1 / 43

The Problem with Integer Programming

The Problem with Integer Programming. H.P.Williams London School of Economics. The Nature of Integer Programming (IP). Is IP like Linear Programming (LP) ? Applications of Integer Programming Mathematical Properties of IP Economic Properties of IP Chv á tal Functions and Integer Monoids.

moke
Download Presentation

The Problem with Integer Programming

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. The Problem with Integer Programming H.P.Williams London School of Economics

  2. The Nature of Integer Programming (IP) • Is IP like Linear Programming (LP) ? • Applications of Integer Programming • Mathematical Properties of IP • Economic Properties of IP • Chvátal Functions and Integer Monoids

  3. A General (Mixed) Integer Programme (IP) Maximise/Minimise ∑j cjxj+∑kdkyk Subject to:∑jaijxj +∑keikyk <=> bi for all i xj>=0 all j, yk>=0 all k and integer

  4. A General (Mixed) Integer Programme • Frequently (but not always) the integer variables are restricted to values 0 and 1 representing (indivisible) Yes/No decisions eg. Investment • Can view as a Logicalstatement about a series of Linear Programmes (LPs) • Leads a to close relationship between Logic and IP

  5. 0-1 Integer Programmes • Any IP with bounded integer variables can be converted to a 0-1 IP • 0-1 IPs can be interpreted as Disjunctions ofLPs • Application of logical methods to formulation and solution

  6. Applications of IP • Extensions to LPs eg Manufacturing, Distribution, Petroleum, Gas and Chemicals • Global Optimisation of non-convex (non-linear) models • Power Systems Loading • Facilities Location • Routing • Telecommunications • Medical Radiation • Statistical Design • Molecular Biology • Genome Sequencing • Archaeological Seriation • Optimal Logical Statements • Computer Design • Aircraft Scheduling • Crew Rostering

  7. Linear Programming v Integer Programming An LP Minimise X2 Subject to: 2X1+X2 >=13 5X1+2X2<=30 -X1+X2 >=5 X1 , X2 >= 0 The Solution • X1 = 22/3 , X2 = 72/3

  8. Linear Programming v Integer Programming An LP Minimise X2 Subject to: 2X1+X2 >=13 5X1+2X2<=30 -X1+X2 >=5 X1 , X2 >= 0 The Solution • X1 = 22/3 , X2 = 72/3 An IP Minimise X2 Subject to: 2X1 + X2 >=13 5X1 + 2X2<=30 -X1 + X2 >=5 X1 , X2 >= 0 and integer

  9. Linear Programming v Integer Programming An LP Minimise X2 Subject to: 2X1+X2 >=13 5X1+2X2<=30 -X1+X2 >=5 X1 , X2 >= 0 The Solution • X1 = 22/3 , X2 = 72/3 The Solution • X1 = 2 , X2 = 9 An IP Minimise X2 Subject to: 2X1 + X2 >=13 5X1 + 2X2<=30 -X1 + X2 >=5 X1 , X2 >= 0 and integer

  10. LP and IP Solutions 9 . . . . . Min x2 c3st 2x1+ x2 >= 13 8 . . c1 . . 5x1 + 2x2 <= 30 x2 -x1 + x2 >= 5 7 . . . . c2 . x1 , x2 >= 0 6 . . . . . 5 . . . . . 0 1 2 3 4 x1

  11. LP and IP Solutions 9 Optimal IP Solution (2 , 9).Min x2 c3st 2x1+ x2 >= 13 8 . . c1 . . 5x1 + 2x2 <= 30 Optimal LP Solution (2 2/3 , 7 2/3) -x1 + x2 >= 5 7 . . . . c2 . x1 , x2 >= 0 x2 6 . . . . . 5 . . . . . 0 1 2 3 4 x1 •

  12. IP Solution after removing constraint 1 Min x2 8 c1 . . . c3 st5x1 + 2x2 <= 30 -x1 + x2 >= 5 . . . . x1 , x2 >= 0 x27 . . . 6 . . . . . c2 Optimal IP Solution (0 , 5) 5 0 1 2 3 4 x1 •

  13. IP Solution 9 Optimal IP Solution (2 , 9).Min x2 c3st 2x1+ x2 >= 13 8 . . c1 . . 5x1 + 2x2 <= 30 Optimal LP Solution (2 2/3 , 7 2/3) -x1 + x2 >= 5 7 . . . . c2 . x1 , x2 >= 0 x2 6 . . . . . 5 . . . . . 0 1 2 3 4 x1 •

  14. IP Solution after removing constraint 2 9 . . . Min x2 c1 c3 st 2x1+ x2 >= 13 8 . . . .Optimal IP Solution (3, 8) -x1 + x2 >= 5 7 . . . . . x1 , x2 >= 0 x2 6 . . . . . 5 . . . . . x1 0 1 2 3 4 •

  15. IP Solution 9 Optimal IP Solution (2 , 9).Min x2 c3st 2x1+ x2 >= 13 8 . . c1 . . 5x1 + 2x2 <= 30 Optimal LP Solution (2 2/3 , 7 2/3) -x1 + x2 >= 5 7 . . . . c2 . x1 , x2 >= 0 x2 6 . . . . . 5 . . . . . x1 0 1 2 3 4 •

  16. IP Solution after removing constraint 3 9 . . . . Min x2 st 2x1+ x2 >= 13 8 . . . . . 5x1 + 2x2 <= 30 c1 c2 x1 , x2 >= 0 7 . . . . . x2 6 . . . . . 5 . . . . .Optimal IP Solution (4 , 5) 0 1 2 3 4 x1

  17. Rounding often not satisfactoryExample: The Alabama Paradox • State Population Fair solution Rounded Solution With 10 Representatives A 621k 4.41 4 B 587k 4.17 4 C 201k 1.43 2 With 11 Representatives A 621k 4.85 5 B 587k 4.58 5 C 201k 1.57 1

  18. IP Formulation of Political Apportionment Problem Vi = Population (Votes cast) for State (Party) i xi = Seats allotted to State (Party) i Choose xi so as to: Min Maxi (xi / vi) st ∑ixi = Total Number of Seats xi >= 0 and integer for all I ie Min y st xi / vi <= y for all i ∑i xi = Total Number of Seats xi >= 0 and integer for all I LP Relaxation gives fractional solution IP Solution give Jefferson/D’Hondt solution

  19. IP Solution • State Population Fractional Rounded Jefferson/ solution solution D’Hondt solution (LP) (IP) With 10 Representatives A 621k 4.41 4 5 B 587k 4.17 4 4 C 201k 1.43 2 1 With 11 Representatives A 621k 4.85 5 5 B 587k 4.58 5 5 C 201k 1.57 1 1

  20. Mathematical Differences between LP and IP • Consider a (Pure) IP in standard form Maximise c1x1+ c2x2 + … + cnxn Subject to: a11x1 + a12x2 + … a1nxn <= b1 a21x1 + a22x2 + … a2nxn <= b2 . . am1x1+ am2x2 + … amnxn <= bm x1 , x2 , … , xn >= 0 and integer

  21. Mathematical Differences between LP and IP • LP IP If has Optimal Solution No limit on number of positive variables there is one with at most m variables positive (a basic solution) Hilbert Basis (no fixed dimension) At most nconstraints At most 2n– 1 binding at optimum constraints binding at optimum There are valuations Chvátal Functions on constraints which close duality gap ie there is a (symmetric) LP No obvious symmetry (dual) model

  22. IPs involve Lattices within Polytopes Eg Max 2x1+x2 st 2x1+9x2<=80 2x1-3x2<=6 -x1 <=0 -x2<=0 2x1+3x2 ≡0(mod12) x1 ≡0(mod1) x2≡0(mod1)

  23. What are the strongest implications? Max 2x1+x2 st 2x1+9x2<=80 2x1+9x2<=80 2x1-3x2<=6 2x1-3x2<=6 -x1 <=0 -x1 <=0 2x1+x2 ? -x2<=0 -x2<=0 2x1+3x2 ≡0(mod12) 2x1+3x2 ≡0(mod12) x1 ≡0(mod1) x1 ≡0(mod1) 2x1+x2 ? x2≡0(mod1) x2 ≡0(mod1)

  24. What are the strongest implications? Dual arguments. Max 2x1+x2 st 2x1+9x2<=80 2x1+9x2<=80⅓ 2x1-3x2<=6 2x1-3x2<=6⅔ -x1 <=0 -x1 <=002x1+x2 <= 302/3 -x2<=0 -x2<=00 2x1+3x2 ≡0(mod12) 2x1+3x2 ≡0(mod12)3 x1 ≡0(mod1) x1 ≡0(mod1)02x1+x2 Ξ0(mod4) x2≡0(mod1) x2 ≡0(mod1)4

  25. IPs involve Lattices within Polytopes 8.. . ..Objective = 302/3 4.. . Objective = 28 . . . . . Objective = 24 0 6 12

  26. IPs involve Lattices within Polytopes Optimisation over polytopes give strongest (LP) bound on objective Optimisation over lattices give strongest congruence relation for objective Combined they give rank 1 cut for objective This may not be adequate

  27. Lattices within Cones give Integer Monoids These are a fundamental structure for IP

  28. Polyhedral and Non-Polyhedral Monoids The integer lattice within the polytope -2x + 7y >= 0 x – 3y >= 0 A Polyhedral Monoid y 4 . . . . . . . . . . . . . . . 3 . . . . . . . . . .. . . . . 2 . . . . . . .. . . . . . . . 1 . . . . . . . . . . . . . . . ……. 0. . . . . . . . . . . . . . . 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 x Projection: A non-polyhedral monoid (Generators 3 and 7) x .. x .. x x . x x. x x x…….

  29. Polyhedral and Non-Polyhedral Monoids 4 . . . . . . . . . . . . . . . 3 . . . . . . . . . .. . . . . 2 . . . . . . .. . . . . . . . 1 . . . . . . . . . . . . . . . ……. 0. . . . . . . . . . . . . . . 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 x Projection: A non-polyhedral monoid (Generators 3 and 7) x.. x .. x x. x x. x x x……. Reverse Head 11 10 9 8 7 6 5 4 3 2 1 0 . x x . x x .. x .. x

  30. Duality in LP and IP The Value Function of an LP Minimise x2 subject to: 2x1 + x2 >= b1 5x1 + 2x2 <= b2 -x1 + x2 >= b3 x1 , x2 >= 0 Value Function of LP is Max(5b1 -2b2 , 1/3( b1 + 2b3) , b3) If b1 = 13, b2 = 30, b3 = 5we have Max( 5, 72/3 , 5 ) = 72/3 , Consistency Tester is Max(2b1 – b2 , -b2 , -b2 + 2b3 ) <= 0 giving Max( -4, -30, -20) <= 0 . (5, -2, 0), (1/3, 0, 2/3), (0, 0, 1) are vertices of Dual Polytope . They give marginal rates of change (shadow prices) of optimal objective with respect to b1, b2, b3 . (5, -2,, 0), (1/3, 0, 2/3), (0, 0, 1) are extreme rays of Dual Polytope . What are the corresponding quantities for an IP ?

  31. LP Solution 9 . . . . Min x2 c3st 2x1+ x2 >= 13 8 . . c1 . . 5x1 + 2x2 <= 30 Optimal LP Solution (2 2/3 , 7 2/3) -x1 + x2 >= 5 7 . . . . c2 . x1 , x2 >= 0 x2 6 . . . . . 5 . . . . . 0 1 2 3 4 x1

  32. IP Solution 9 Optimal IP Solution (2 , 9). . . Min x2 c3st 2x1+ x2 >= 13 8 . . c1 . . 5x1 + 2x2 <= 30 Optimal LP Solution (2 2/3 , 7 2/3) -x1 + x2 >= 5 7 . . . . c2 . x1 , x2 >= 0 x2 6 . . . . . 5 . . . . . 0 1 2 3 4 x1

  33. Duality in LP and IP The Value Function of an IP Minimise x2 subject to: 2x1 + x2 >= b1 5x1 + 2x2 <= b2 -x1 + x2 >= b3 x1 , x2 >= 0 and integer Value Function of IP is Max( 5b1 -2b2 ,┌1/3( b1 + 2b3) ┐ , b3 , b1+ 2 ┌ 1/5 (-b2+ 2┌1/3(b1 + 2b3) ┐ ) ┐ ) This is known as a Gomory Function. The component expressions are known as Chvάtal Functions . Consistency Tester same as for LP (in this example)

  34. Gomory and Chvátal Functions Max( 5b1-2b2, ┌1/3(b1 + 2b3) ┐, b3 , b1+ 2 ┌1/5 (-b2+ 2┌1/3(b1 + 2b3) ┐ ) ┐ ) If b1=13, b2=30, b3=5 we have Max(5,8,5,9)=9 Chvátal Function b1+ 2 ┌1/5 (-b2+ 2┌1/3(b1 + 2b3) ┐ )┐determines the optimum. LP Relaxation is 19/15 b1 - 2/5 b2 +8/15 b2 (19/15, -2/5, 8/15) is an interior point of dual polytope but (5, -2, 0) and (1/3, 0, 2/3) are vertices of dual corresponding to possible LP optima (for different bi )

  35. Why are valuations on discrete resources of interest ? Allocation of Fixed Costs Maximise ∑j pi xi - f y stxi - Di y<= 0 for all I y ε {0,1} depending on whether facility built. f is fixed cost. xi is level of service provided to i (up to level Di ) pi is unit profit to i. A ‘dual value’ vion xi - Di y<= 0 would result in Maximise ∑j (pi – vi ) xi - (f – (∑j D i v i) y Ie an allocation of the fixed cost back to the ‘consumers’

  36. A Representation for Chvátal Functions b1 b3 -b2 1 2 Multiply and add on arcs 1 1 Divide and round up on nodes 2 2 Giving b1+ 2 ┌1/5( -b2+ 2┌1/3( b1 + 2b3) ┐ ) ┐ LP Relaxation is19/15 b1 - 2/5 b2 +8/15 b3 3 5 1

  37. Simplifications sometimes possible • ┌ 2/7 ┌7/3n┐┐ ≡ ┌2/3n┐ • But ┌ 7/3 ┌2/7n┐┐ ≠ ┌2/3n┐ eg n = 1 • ┌ 1/3 ┌ 5/6n┐┐ ≡ ┌5/18n┐ • But ┌ 2/3 ┌ 5/6n┐┐ ≠ ┌5/9n┐ eg n = 5 Is there a Normal Form ?

  38. Properties of Chvátal Functions • They involve non-negative linear combinations (with possibly negative coefficients on the arguments) and nested integer round-up. • They obey the triangle inequality. • They are shift-periodic ie value is increased in cyclic pattern with increases in value of arguments. • They take the place of inequalities to define non-polyhedral integer monoids.

  39. The Triangle Inequality • ┌a┐ + ┌b┐ >= ┌a + b┐ • Hence of value in defining Discrete Metrics

  40. A Shift Periodic Chvátal Function of one argument ┌ ½ ( x + 3 ┌ x /9 ┐ ) ┐ is (9, 6) Shift Periodic. 2/3is ‘long-run marginal value’ 14 13 12 11 10 9 8 7 6 5 4 3 2 1 . 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 --- x

  41. Polyhedral and Non-Polyhedral Monoids The integer lattice within the polytope -2x + 7y >= 0 x – 3y >= 0 A Polyhedral Monoid 4 . . . . . . . . . . . . . . . 3 . . . . . . . . . .. . . . . 2 . . . . . . .. . . . . . . . 1 . . . . . . . . . . . . . . . ……. 0. . . . . . . . . . . . . . . 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Projection: A Non-Polyhedral Monoid (Generators 3 and 7) x .. x .. x x . x x . x x x ……. Defined by ┌-x /3┐ +┌2x /7┐ < = 0

  42. Finally • We should be Optimising Chvátal Functions over Integer Monoids

  43. References • CE Blair and RG Jeroslow, The value function of an integerprogramme, Mathematical Programming 23(1982) 237-273. • V Chvátal, Edmonds polytopes and a hierarchy of combinatorialproblems, Discrete Mathematics 4(1973) 305-307. • D.Kirby and HP Williams, Representing integral monoids by inequalities Journal of Combinatorial Mathematicsand Combinatorial Computing 23 (1997) 87-95. • F Rhodes and HP Williams Discrete subadditive functions as Gomory functions, Mathematical Proceedings of the CambridgePhilosophicalSociety 117 (1995) 559-574. • HP Williams, A Duality Theorem for Linear Congruences, Discrete Applied Mathematics 7 (1984) 93-103. • HP Williams, Constructing the value function for an integer linear programme over a cone, Computational Optimisation andApplications 6 (1996) 15-26. • LA Wolsey, The b-hull of an integer programme, Discrete Applied Mathematics 3(1981) 193-201.

More Related