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15-853:Algorithms in the Real World

15-853:Algorithms in the Real World. Linear and Integer Programming I Introduction Geometric Interpretation Simplex Method. Linear and Integer Programming. Linear or Integer programming minimize z = c T x cost or objective function

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15-853:Algorithms in the Real World

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  1. 15-853:Algorithms in the Real World Linear and Integer Programming I Introduction Geometric Interpretation Simplex Method 15-853

  2. Linear and Integer Programming • Linear or Integer programming • minimize z = cTx cost or objective function • subject to Ax = b equalities • x ¸ 0 inequalities • c 2 Rn, b 2 Rm, A 2 Rn x m • Linear programming: • x 2 Rn (polynomial time) • Integer programming: • x 2 Zn (NP-complete) • Extremely general framework, especially IP 15-853

  3. Related Optimization Problems • Unconstrained optimization min{f(x) : x 2 Rn} • Constrained optimization min{f(x) : ci(x) · 0, i 2 I, cj(x) = 0, j 2 E} • Quadratic programming min{1/2xTQx + cTx : aiTx · bi, i 2 I, aiTx = bj, j 2 E} • Zero-One programming min{cTx : Ax = b, x 2 {0,1}n, c 2 Rn, b 2 Rm} • Mixed Integer Programming min{cTx : Ax = b, x ¸ 0, xi2 Zn, i 2 I, xr2 Rn, r 2 R} 15-853

  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. 15-853

  5. Linear+Integer Programming Outline • Linear Programming • General formulation and geometric interpretation • Simplex method • Elipsoid method • Interior point methods • Integer Programming • Various reductions of NP hard problems • Linear programming approximations • Branch-and-bound + cutting-plane techniques • Case study from Delta Airlines 15-853

  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 15-853

  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 15-853

  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 15-853

  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-Barrirer 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 15-853

  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. 15-853

  11. Comparisons, 1994 15-853

  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 15-853

  13. Formulations • The two most common formulations: 1 2 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. 15-853

  14. Geometric View • 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 15-853

  15. Geometric View x2 • An intersection of 5 halfspaces x2· 10 FeasibleSet Corners Objective Function -2x1 – 3x2 x1 x1 – 2x2· 4 2x1 + x2· 18 15-853

  16. Notes about higher dimensions • For n dimensions and no degeneracy • Each corner (extreme point) consists of: • n intersecting n-1 dimensional hyperplanese.g. 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 15-853

  17. 1 z ri = z ¢ ei ei pi Optimality and Reduced Cost • The Optimal solution must include a corner. • 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 cost are non-negative, then we are at an optimal solution. Finding the most negative reduced cost is a heuristic for choosing an edge to leave on 15-853

  18. 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. (2,1) (0,1) 15-853

  19. 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 15-853

  20. Step 2 x2 Departing z = -2x1 – 3x2 Step 1 x1 Entering Start Example 15-853

  21. 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 15-853

  22. 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 15-853

  23. 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 a n dimensional subspace. 15-853

  24. Gauss-Jordan Elimination m A_{ij} l Gauss-Jordan elimination 0 0 0 0 1 0 i j 15-853

  25. 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 15-853

  26. Tableau Method • Find a corner of the feasible region A b z 0 Using Gauss-Jordan Elimination 1 0 0 0 1 0 0 0 1 I F b’ values of basic variablesb’ ¸ 0 m 0 r z Basic Variables Free Variables values are 0 15-853

  27. Tableau Method • Repeat • Find reduced costs • Quit if all ¸ 0 • Find entering plane (variable) • Find leaving plane (variable) and move along edge between them 15-853

  28. Tableau Method • For each of the n hyperplanes intersecting at the corner, calculate its reduced cost • If they 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 15-853

  29. 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 15-853

  30. 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 15-853

  31. Tableau Method • Continued x b’ No longer in proper form x x x x r z Gauss-Jordan Elimination swap I Fi+1 bi+1’ Back to proper form 0 ri+1 zi+1 15-853

  32. x5 x1 x4 x3 x2 Example Find corner x1 = x2 = 0 (start) 15-853

  33. Example 18 10 x5 x1 x4 x3 x2 -2 min positive 15-853

  34. 18 10 x5 x1 x4 x3 x2 -2 Example swap Gauss-Jordan Elimination 15-853

  35. 18 10 x5 x1 x4 x3 x2 -2 Example 15-853

  36. Example 18 swap 10 x5 x1 x4 x3 x2 -2 Gauss-Jordan Elimination 15-853

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