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Projection and inverse projection as a method of reformulating linear and integer programmes

Projection and inverse projection as a method of reformulating linear and integer programmes. H.P. Williams London School of Economics. h.p.williams @ lse.ac.uk www.lse.ac.uk/depts/op-research/personal/Williams. Example (Maximise z ). Subject to: . Project out x 1.

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Projection and inverse projection as a method of reformulating linear and integer programmes

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  1. Projection and inverse projection as a method of reformulating linear and integer programmes H.P. Williams London School of Economics h.p.williams @ lse.ac.uk www.lse.ac.uk/depts/op-research/personal/Williams

  2. Example(Maximise z) Subject to: Project out x1

  3. How to carry out Projection The following statements are equivalent: Immediate Proof Take (Decision Procedure of Langford for Theory of Dense Linear Order)

  4. We can take fi and g i as linear expressions in the other variables (apart from x) The constraints of any linear programme can be put in this form and x eliminated (projected out) But need to combine every inequality of form With every inequality of form Can lead to combinatorial explosion in number of inequalities. Equations and associated variables can be eliminated prior to this by Gaussian Elimination.

  5. Example(Maximise z) Subject to:

  6. Write in form

  7. Eliminate x1

  8. i.e. Have added (in suitable multiples) every inequality in which x1 has a positive coefficient to every inequality in which x1 has a negative coefficient

  9. C0 C1 C2 C3 C5 C4 4 1 1 4 1 1 1 1 1 1 Can continue process to eliminate x2and x3 This is Fourier-Motzkin.

  10. - - + + 0 0 C0 C1 C2 C3 C5 C4 0 - + - + - x2 - - - + + + - - + + - + - - x3 Can be shown (Kohler) that if, after n variables have been eliminated, an inequality depends on more than n+1 of original inequalities it is redundant.

  11. Can choose which variables to project out and order in which to do so.

  12. EXAMPLE OF PARTIAL PROJECTION Benders’ Decomposition Partially project out some variables (for MIP, continuous variables) to give bound on objective (Benders Cut) Potentially very large increase in constraints. Performed partially and iteratively.

  13. Application of Projection to Non-Exponential Formulations of the Travelling Salesman Problem Example Sequential Formulation (Miller, Tucker and Zemlin) Constraints and Variables

  14. Project out variables all directed cycles Gives A relaxation of Conventional Formulation all proper subsets

  15. Other non-exponential formulations give modifications of subtour elimination constraints of (exponential) conventional formulation Single Commodity Flow Formulation gives Modified Single Commodity Flow Formulation

  16. Multi Commodity Flow Formulation i.e. of equal strength to Conventional Flow Formulation Time Staged Formulation

  17. Minimise 2y1 + 3y2 subject to: -y1 + y2  4 y1 + y2  5 -y1 + 2y2  3 y1, y2  0 INVERSE PROJECTION Apply the dual procedures to eliminate constraints (as opposed to variables). -

  18. Write in form Minimise 2y1 + 3y2 subject to: 4y0 - y1 + y2 - y3 = 0 -5y0 + y1 + y2 - y4 = 0 -3y0 - y1 + 2y2 - y5 = 0 y0 = 1 y1, y2, y3,y4,y5 0  Eliminate homogeneous constraints by adding columns (in suitable multiples) where coefficients have opposite sign.

  19. y1 4u1 4y0 4u2 u3 y3 y2 u4 Implemented by a transformation of variables. First homogeneous constraint vanishes.

  20. Substitute 4y0 = 4u1 + 4u2 y1 = 4u1 + u3 y2 = u3 + u4 y3 = 4u2 + u4

  21. Minimise 8u1 + 5u3 + 3u4 subject to - u1 + 5u2 + 2u3 + u4 - y4 = 0 - 7u1 - 3u2 + u3 + 2u4 - y5 = 0 u1 + u2 = 1 u1, u2, u3, u4, y4, y5 0 Model becomes

  22. Minimise 43w1 + 5w2 + 38/3w3 + 15w4 + 3w5 w1 + w3 + w4 = 1 w1, w2, w3, w4, w5 0 Repeat with transformations to eliminate second and third homogeneous constraints. Results in

  23. w1 w2 w3 w4 w5 1 y0 1 0 0 1 0 0 y1 11 1 7/3 1 5 8/3 1 y2 7 13/3 9 1 y3 0 0 0 1 0 y4 13 2 0 7 2 y5 0 1 Applying corresponding transformations to identity matrix gives relation between final and original variables

  24. y1 y2 C: w1 = 1 gives (11 7) Objective = 43 w2 gives extreme ray ( 1 1) B: w3 = 1 gives (7/3 8/3) Objective = 38/3 A: w4 = 1 gives ( 0 5) Objective = 15 w5 gives extreme ray ( 0 1) Gives all vertices and extreme rays of model

  25. NB: Transformations mirror those of (Primal) Projection. Redundant transformations if variable depends on more than n+1 of original variables after n constraints eliminated.

  26. Example of Inverse Projection (On some constraints) Dantzig-Wolfe Decomposition . . .

  27. Project out subproblems into single (convexity) constraints 1111….1 1111…. . . . 1111….1

  28. Also Modal Formulations i.e. Variables represent extreme modes of processes rather than quantities NB Inverse Projection is not the same as Reverse Projection (not well defined).

  29. INTEGER PROJECTION The Elimination of Integer Variables a, bpositive integers x an integer variable f and g linear expressions Need to state condition “a multiple of ab lies between bfand–ag”

  30. Can be done in a finite way by Alternatively Presburger Arithmetic I.e. Arithmetic “without multiplication”)

  31. Projection of an IP may not produce an IP in lower dimension e.g. Project out x2

  32. But if coefficient of x is unity in one of inequalities can apply F-M elimination NB The projection of an IP generally does not result in an IP in a lower dimension

  33. INTEGER PROJECTION Example Minimize z i.e.

  34. Eliminate

  35. Eliminate x2(taking into account congruence relations)

  36. Optimal Solution Optimal Solution a Lattice Point within Polytope. Reduces problem from an infinite number to a finite number of solutions.

  37. C2 C3 C1

  38. Inverse Integer Projection Example Minimize z Subject to:

  39. Eliminate constraint 1 by a transformation of variables

  40. Leads to: Minimise Subject to: Eliminate constraints

  41. Leads to: Minimise Subject to: Optimal LP Solution. w4 = 3220 Objective = 4.5 N.B. Could replace congruence conditions by Mixed Integer Constraints. Feasible Solutions defined by Convexity Constraint + Congruence Conditions Illustrates Hilbert Basis result

  42. Alternatively we could scale variables and write in form Minimise Subject to: corresponds to the extreme ray

  43. Projection and Sign Patterns Can remove columns and all constraints in which non-negative entry. + - + - : : + or - 0 0 0 0 : : 0 0 Can remove constraints and all variables in which non-negative entry

  44. Eg Network Flow can remove node Can add constraints in suitable multiple and remove variable Integer Variable (and other variables integer) Can regard variable as continuous And generalisations of above

  45. REFERENCES • K.P. Martin Large Scale Linear and Integer Optimization : A Unified Approach, Kluwer 1999 • Porta Version 2, Free Software Foundation, Boston 1991 • H.P. Williams (1986) Fourier’s Method of Linear Programming and its Dual, Am.Math, Monthly, 93, 681-94 • H.P. Williams (1976) Fourier-Motzkin Extension to Integer Programming Problems, Journal ofCombinatorial Theory, 21, 118-123 • H.P. Williams (1983) A Characterisation of all Feasible Solutions to an Integer Programme, Discrete Applied Mathematics, 5, 147-155

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