ADVANCED ALGORITHMS Lecture 20: Linear Programming 1 ANNOUNCEMENTS - PowerPoint PPT Presentation

ADVANCED ALGORITHMS Lecture 20: Linear Programming � 1

ANNOUNCEMENTS ➤ HW 4 is due on Monday, November 5 ➤ Project meetings … � 2

LAST CLASS ➤ Optimization ➤ {x 1 , x 2 , …, x n } are variables — values in some domain D ➤ find maximum value of f(x) subject to 
 g 1 ( x ) ≥ 0 g 2 ( x ) ≥ 0 …. ➤ Can phrase many natural problems as optimization — e.g. scheduling, matching, shortest paths, … � 3

WHEN CAN WE SOLVE OPTIMIZATION? domain is discrete optimization a solitude ➤ The bad news: ➤ all the formulations we wrote so far are intractable! 
 programming linear ➤ The good news: ➤ Continuous optimization with linear constraints, objective ➤ Convex optimization 
 usually Main challenge: can we express problem of interest as an optimization we can solve? � 4

Linear Programs In Xu EIR Vars x 1 2 g t Cz N C H min t Cnxn t to subject a Tn 3 b X t Akka t b A tansen gin 3 b n ain's.sn It Frivialobservations air b is a hyperplane a half space a.tn b is E

t Ry t 2 42 I N t X LAST CLASS (CONTD.) Every linear constrain ➤ Linear and convex optimization µ is a half space ➤ Visualizing linear optimization a zXzt A Tx b t 9 nXn3 A t X I � 5

LINEAR FUNCTIONS — “LEVEL SETS” Yotz not Go Yo f cT no yo kot2y y Elnothyoth Is x + 2y = 2 x + 2y = 1.5 I 2 x + 2y = 1 � 6

In in R Observation minimizing p R finding the furthest point in R in the direction c Ty d a Tn Eb Finder ain LEI solve for the n

OBSERVATIONS EEE ➤ Theorem: optimum value is always achieved at a “vertex” or a “corner 77 point” of the polytope ➤ Feasible region (and hence opt value) can be unbounded ffeanhi.Y.IE aD ➤ Saw approach, not algorithm! ➤ Brute-force algorithm: check all corners — not all corners are feasible! paenuaearnuiicheaEkh'bimff g ➤ Simplex algorithm (Dantzig 1949): systematic way to move from one corner point to a neighbor — local search Fia melon 7 � 7

exp time local search on set Algorithm SIMPLEX of corner points corner ft start with v one basic feasible solution that has a mom random pivot lowercturalmee no such neighbor stop if there is I return solution am x U l v a pti that has neighbor of is v In General a equations in common with je precisely Cn l Ain Eb one of air bi repaced o by another equation ant's bn

DOES “NO NEIGHBOR” => GLOBAL OPTIMUM? i.e., can we get stuck? ➤ Note: Fomenktopht the ➤ this is NOT the same as local opt = global opt! 
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MATH OF LINEAR PROGRAMS — NEIGHBORING DIRECTIONS iix b n different neighbors of we get aIn bz o V fit obtained by dropping each V Hygagusataint ain bio replacing fight ai'n bn Z M 1.5 o nffittia ai i jthentry Gen whatis AX ai the � 9

MATH OF LINEAR PROGRAMS — NEIGHBORING DIRECTIONS n'H fit v if I bi bi 0 ayy o a diagonal matrix is AX a Can think of A X I Ax are simply the columns the directions UH Moira A of � 10

CORRECTNESS OF THE SIMPLEX ALGORITHM o o v ctfu.jo ctu cTu c if theorem Iv I 3 j s't v then f Iu co For any point atR U'd't when dj can be expressed as U claim n dj 30 � 11

afs.bg f f ai um g ane di Uli because u U N form a basis for pi di Cafu g ajT g.tv ajtx 11 I I bj 1 bj because u were Y chosen tobe columns of the inverse dj 70

GENERAL RESULTS ➤ [Dantzig 1949]: simplex algorithm — known to be exp-time ➤ Khachiyan’s “ellipsoid” algorithm — 1979 1 methods interior point Karmarkow 843 � 12

MATCHING AS LINEAR PROGRAM � 13

MATCHING AS LINEAR PROGRAM � 14

MATCHING AS LINEAR PROGRAM � 15