Linear Programming! References! Introduction to Algorithms.! Dasgupta, Papadimitriou, Vazirani. Algorithms.! Cormen, Leiserson, Rivest, and Stein.
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1 Liear Programmig! Refereces! Dasgupta, Papadimitriou, Vazirai. Algorithms.! Corme, Leiserso, Rivest, ad Stei. Itroductio to Algorithms.!
2 Slack form! For each costrait i, defie a oegative slack variable x +i :! x +i = 0 costrait i is tight.! At all times, express costraits as a set of equalities where the variables i RHS are 0 (tight), while those o LHS are ot.! Tight variables defie a vertex.! maximize subject to j = 1 j = 1 c x j j a x b for i = 1,, m ij j i x 0 for j = 1,, j z = 1 j = 1 c x x = b a x for i = 1,, m + i i ij j j = 1 j j ( x,, x + 0) m 2!
3 Slack form! max 2x + 5x 2x x 4 x + 2x 9 x + x x, x 0 3 z = 2x + 5x x = 4 2x + x 3 x = 9 x 2x 4 x = 3 + x x 5 ( x, x, x, x, x 0) x, x, x are the slack variables !
4 Simplex: Steps 1 ad 2! z = 2x + 5x x = 4 2x + x 3 x = 9 x 2x 4 x = 3 + x x 5 Curret vertex: (x 1,x 2 ) = (0,0)! Objective value: 0! Move: Icrease x 2! Tightest costrait: #3! Leavig: x 5! Eterig: x 2! z = x 5x x = 7 x x x = 3 3x + 2x 1 5 x = 3 + x x Curret vertex: (x 1,x 2 ) = (0,3) Objective value: 15 Move: Icrease x 1 Tightest costrait: #2 Leavig: x 4 Eterig: x 1 4!
5 Simplex: Steps 2 ad 3! z = x 5x x = 7 x x x = 3 3x + 2x 1 5 x = 3 + x x z = 22 x x x = 6 + x x x = 1 x + x x = 4 x x Curret vertex: (x 1,x 2 ) = (0,3) Objective value: 15 Move: Icrease x 1 Tightest costrait: #2 Leavig: x 4 Eterig: x 1 Curret vertex: (x 1,x 2 ) = (1,4) Objective value: 22 Move: Noe: Optimal 5!
6 Dictioaries! At all times, simplex maitais a dictioary:! z = v ' + c ' x j N j x = b' a ' x for i B i i ij j j N Basic variables: The x j s o the LHS! No-basic variables: The other x j s.! Basic feasible solutio: Determied by settig o-basic variables to 0.! Correspods to a vertex of feasible regio! Pivotig: Movig from oe bfs to aother! Equivalet to movig from oe dictioary to aother! j 6!
7 A added bous! Origial LP:! Fial dictioary: Multipliers of slack variables x 3, x 4, x 5 i obj. fuctio are 0, -7/3, -1/3.! max 2x + 5x 2x x 4 x + 2x 9 x + x 3 x, x 0 z = 22 x x x = 6 + x x x = 1 x + x x = 4 x x Compute 0 (Ieq. 1) + 7/3 (Ieq. 2) + 1/3 (Ieq. 3):! 2x + 5x 22 Not a coicidece! 7!
8 The iitial feasible solutio! The origi is ot always feasible! E.g., add costrait x 1 x 2 1 to previous LP! Problem arises whe oe of the umbers i the RHS of a costrait is egative.! Idea: Solve a auxiliary LP! maximize subject to j = 1 j = 1 c x j j a x b for i = 1,, m ij j i x 0 for j = 1,, j maximize subject to j = 1 x 0 a x x b for i = 1,, m ij j 0 i x 0 for j = 0,, j 8!
9 The auxiliary LP! maximize x subject to j = 1 0 a x x b for i = 1,, m ij j 0 i x 0 for j = 0,, j Observatio. Auxiliary LP is always feasible:! Set x i = 0 for j = 1,...,, ad x 0 = mi i b i.! (Recall: Ca assume that oe of the b i s is egative.)! Fact. The origial LP is feasible if ad oly if optimum objective value for the auxiliary LP is 0.! 9!
10 Example! max 2x + 5x 2x x 4 x + 2x 9 x + x 3 x x 1 x, x 0 max x 2x x x x + 2x x 9 0 x + x x 3 0 x x x 1 0 x, x, x 0 0 z = x 0 x = 4 2x + x + x 3 0 x = 9 x 2x + x 4 0 x = 3 + x x + x 5 0 x = 1 + x + x + x 6 0 z = 1 + x + x x x = 5 3x + x x = 10 2x 3x + x 4 x = 4 2x + x x = 1 x x + x !
11 Uboudedess! Assume maximizatio.! What if objective fuctio ca be made arbitrarily large?! Fact. Simplex will discover this.! Reaso: At some poit, it will fid a o-basic x i s.t.! x i has a positive coefficiet i the objective fuctio! Noe of the costraits bids x i.! 11!
12 Degeeracy! Two successive pivots may ecouter solutios with same objective value.! Reaso: the same vertex may be defied by differet sets of costraits.! Dager: Cyclig! Blad s rule: Whe choosig eterig or leavig variable, always pick the oe with smallest idex.! Fact. Usig Blad s rule, Simplex will ever cycle.! 12!
13 Ruig time of simplex! Time per pivot is polyomial.! Number of basic feasible solutios is! m + I worst case, simplex may go through most of them.! Ru time of simplex is expoetial i worst case.! However, this rarely happes, ad i practice Simplex is fast! Note: LP is solvable i polyomial time (Khachiya 1979, Karmarkar 1984)! 13!
14 The Geometry of Liear Programmig! 14!
15 The Geometry of Simplex! 15!
16 The Geometry of Simplex i 3D! 16!
17 The Geometry of Simplex i 3D! (0, 0, 0)! $0!! (200, 0, 0)! $200!! (200, 200, 0)! $1400!! (200, 0, 200)! $2800!! (0, 300, 100)! $3100! 17!
18 Duality! Every liear maximizatio problem (the primal) has a dual miimizatio problem.! The primal ad the dual relate to each other i the same way as flows ad cuts:! The value of ay primal feasible solutio is a lower boud for the value of ay dual feasible solutio.! The value of ay dual feasible solutio is a upper boud for the value of ay primal feasible solutio.! I fact, the values of the primal ad dual optimum solutios are exactly the same.! 18!
19 Example! The optimum solutio is (x 1, x 2 ) = (100,300), with objective value 1900.! We verify that this is optimum by formulatig the dual l.p.! 19!
20 Costructig the dual! 1. Assig a multiplier to each iequality.! 2. Multiply ad add:! 3. Derive costraits o the multipliers:! 4. Write objective fuctio:! 20!
21 Summary! Primal Dual Ay feasible value of the dual is a upper boud o the primal.! If we fid a pair of primal ad dual feasible values that are equal, the they must both be optimal.! The primal solutio (x 1, x 2 ) = (100, 300) ad the dual solutio (y 1, y 2, y 3 ) = (0, 5, 1) have the same value, Therefore, each is optimal for its respective problem.! 21!
22 The geeral case! 22!
23 The Duality Theorem! If a liear program has a bouded optimum, the so does its dual, ad the two optimum values coicide.! 23!
24 A eve more geeral versio of the dual! 24!
25 Visualizig duality: Shortest s t A miimizatio problem.! Solve by buildig a physical model:! Each edge is a strig with legth = weight of edge.! Each ode is a kot.! Fid the shortest s t path by pullig s away from t util the gadget is taut.! This is a maximizatio problem:! stretch s ad t as far apart as possible, subject to the costrait that the edpoits of ay edge (u, v) are at distace at most w uv.! paths! 25!
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