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296.3:Algorithms in the Real World

296.3:Algorithms in the Real World. Linear and Integer Programming I Introduction Geometric Interpretation Simplex Method Dual Formulation. Linear and Integer Programming. Linear or Integer programming (one formulation) find x to minimize z = c T x cost or objective function

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296.3:Algorithms in the Real World

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  1. 296.3:Algorithms in the Real World Linear and Integer Programming I Introduction Geometric Interpretation Simplex Method Dual Formulation 296.3

  2. Linear and Integer Programming Linear or Integer programming (one formulation) find x to minimize z = cTx cost or objective function subject to Ax ≤ b inequalities x ≥ 0 c  Rn, b  Rm, A  Rn x m Linear programming: x  Rn (polynomial time) Integer programming: x  Zn (NP-complete) Extremely general framework, especially IP 296.3

  3. Related Optimization Problems Unconstrained optimization min{f(x) : x  Rn} Constrained optimization min{f(x) : ci(x) ≤ 0, i  I, cj(x) = 0, j  E} Quadratic programming min{1/2xTQx + cTx : aiTx ≤ bi, i  I, aiTx = bj, j  E} Zero-One programming min{cTx : Ax = b, x  {0,1}n, c  Rn, b  Rm} Mixed Integer Programming min{cTx : Ax ≤ b, x ≥ 0, xi Zn, i  I, xr Rn, r  R} 296.3

  4. How important is optimization? • 50+ packages available • 1300+ papers just on interior-point methods • 100+ books in the library • 10+ courses at most universities • 100s of companies • All major airlines, delivery companies, trucking companies, manufacturers, … make serious use of optimization. 296.3

  5. Linear+Integer Programming Outline Linear Programming • General formulation and geometric interpretation • Simplex method • Ellipsoid method • Interior point methods Integer Programming • Various reductions from NP hard problems • Linear programming approximations • Branch-and-bound + cutting-plane techniques • Case study from Delta Airlines 296.3

  6. Applications of Linear Programming • A substep in most integer and mixed-integer linear programming (MIP) methods • Selecting a mix: oil mixtures, portfolio selection • Distribution: how much of a commodity should be distributed to different locations. • Allocation: how much of a resource should be allocated to different tasks • Network Flows 296.3

  7. x1 x1’ x2 x1 x3 Linear Programming for Max-Flow 1 Create two variables per edge: Create one equality per vertex: x1 + x2 + x3’ = x1’ + x2’ + x3 and two inequalities per edge: x1≤ 3, x1’ ≤ 3 add edge x0 from out to in maximize x0 7 5 2 2 in out 7 3 3 6 296.3

  8. In Practice In the “real world” most problems involve at least some integral constraints. • Many resources are integral • Can be used to model yes/no decisions (0-1 variables) Therefore “1. A subset in integer or MIP programming” is the most common use in practice 296.3

  9. Algorithms for Linear Programming • Simplex (Dantzig 1947) • Ellipsoid (Kachian 1979) first algorithm known to be polynomial time • Interior Pointfirst practical polynomial-time algorithms • Projective method (Karmakar 1984) • Affine Method (Dikin 1967) • Log-Barrier Methods (Frisch 1977, Fiacco 1968, Gill et.al. 1986) Many of the interior point methods can be applied to nonlinear programs. Not known to be poly. time 296.3

  10. State of the art 1 million variables 10 million nonzeros No clear winner between Simplex and Interior Point • Depends on the problem • Interior point methods are subsuming more and more cases • All major packages supply both The truth: the sparse matrix routines, make or break both methods. The best packages are highly sophisticated. 296.3

  11. Comparisons, 1994 296.3

  12. Formulations There are many ways to formulate linear programs: • objective (or cost) functionmaximize cTx, orminimize cTx, orfind any feasible solution • (in)equalitiesAx ≤ b, orAx ≥ b, orAx = b, or any combination • nonnegative variablesx ≥ 0, or not Fortunately it is pretty easy to convert among forms 296.3

  13. Formulations The two most common formulations: Canonical form Standard form slack variables minimize cTx subject to Ax ≥ b x ≥ 0 minimize cTx subject to Ax = b x ≥ 0 e.g. y1 7x1 + 5x2 ≥ 7 x1, x2 ≥ 0 7x1 + 5x2 - y1 = 7 x1, x2, y1 ≥ 0 More on slack variables later. 296.3

  14. Geometric View of Canonical Form A polytope in n-dimensional space Each inequality corresponds to a half-space. The “feasible set” is the intersection of the half-spaces This corresponds to a polytope Polytopes are convex: if x,y is in the polytope, so is the line segment joining them. The optimal solution is at a vertex (i.e., a corner). 296.3

  15. Geometric View of Canonical Form x2 An intersection of 5 halfspaces x2≤ 10 FeasibleSet Direction of “Goodness” Corners Objective Function -2x1 – 3x2 x1 x1 – 2x2≤ 4 2x1 + x2≤ 18 296.3

  16. Optimum (max problem) is at a Vertex Holds even if the objective function is merely convex: f is convex if for all vectors x, y  S and β [0,1] f(βx+(1- β)y) ≤βf(x) + (1- β)f(y) f(y) f(x) βf(x) + (1- β)f(y) f(βx+(1- β)y) 296.3

  17. Optimum (max problem) is at a Vertex Vertices vi Every point q in P is a convex combination of vertices of P. There exist βi q = ∑ βivi P q Fix convex f, then f(q) = f(∑ βivi) ≤ ∑ βif(vi) ≤ maxi f(vi) If f is linear, then f(q) = f(∑ βivi) = ∑ βif(vi) so mini f(vi) ≤ f(q) ≤ maxi f(vi) 296.3

  18. Geometric View of Canonical Form A polytope in n-dimensional space Each inequality corresponds to a half-space. The “feasible set” is the intersection of the half-spaces. This corresponds to a polytope The optimal solution is at a corner. Simplex moves around on the surface of the polytope Interior-Point methods move within the polytope 296.3

  19. Notes about higher dimensions For n dimensions and no degeneracy (i.e., A has full row rank) Each corner (extreme point) consists of: • n intersecting (n-1)-dimensional hyperplanese.g. n = 3, 2d planes in 3d • n intersecting edges Each edge corresponds to moving off of one hyperplane (still constrained by n-1 of them) Simplex will move from corner to corner along the edges 296.3

  20. The Simple Essense of Simplex Input: min f(x) = cx s.t. x in P = {x: Ax ≤ b, x ≥ 0} Polytope P Consider Polytope P from canonical form as a graph G = (V,E) with V = polytope vertices, E = polytope edges. • Find any vertex v of P. • While there exists a neighbor u of v in G with • f(u) < f(v), update v to u. • 3) Output v. Choice of neighbor if several u have f(u) < f(v)? Termination? Correctness? Running Time? 296.3

  21. 1 z ri = z  ei ei pi Optimality and Reduced Cost The Reduced cost for a hyperplane at a corner is the cost of moving one unit away from the plane along its corresponding edge. For minimization, if all reduced costs are non-negative, then we are at an optimal solution. Finding the most negative reduced cost is one often used heuristic for choosing an edge to leave on 296.3

  22. x2 z = -2x1 – 3x2 x1 Reduced cost example In the example the reduced cost of leaving the plane x1 is (-2,-3)  (2,1) = -7 since moving one unit off of x1 will move us (2,1) units along the edge. We take the dot product of this and the cost function. ei 1 x1 – 2x2≤ 4 ei = (2, 1) 296.3

  23. Simplex Algorithm • Find a corner of the feasible region • Repeat • For each of the n hyperplanes intersecting at the corner, calculate its reduced cost • If they are all non-negative, then done • Else, pick the most negative reduced costThis is called the entering plane • Move along corresponding edge (i.e. leave that hyperplane) until we reach the next corner (i.e. reach another hyperplane)The new plane is called the departing plane 296.3

  24. Step 2 x2 Departing z = -2x1 – 3x2 Step 1 x1 Entering Start Example 296.3

  25. Simplifying Problem: • The Ax ≤ b constraints not symmetric with the x ≥ 0 constraints.We would like more symmetry. Idea: • Make all inequalities of the form x ≥ 0. Use “slack variables” to do this. Convert into form: minimize cTx subject to Ax = b x ≥ 0 296.3

  26. x5 x1 x4 x3 x2 Standard Form Standard Form slack variables minimize cTx subject to Ax ≤ b x ≥ 0 minimize cTx’ subject to A’x’ = b x’ ≥ 0 |A| = m x n i.e. m inequalities, n variables |A’| = m x (m+n) i.e. m equations, m+n variables x2≤ 10 2x1 + x2 + x4 = 18 2x1 + x2≤ 18 x1 x1 – 2x2≤ 4 x2 296.3

  27. Example, again x2 x5 x1 x4 x3 x2 x1 The equality constraints impose a 2d plane embedded in 5d space, looking at the plane gives the figure above 296.3

  28. Using Matrices If before adding the slack variables A has size m x n then after it has size m x (n + m) m can be larger or smaller than n n m 1 0 0 … 0 1 0 … 0 0 1 … … A = m slack vrs. Assuming rows are independent, the solution space of Ax = b is an n-dimensional subspace on n+m variables. 296.3

  29. Gauss-Jordan Elimination Gauss-Jordan elimination 0 0 0 0 1 0 i j 296.3

  30. Simplex Algorithm, again • Find a corner of the feasible region • Repeat • For each of the n hyperplanes intersecting at the corner, calculate its reduced cost • If they are all non-negative, then done • Else, pick the most negative reduced costThis is called the entering plane • Move along corresponding line (i.e. leave that hyperplane) until we reach the next corner (i.e. reach another hyperplane)The new plane is called the departing plane 296.3

  31. Simplex Algorithm (Tableau Method) n This form is called a Basic Solution • the n “free” variables are set to 0 • the m “basic” variables are set to b’ A valid solution to Ax = b if reached using Gaussian Elimination Represents n intersecting hyperplanes If feasible (i.e., b’ ≥ 0), then the solution is called a Basic Feasible Solution and is a corner of the feasible set I F b’ m current cost 0 r -z reduced costs Basic Vars. Free Variables 296.3

  32. Corner basic variables free variables x5 x1 x4 x3 x2 free variables indicate corner 296.3

  33. Corner x5 x1 x4 x3 x2 Exchange free/basic variables by swapping columns, using GE to restore to tableau format. (“corner” not necessarily feasible) 296.3

  34. Corner x5 x1 x4 x3 x2 296.3

  35. Corner x5 x1 x4 x3 x2 296.3

  36. Corner x5 x1 x4 x3 x2 Note that in general there are n+m choose m corners 296.3

  37. Simplex Method Again Once you have found a basic feasible solution (a corner), we can move from corner to corner by swapping columns and eliminating. ALGORITHM • Find a basic feasible solution • Repeat • If r (reduced cost ) ≥ 0 , DONE • Else, pick column i with most negative r (nonbasic gradient heuristic) • Pick row j with least non-negative bj’/(j’th entry in column i) • Swap columns • Use Gaussian elimination to restore form 296.3

  38. Tableau Method • If r are all non-negative then done n I F b’ current cost 0 r z Basic Variables Free Variables values are 0 reduced costs if all ≥ 0 then done 296.3

  39. Tableau Method • Else, pick the most negative reduced costThis is called the entering plane n I F b’ 0 r z min{ri} entering variable 296.3

  40. Tableau Method • Move along corresponding line (i.e., leave that hyperplane) until we reach the next corner (i.e. reach another hyperplane)The new plane is called the departing plane u I F b’ min positive bj’/uj 1 0 r z departing variable 296.3

  41. Tableau Method • Swap columns • Gauss-Jordan elimination x b’ No longer in proper form x x x x r z swap I Fi+1 bi+1’ Back to proper form 0 ri+1 zi+1 296.3

  42. x5 x1 x4 x3 x2 Example Find corner x1 = x2 = 0 (start) 296.3

  43. Example 18 10 x5 x1 x4 x3 x2 -2 min positive 296.3

  44. 18 10 x5 x1 x4 x3 x2 -2 Example swap Gauss-Jordan Elimination 296.3

  45. 18 10 x5 x1 x4 x3 x2 -2 Example 296.3

  46. Example 18 swap 10 x5 x1 x4 x3 x2 -2 Gauss-Jordan Elimination 296.3

  47. Problem?: Unbounded Solution • What if all bj/uj are negative (i.e., all bj are positive and all uj negative)? • Then there is no bounded solution. • Can move an arbitrary distance from the current corner, reducing cost by an arbitrary amount. • Not really a problem. LP is solved. 296.3

  48. Problem: Cycling • If basic variable swapped in is already zero (b’j=0), don’t reduce total cost. • Can cycle back to already seen vertex! • Solution: Bland’s anticycling rule for tie breaking among columns & rows. 296.3

  49. Simplex Concluding remarks For dense matrices, takes O(n(n+m)) time per iteration Can take an exponential number of iterations. In practice, sparse methods are used for the iterations. 296.3

  50. Duality Primal (P): maximize z = cTx subject to Ax ≤ b x ≥ 0 (n equations, m variables) Dual (D): minimize z = yTb subject to ATy ≥ c y ≥ 0 (m equations, n variables) Duality Theorem: if x is feasible for P and y is feasible for D, then cTx ≤ yTb and at optimality cTx = yTb. 296.3

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