section5

# section5 - 5 The tableau Solving simple linear programs...

This preview shows pages 1–2. Sign up to view the full content.

5 The tableau Solving simple linear programs graphically suggests the importance of ex- treme points of the feasible region. In the last section, we related the ge- ometric idea of an extreme point to the algebraic idea of a basic feasible solution. Our aim now is to capitalize on this algebraic idea to design a solution algorithm for linear programs. Consider a linear programs in standard equality form: maximize c T x subject to Ax = b x 0 . By deﬁnition, a basis for the matrix A is a list of indices such that the matrix of corresponding columns A B is invertible. We know from linear algebra 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 operations to the system Ax = b . As we saw before, we can rewrite the original system A B x B + A N x N = b, (5.1) 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 Ix B + ¯ Ax N = ¯ b, (5.2) 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 well-suited 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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 5

section5 - 5 The tableau Solving simple linear programs...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online