear programming problems graphically suggests the impor tance of extreme points of the feasible region. In the last section, we related the geometric idea of an extreme point to the algebraic idea of a basic feasi ble solution. Our aim now is to capitalize on this algebraic idea to design a solution algorithm for linear programming problems. Consider a linear programming problem in standard equality form: maximize c T x subject to Ax = b x . By definition, a basis B for the matrix A is a list of indices such that the matrix of corresponding columns A B is invertible. We know from linear alge bra that, since A B is invertible, some sequence of elementary row operations (adding multiples of rows to other rows and interchanging rows) reduces A B to the identity matrix. Suppose we apply exactly that sequence of row oper ations to the system Ax = b . As we saw before, we can rewrite the original system (5.1) A B x B + A N x N = b, where the list N consists of the nonbasic indices (those indices not in B ). The vectors x B and x N have components respectively x i for i B (the basic variables) and x j for j N (the nonbasic variables). Applying our sequence of row operations, we must arrive at an equivalent system, having the form (5.2) Ix B + Ax N = b, or in other words x i + X j N a ij x j = b i ( i B ) . This form of the system of constraints is particularly wellsuited for un derstanding the basis B . Given any values of the nonbasic variables x j (for j N ), we can easily read off the corresponding values of the basic variables x i (for i B ). In particular, if we set the nonbasic variables to zero, we obtain the corresponding basic solution x B = b and x N = 0. This basic solu tion is feasible exactly when b 0. Notice that each basic variable appears exactly once in the system....
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This note was uploaded on 10/31/2010 for the course ORIE 5300 at Cornell University (Engineering School).
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