1 / 21

Linear Programs with Totally Unimodular Matrices

Linear Programs with Totally Unimodular Matrices. updated 19 March 2009. Basic Feasible Solutions. Standard Form. Basic Feasible Solutions. Vector-Matrix Representation. Example MCNFP. -2. (3, 2,5). (4, 1,3). 2. 5. -3. 1. 4. (1, 0,2). (4, 0,3). (2, 0,2). 3. 0.

temima
Download Presentation

Linear Programs with Totally Unimodular Matrices

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Linear Programs with Totally Unimodular Matrices updated 19 March 2009

  2. Basic Feasible Solutions Standard Form

  3. Basic Feasible Solutions

  4. Vector-Matrix Representation

  5. Example MCNFP -2 (3, 2,5) (4, 1,3) 2 5 -3 1 4 (1, 0,2) (4, 0,3) (2, 0,2) 3 0

  6. LP for Example MCNFP Min 3x12 + 2 x13 + x23 + 4 x24 + 4 x34 s.t. x12 + x13 = 5 {Node 1} x23 + x24 – x12 = -2 {Node 2} x34 – x13 - x23 = 0 {Node 3} – x24 - x34 = -3 {Node 4} 2  x12 5, 0  x13 2, 0  x23 2, 1  x24 3, 0  x34 3,

  7. Matrix Representation of Flow Balance Constraints

  8. Solving for a Basic Feasible Solution

  9. Cramer’s Rule Use determinants to solve x=A-1b. Take the matrix A and replace column j with the vector b to form matrix Bj.

  10. Using Cramer’s Rule to Solve for x12

  11. Total Unimodularity • A square, integer matrix is unimodular if its determinant is 1 or -1. • An integer matrix A is called totally unimodular (TU) if every square, nonsingular submatrix of A is unimodular.

  12. Total Unimodularity • A square, integer matrix is unimodular if its determinant is 1 or -1. • An integer matrix A is called totally unimodular (TU) if every square, nonsingular submatrix of A is unimodular.

  13. Sufficient Conditions for TU An integer matrix A is TU if • All entries are -1, 0 or 1 • At most two non-zero entries appear in any column • The rows of A can be partitioned into two disjoint sets M1 and M2 such that • If a column has two entries of the same sign, their rows are in different sets. • If a column has two entries of different signs, their rows are in the same set.

  14. The Matrix of Flow Balance Constraints • Every column has exactly one +1 and exactly one -1. • Conditions 1 and 2 are satisfied. • Let the row partition be M1 = {all rows} and M2 = {}. • Condition 3 is satisfied. • Therefore, the flow balance constraint matrix is TU.

  15. Using Cramer’s Rule to Solve for x12

  16. Expansion by Minors: 4-by-4 Matrix

  17. Expansion by Minors: 3-by-3 Matrix

  18. Using Cramer’s Rule to Solve for x12

  19. Using Cramer’s Rule to Solve for x12 • When we expand along minors, the determinants of the submatrices will be +1, -1, or 0. • Therefore, the determinant will be an integer: (5)(+1, -1, or 0) + (-2) (+1, -1, or 0) + 0 + (-3) (+1, -1, or 0).

  20. Using Cramer’s Rule to Solve for x12

  21. TU Theorems • Matrix A is TU if and only if AT is TU. • Matrix A is TU if and only if [A, I] is TU. • I is the identity matrix. • If the constraint matrix for an IP is TU, then its LP relaxation has an integral optimal solution. • The BFSs of an MCNF LP are integer valued.

More Related