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Outline. relationship among topics secrets LP with upper bounds by Simplex method basic feasible solution ( BFS ) by Simplex method for bounded variables extended basic feasible solution ( EBFS ) optimality conditions for bounded variables ideas of the proof examples

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  1. Outline • relationship among topics • secrets • LP with upper bounds • by Simplex method • basic feasible solution (BFS) • by Simplex method for bounded variables • extended basic feasible solution (EBFS) • optimality conditions for bounded variables • ideas of the proof • examples • Example 1 for ideas but inexact • Example 2 for the exact procedure 1

  2. A Depot for Multiple Products • multi-product by a fleet of trucks Possible Formulation: objective function common constraints, e.g., trucks, DC capacity, etc. network constraints for type-1 product network constraints for type-1 product depot .... network constraints for type-1 product non-negativity constraints 2

  3. A General Type of Optimization Problems • structure of many problems: • network constraints: easy • other constraints: hard • making use of the easy constraints to solve the problems • solution methods: large-scale optimization • column generation, Lagrangian relaxation, Dantzig-Wolfe decomposition … • basis: linear programming, network optimization (and also non-linear optimization, integer optimization, combinatorial optimization) objective function network constraints hard constraints non-negativity constraints 3

  4. Relationship of Solution Techniques linear prog. • two directions of theoretical development for network programming • from special structures of networks • from linear programming • ideal: understanding development in both directions network prog. int. prog. non-linear prog. dynamic prog. … 4

  5. Relationship of Solution Techniques minimum cost flow column generation, Dantzig-Wolfe decomposition network algorithms network simplex revised simplex method shortest-path algorithms simplex method Lagrangian relaxation linear algebra non-linear optimization 5

  6. Our Topics • simplex method for bounded variables • linkage between LP and network simplex • optimality conditions for minimum cost flow networks • minimum cost algorithms • standard, and successive shortest path • equivalence among network and LP optimality conditions • revised simplex • column generation • Dantzig-Wolfe decomposition • Lagrangian relaxation It takes more than one semester to cover these topics in detail! We will only cover the ideas. 6

  7. Secrets 7

  8. The Most Beautiful … 8

  9. Maybe the Most Beautiful of All… • linear algebra geometric properties algebraic properties matrix properties 9

  10. LP with Upper Bounds 10

  11. LP with Upper Bounds • upper bounds: common in network problems, e.g., an arc with finite capacity • quite some theory of network optimization being from LP 11

  12. To Solve LP with Upper Bounds • incorporate the upper-bound constraints into the set of functional constraints and solve accordingly  12

  13. To Solve LP with Upper Bounds • In the simplex method the lower bound constraints 0  x do not appear in A. • Is it possible to work only with A even with upper-bound constraints? • Yes.  13

  14. BFS for Standard LP • Amn, m  n, of rank m • basic feasible solution (BFS)x of LP, i.e., • feasible: Ax  b, 0 x • basic • non-basic variables: (at least) n-m variables = 0 • basic variables: m non-negative variables with linearly independent columns 14

  15. Extended Basic Feasible Solution of LP with Bounded Variables • Amn, m  n, of rank m • extended basic feasible solution ( EBFS ) x of LP with bounded variables, i.e., • feasible: Ax  b, 0 x u • basic solution • non-basic variables: (at least) n-m variables = 0, or = their upper bounds • Basic variables: m variables of the form 0  xi  ui, with linearly independent columns 15

  16. Optimality Conditions of Standard LP • Maximum Conditions: BFSx is maximal if •  0 for all non-basic variable xj = 0 • MinimumConditions: BFSx is minimal if •  0 for all non-basic variable xj = 0 • intuition • : increase of the objective function by unit increase in xj • maximum condition: no good to increase non-basic xj • minimum condition: no good to decrease non-basic xj 16

  17. Optimality Conditions of LP with Bounded Variables • Maximum Conditions: EBFSx is maximal if •  0 for all non-basic variable xj = 0, and •  0 for all non-basic variable xj = uj • MinimumConditions: EBFSx is minimal if •  0 for all non-basic variable xj = 0, and •  0 for all non-basic variable xj = uj 17

  18. How to Prove? 18

  19. General Idea • optimality conditions of the EBFS • from duality theory and complementary slackness conditions 19

  20. Complementary Slackness Conditions • primal-dual pair • Theorem 1 (Complementary Slackness Conditions) • if x primal feasible and ydual feasible • then xprimal optimal and ydual optimal iffxj(yTAjcj) = 0 for all j, and yi(biAix) = 0 for all i 20

  21. Complementary Slackness Conditions • primal-dual pair • Theorem 2 (Necessary and Sufficient Condition) • if x primal feasible • then x primal optimal iff there exists dual feasible y such that x and y satisfy the Complementary Slackness Conditions 21

  22. Complementary Slackness Conditions for LP with Bounded Variables • by Theorem 2, primal feasible x and dual feasible (yT, T) are optimal iff • xj(yTAj+ j - cj­) = 0, j • yi(bi- Aix) = 0, i • j(uj - xj­) = 0, j 22

  23. General Idea of the Proof • optimality conditions of the EBFS • from duality theory and complementary slackness conditions • ideas of the proof • given an EBFSx satisfying the upper-bound optimality conditions • then possible to find dual feasible variables (yT, T)Tsuch that x and (yT, T)Tsatisfy the complementary slackness conditions 23

  24. Example 1. Upper-Bound Constraints as Functional Constraints • max 2x + 5y,  min 2x 5y, • s.t. • x + 2y 20, • 2x + y 16, • 0 x 2, 0 y 8. 24

  25. Examples of LP with Bounded Variables 25

  26. Example 1. Upper-Bound Constraints as Functional Constraints • min 2x 5y, • s.t. • x + 2y 20, • 2x + y 16, • 0 x 2, 0 y 8. • max. value = 44 • x* = 2 and y* = 8 26

  27. The following procedure is not exactly the Simplex Method for Bounded Variables. It primarily brings out the ideas of the exact method. 27

  28. Example 1. Upper-Bound Constraints by Optimality Conditions of Bounded Variables • y as the entering variable • 2y + s1 = 20 • y + s2 = 16 • y 8 -5 min 2x 5y, s.t. x + 2y 20, 2x + y 16, 0 x 2, 0 y 8. 28

  29. Example 1. Upper-Bound Constraints by Optimality Conditions of Bounded Variables • mark the non-basic variable y at its upper bound • for y = 8 • obj. fun.: -2x – 5y – z = 0  -2x - z = 40 • eqt. (1): x + 2y + s1 = 20  x + s1 = 4 • eqt. (2): 2x + y + s2 = 16  2x + s2 = 8 29

  30. Example 1. Upper-Bound Constraints by Optimality Conditions of Bounded Variables • x as the entering variable • x + s1 = 4 • 2x + s2 = 8 • x 2 min 2x 5y, s.t. x + 2y 20, 2x + y 16, 0 x 2, 0 y 8. 30

  31. Example 1. Upper-Bound Constraints by Optimality Conditions of Bounded Variables • for x at its upper bound 2, mark x, and • obj. fun.: -2x – z = 40  -z = 44 • eqt. (1): x + s1 = 4  s1 = 2 • eqt. (2): 2x + s2 = 8  s2 = 4 min 2x 5y, s.t. x + 2y 20, 2x + y 16, 0 x 2, 0 y 8. 31

  32. Example 1. Upper-Bound Constraints by Optimality Conditions of Bounded Variables • satisfying the optimality condition for bounded variables •  0 for all non-basic variable xj = 0, and •  0 for all non-basic variable xj = uj • z* = -44, with x* = 2 and y* = 8 32

  33. Example 1 Being Too Specific • in general, variables swapping among all sorts of status • non-basic at 0 • basic at 0 • basic between 0 and upper bound • basic at upper bound • non-basic at upper bound • Simplex method for bounded variables: a special algorithm to record all possibilities 33

  34. The following example follows the exact procedure of the Simplex Method for Bounded Variables. 34

  35. Example 2 • max 3x1 + 5x2 + 2x3 min 3x1 5x2 2x3, • s.t. • x1 + x2 + 2x3 7, • 2x1 + 4x2 + 3x3 15, • 0 x1 4, 0 x2 3, 0 x3 3. 35

  36. Example 2 by Simplex Method for Bounded Variables • potential entering variable: x2 • bounded by upper bound 3 • define = u2-x2 = 3-x2 min 3x1 5x2 2x3, s.t. x1 + x2 + 2x3 7, 2x1 + 4x2 + 3x3 15, 0 x1 4, 0 x2 3, 0 x3 3. 36

  37. Example 2 by Simplex Method for Bounded Variables 37

  38. Example 2 by Simplex Method for Bounded Variables • x1 as the (potential) entering variable • s2 as the leaving variable • a pivot operation as in standard Simplex Method min 3x1 5x2 2x3, s.t. x1 + x2 + 2x3 7, 2x1 + 4x2 + 3x3 15, 0 x1 4, 0 x2 3, 0 x3 3. 38

  39. Example 2 by Simplex Method for Bounded Variables • which can be an entering variable? • can s1 be a leaving variable? Yes • can x1 be a leaving variable? Yes min 3x1 5x2 2x3, s.t. x1 + x2 + 2x3 7, 2x1 + 4x2 + 3x3 15, 0 x1 4, 0 x2 3, 0 x3 3. 39

  40. Example 2 by Simplex Method for Bounded Variables • when = 1.25, x1 reaches its upper bound 4 • replace x1 by and is a basic variable = 0 • result min 3x1 5x2 2x3, s.t. x1 + x2 + 2x3 7, 2x1 + 4x2 + 3x3 15, 0 x1 4, 0 x2 3, 0 x3 3. 40

  41. Example 2 by Simplex Method for Bounded Variables • . • a “normal” pivot operation with aij < 0 min 3x1 5x2 2x3, s.t. x1 + x2 + 2x3 7, 2x1 + 4x2 + 3x3 15, 0 x1 4, 0 x2 3, 0 x3 3. 41

  42. Example 2 by Simplex Method for Bounded Variables • minimum • z* = -20.75, x1* = 4, x2* = 1.75, x3* = 0 min 3x1 5x2 2x3, s.t. x1 + x2 + 2x3 7, 2x1 + 4x2 + 3x3 15, 0 x1 4, 0 x2 3, 0 x3 3. 42

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