ADVANCED ALGORITHMS Lecture 19: optimization, linear programming � 1
ANNOUNCEMENTS ➤ HW 4 is due on Monday, November 5 lb mod 2 ➤ Project meetings … air n L � 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, … [[why?]] � 3
EXAMPLE: MATCHING � 4
EXAMPLE: SHORTEST PATHS ➤ What are variables? O't AE I ➤ What are constraints? V ➤ What is the objective? Variables: linear constrain { x e } e ∈ E j exponentially their many of Three different ways of writing constraints … (“matrix powers”, flow based, cut based) polynomial constraints Objective: minimize ∑ w e x e introducing directed edges e out degree I in deg � 5
EXAMPLE: SPANNING TREE WE G Pro I a is.to a aoe Ledges of min sub the so that 0 if total weight e is not selected graph is connected see l if e is selected min Exe To we if Obelix method we saw for enforcing a take any use such constraints As 1 ideas s t path between � 6
EXAMPLE: SPANNING TREE t n see idea f no cycles along with E 1st I E ne Hs e with bothendpb u.rs V stfo.BE He 7 m Cut e in the cut 15,5 � 7
ZOOMING OUT Argue about constraints ➤ What are variables? ➤ What are constraints? capturing the problem ➤ What is the objective? � 8
WHEN CAN WE SOLVE OPTIMIZATION? ➤ The bad news: ➤ all the formulations we wrote so far are intractable! Discrete optimization ni E 0113 D is generally hard � 9
WHEN CAN WE SOLVE OPTIMIZATION? ➤ The bad news: ➤ all the formulations we wrote so far are intractable! Linear programs ➤ The good news: i ➤ Continuous optimization with linear constraints, objective ➤ Convex optimization local search Main challenge: can we express problem of interest as an optimization we can solve? � 10
TODAY ➤ Linear and convex optimization ➤ Visualizing linear optimization ➤ Express problems as linear optimization ➤ What to do with a solution? � 11
LINEAR PROGRAMMING a fx xn EIR Variables I Ise 1 Cnn n t Gaz t min n c Objedive I b t Ain Xn Ann t Aizazt Constraints A Tn Sb I � 12
LINEAR PROGRAMMING — TWO DIMENSIONS na EIR q ni m.nu Tsnbiea to II half space Each constraint is sn Gen Overall set of feasible solutions is an intersection of half spaces � 13 I
t.io I.T t Xz L j 2 X X X 1 0 X 2 FE c a a a direction Objective function defines the optimum point is the'furthestpoint Qection in the fean.to
LINEAR PROGRAMMING — TWO DIMENSIONS � 14
OPTIMAL POINT ALWAYS A “CORNER” feasible set is the at Aaa'In I suppose mdd theanoptimalpoint e EIR Then for any F corner point of this feasible set I c Hand warytof � 15
ALGORITHM FOR LINEAR PROGRAMMING — 2 DIMENSIONS � 16
GENERAL ALGORITHM & RUN TIME � 17
IMPROVEMENT — ALWAYS MOVE TO “NEIGHBOR”? � 18
CAN WE GET STUCK? � 19
CONVEX OPTIMIZATION � 20
GENERAL RESULTS ➤ [Dantzig]: simplex algorithm ➤ Khachiyan’s “ellipsoid” algorithm � 21
MATCHING AS LINEAR PROGRAMS � 22
MATCHING AS LINEAR PROGRAMS � 23
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