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Constraint Optimization Presentation by Nathan Stender Chapter 13 of Constraint Processing by Rina Dechter 3/25/20131Constraint Optimization

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Motivation 3/25/20132Constraint Optimization Real-life problems often have both hard and soft constraints – Hard constraints must be satisfied – Soft constraints would like to be satisfied Goal is to – optimize soft constraints – while satisfying hard constraints Applications – Planning – Scheduling – Design

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Example: Combinatorial Auction Bidders place a bid on a set of items Hard constraints: no two bids sharing an item can be selected Soft constraints: cost of selecting each bid Auctioneer must select bids to maximize revenue 3/25/2013Constraint Optimization3

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Outline 3/25/2013Constraint Optimization4 Motivation Constraint Optimization and Cost Networks Section 13.1 Branch-and-Bound Search Section 13.2 Bucket Elimination for Optimization Section 13.3 Mini-bucket Elimination Section 13.4 Search with Mini-bucket Heuristics Section 13.5 Summary

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Constraint Optimization Problem COP = Constraint network + a cost function Global cost function – Real-valued functional components: F 1,…,F l – F 1,…,F l defined over scopes: Q 1,…,Q l, Q j ∈X – Assignment: ā = (a 1,…, a n ), a i in the domain of x i 3/25/2013Constraint Optimization5

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Example: Combinatorial Auction Boolean variables b i for bids – Each bid has an associated cost r i – b i = 1 if bid is selected, b i =v0 if bid is rejected Hard constraint R ij between b i and b j – If they share an item: ( b i =1,b i =1)∉R ij Soft constraint F(b i ) – F(b i )=r i if b i =1, F(b i )=0 if b i =0 Goal is to maximize: 3/25/2013Constraint Optimization6 R ij bibi bjbj cost riri 01rjrj

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Example: Combinatorial Auction Consider the following set of bids: 3/25/2013Constraint Optimization7 b1b1 b2b2 b3b3 b4b4 b5b5 Optimal solution: b 1 = 0, b 2 = 1, b 3 = 1, b 4 = 0, b 5 = 0 b 1 = {1,2,3,4} D b 1 = {0,1} r 1 = 8 b 2 = {2,3,6} D b 2 = {0,1} r 2 = 6 b 3 = {1,4,5} D b 3 = {0,1} r 3 = 5 b 4 = {2,8} D b 4 = {0,1} r 4 = 2 b 5 = {5,6} D b 5 = {0,1} r 5 = 2 R 12, R 13, R 14, R 24, R 25, R 35

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4-tuple: – (X,D,C h ) is a constraint network – C s = {F Q 1,…, F Q l } is a set of cost components – C h and C s also called hard and soft constraints – Graph representation – Variables are nodes – Arcs connect variables in the same hard or soft constraint 3/25/2013Constraint Optimization8 Cost Network

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Solving COP as a Series of CSPs It is always possible to a solve COP as a series of CSPs Add a single hard constraint: Increase the cost-bound: Eventually the cost-bound so high no solution exists, the previous solution is optimal 3/25/2013Constraint Optimization9

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Solving COP as a Series of CSPs With cost C j = 1, we have 4 solutions C 2 = 11, 1 solution C 1 = 12, 0 solutions Backtrack to previous solution, which is optimal The cost of solving so many CSPs is prohibitive Instead, we use search & inference 3/25/2013Constraint Optimization10

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Outline 3/25/2013Constraint Optimization11 Motivation Constraint Optimization and Cost Networks Section 13.1 Branch-and-Bound Search Section 13.2 Bucket Elimination for Optimization Section 13.3 Mini-bucket Elimination Section 13.4 Search with Mini-bucket Heuristics Section 13.5 Summary

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Branch-and-Bound Search Extends backtracking search – Traverses hard constraint search tree with DFS – Keeps track of two bounds using cost function Bounds – Lower bound : cost of best solution so far – Upper bound using bounding evaluation function – For partial assignment, : – If, backtrack 3/25/2013Constraint Optimization12

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Example: Combinatorial Auction 3/25/2013Constraint Optimization r 1 = 8 r 2 = 6 r 3 = 5 r 4 = 2 r 5 = 2 R 12, R 13, R 14, R 24, R 25, R 35 b1b2b3b4b5b1b2b3b4b5

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Russian Doll Search Run n successive BnB searches – Each search involves a single additional variable – First subproblem includes just nth variable – The (n-i+1)th subproblem involves x i,…,x n Bounds are augmented by previous searches Previous solutions can also heuristically guide search Extra pruning power is often worth extra cost of doing n searches 3/25/2013Constraint Optimization14

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Example: Combinatorial Auction 3/25/2013Constraint Optimization =19 8+7=15 8+4=12 8+2= Russian Doll: Previous searches: Over b 2,b 3,b 4,b 5 : b 2 =1,b 3 =1,b 4 =0,b 5 =0 11 Cost: Initial lower bound given by previous optimal solution Over b 3,b 4,b 5 : b 3 =1,b 4 =0,b 5 =1 7 Over b 4,b 5 : b 4 =1,b 5 =1 4 Over b 5 : b 5 =1 2 Optimal solutions from previous searches estimate upper bound r 1 = 8, r 2 = 6, r 3 = 5, r 4 = 2, r 5 = 2

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Example: Combinatorial Auction 3/25/2013Constraint Optimization Russian Doll: Original: b1b2b3b4b5b1b2b3b4b5

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Improving Branch-and-Bound Substantial work on BnB has been done for integer programs by OR community Primary way to improve BnB is to improve accuracy of bounding function Constraint community extends same ideas to general constraints and cost functions Generate bounding functions using Bucket Elimination 3/25/2013Constraint Optimization17

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Outline 3/25/2013Constraint Optimization18 Motivation Constraint Optimization and Cost Networks Section 13.1 Branch-and-Bound Search Section 13.2 Bucket Elimination for Optimization Section 13.3 Mini-bucket Elimination Section 13.4 Search with Mini-bucket Heuristics Section 13.5 Summary

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Example: All soft constraints Consider the following cost network: 3/25/2013Constraint Optimization19 a b c f g d and ordering: a,c,b,f,d,g g d f b c a

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Example: All soft constraints 3/25/2013Constraint Optimization20 a c b f d g

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Derivation of Bucket Elimination 3/25/2013Constraint Optimization21 Cost function: Move each function into lowest (leftmost) bucket: Maximize functions in highest bucket for bucket variable and move to new lowest bucket: Repeat:

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Bucket Elimination for Optimization Inference algorithm for COP Analogous to normal bucket-elimination – Each bucket holds a set of cost functions – Join operation replaced with summation – Projection replaced with maximization (or min) Forward direction generates an augmented cost network representation Backward direction generates optimal solution in linear time 3/25/2013Constraint Optimization22

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Joining Soft Constraints Joining is replaced by summation: 3/25/2013Constraint Optimization23 x1x1 x2x2 Cost x2x2 x3x x1x1 x2x2 x3x … 5

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Projecting Soft Constraints Projection is replaced by maximization: 3/25/2013Constraint Optimization24 x1x1 x2x2 x3x3 Cost … x1x1 x3x … 6

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Bucket Elimination for Opt: Properties Can incorporate hard constraints by adding back join and project operations It is always possible to treat hard constraints as cost functions, but not optimal – Hard constraints have stronger properties that can only be exploited if expressed explicitly – Hard constraints permit use of local consistency enforcing Complexity is based on induced width – For r hard and soft constraints and ordering d with induced width w*(d) – Time and space: O(r exp w*(d)+1 ) 3/25/2013Constraint Optimization25

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Example: Combinatorial Auction 3/25/2013 Constraint Optimization 26 b1b1 b2b2 b3b3 b4b4 b5b5 Opt

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R 41 b1b1 b4b r(b4)r(b4) b4b4 C R 42 b2b2 b4b R 124 b1b1 b2b2 b4b h4(b1,b2)h4(b1,b2) b1b1 b2b2 C h4(b1,b2,b4)h4(b1,b2,b4) b1b1 b2b2 b4b4 C Example: Combinatorial Auction 3/25/2013Constraint Optimization27 R 21, R 42, r(b 4 )h 4 (b 1,b 2 ) Join hard constraints Sum soft constraints (cost) Maximize over bucket variable Opt b1b1 b2b2 b3b3 b4b4 b5b5

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h3(b1,b3,b5)h3(b1,b3,b5) b1b1 b3b3 b5b5 C h3(b1,b5)h3(b1,b5) b1b1 b5b5 C R 135 b1b1 b3b3 b5b r(b3)r(b3) b3b3 C R 31 b1b1 b3b R 35 b3b3 b5b Example: Combinatorial Auction 3/25/2013Constraint Optimization28 R 31, R 35, r(b 3 )h 3 (b 1,b 5 ) Join hard constraints Sum soft constraints (cost) Maximize over bucket variable Opt b1b1 b2b2 b3b3 b4b4 b5b5

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R 125 b1b1 b2b2 b5b h 2 (b 1, b 2,b 5 ) b1b1 b2b2 b5b5 C h4(b1,b2)h4(b1,b2) b1b1 b2b2 C h2(b1,b5)h2(b1,b5) b1b1 b5b5 C r(b2)r(b2) b2b2 C R 12 b1b1 b2b R 25 b2b2 b5b Example: Combinatorial Auction 3/25/2013Constraint Optimization29 R 12, R 25, r(b 2 ), h 4 (b 1,b 2 )h 2 (b 1,b 5 ) Join hard constraints Sum soft constraints (cost) Maximize over bucket variable Opt b1b1 b2b2 b3b3 b4b4 b5b5

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h5(b1)h5(b1) b1b1 C h5(b1,b5)h5(b1,b5) b1b1 b5b5 C r(b5)r(b5) b5b5 C h2(b1,b5)h2(b1,b5) b1b1 b5b5 C h3(b1,b5)h3(b1,b5) b1b1 b5b5 C Example: Combinatorial Auction 3/25/2013Constraint Optimization30 r(b 5 ), h 3 (b 1,b 5 ), h 2 (b 1,b 5 )h 5 (b 1 ) Sum soft constraints (cost) Maximize over bucket variable Opt b1b1 b2b2 b3b3 b4b4 b5b5

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11 h1(b1)h1(b1) b1b1 C r(b1)r(b1) b5b5 C h5(b1)h5(b1) b1b1 C Example: Combinatorial Auction 3/25/2013Constraint Optimization31 r(b 1 ), h 5 (b 1 )Opt Sum soft constraints (cost) Maximize over bucket variable Opt b1b1 b2b2 b3b3 b4b4 b5b5

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h1(b1)h1(b1) b1b1 C h5(b1,b5)h5(b1,b5) b1b1 b5b5 C h 2 (b 1, b 2,b 5 ) b1b1 b2b2 b5b5 C h3(b1,b3,b5)h3(b1,b3,b5) b1b1 b3b3 b5b5 C h4(b1,b2,b4)h4(b1,b2,b4) b1b1 b2b2 b4b4 C Example: Combinatorial Auction 3/25/2013Constraint Optimization32 Go in the reverse direction to generate a solution: b 1 =0b 5 =0b 2 =1b 3 =1b 4 =0 Cost of optimal solution = = 11

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Outline 3/25/2013Constraint Optimization33 Motivation Constraint Optimization and Cost Networks Section 13.1 Branch-and-Bound Search Section 13.2 Bucket Elimination for Optimization Section 13.3 Mini-bucket Elimination Section 13.4 Search with Mini-bucket Heuristics Section 13.5 Summary

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Example: All Soft Constraints 3/25/2013Constraint Optimization34 Bucket: Mini-bucket (i=3): Upper Bound (constraints are ignored, thus, the bound is optimistic)

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Mini-bucket Elimination BE is exponential in separator size (space) Instead use mini-bucket elimination – Bucket are partitioned to restrict arity of projected constraints to at most i (variables) – Result generates an estimation of optimal solution – Size of mini-buckets limits complexity and accuracy of estimation – Selection of partition will also affect results, so a good heuristic is needed 3/25/2013Constraint Optimization35

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Example: Combinatorial Auction 3/25/2013Constraint Optimization36 b1b1 b2b2 b3b3 b4b4 b5b5 Upper Bound

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h4(b2,b4)h4(b2,b4) b2b2 b4b4 C h4(b2)h4(b2) b2b2 C r(b4)r(b4) b4b4 C R 42 b2b2 b4b Example: Combinatorial Auction 3/25/2013Constraint Optimization37 R 42, r(b 4 )h4(b2)h4(b2) Sum soft constraints (cost) Maximize over bucket variable Bound b1b1 b2b2 b3b3 b4b4 b5b5

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h3(b3,b5)h3(b3,b5) b3b3 b5b5 C h3(b5)h3(b5) b5b5 C r(b3)r(b3) b3b3 C R 35 b3b3 b5b Example: Combinatorial Auction 3/25/2013Constraint Optimization38 R 35, r(b 3 )h3(b5)h3(b5) Sum soft constraints (cost) Maximize over bucket variable Bound b1b1 b2b2 b3b3 b4b4 b5b5

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h2(b2,b5)h2(b2,b5) b2b2 b5b5 C h2(b5)h2(b5) b5b5 C r(b2)r(b2) b2b2 C R 25 b2b2 b5b h4(b2)h4(b2) b2b2 C Example: Combinatorial Auction 3/25/2013Constraint Optimization39 R 25, r(b 2 ), h 4 (b 2 )h2(b5)h2(b5) Sum soft constraints (cost) Maximize over bucket variable Bound b1b1 b2b2 b3b3 b4b4 b5b5

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h5(b5)h5(b5) b5b5 C h5h5 r(b5)r(b5) b5b5 C h3(b5)h3(b5) b5b5 C h2(b5)h2(b5) b5b5 C Example: Combinatorial Auction 3/25/2013Constraint Optimization40 r(b 2 ), h 3 (b 5 ), h 2 (b 5 )h5h5 Sum soft constraints (cost) Maximize over bucket variable Bound b1b1 b2b2 b3b3 b4b4 b5b5

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19 h1(b1)h1(b1) b1b1 C r(b1)r(b1) b1b1 C h5h5 11 Example: Combinatorial Auction 3/25/2013Constraint Optimization41 r(b 1 ), h 5 Bound Sum soft constraints (cost) Maximize over bucket variable Bound b1b1 b2b2 b3b3 b4b4 b5b5

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h5(b5)h5(b5) b5b5 C h2(b2,b5)h2(b2,b5) b2b2 b5b5 C h3(b3,b5)h3(b3,b5) b3b3 b5b5 C h4(b2,b4)h4(b2,b4) b2b2 b4b4 C h1(b1)h1(b1) b1b1 C Example: Combinatorial Auction 3/25/2013Constraint Optimization42 Go in the forward direction to generate a solution (lower bound): b 1 =1b 5 =0b 2 =0b 3 =0b 4 =0 b1b1 b2b2 b3b3 b4b4 b5b5 Cost of solution/lower bound: 8 Interval bound on optimal solution: [8,19]

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Mini-bucket Elimination: Properties Partitioning buckets amounts to moving maximization inside summation: Going up the elimination ordering (backward direction) – Maximization over mini-buckets gives an upper bound Going down the instantiation ordering (forward direction) – Finding a solution provides a lower bound Complexity depends on i, assuming i < w* – If : Time and space complexity = O(exp(i)) – Otherwise: Time = O(r exp(i)), Space = O(r exp(i-1)), where r is the number of cost functions 3/25/2013Constraint Optimization43

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Outline 3/25/2013Constraint Optimization44 Motivation Constraint Optimization and Cost Networks Section 13.1 Branch-and-Bound Search Section 13.2 Bucket Elimination for Optimization Section 13.3 Mini-bucket Elimination Section 13.4 Search with Mini-bucket Heuristics Section 13.5 Summary

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Search with Mini-Bucket Heuristics Mini-bucket allows us to bound error, however we may need the optimal solution We can improve search with estimations from mini-buckets – Search using BnB, use mini-bucket to improve accuracy of bounding function – At each step, estimate the cost of optimal solution given a partial assignment using mini-bucket – Assuming a static variable ordering, buckets can be computed before search 3/25/2013Constraint Optimization45

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8 f mb (b 1 =0,b 5 =0,b 2 =1,b 3 =0) = 6+0+h 4 (b 2 ) = = 6 f mb (b 1 =0,b 5 =0,b 2 =0) = 0+h 4 (b 2 ) +h 3 (b 5 ) = = 7 f mb (b 1 =1,b 5 =0) = 8+0+h 2 (b 5 ) +h 3 (b 5 ) = = 19 f mb (b 1 =0) = 0+h 5 = 0+11 = 11 f mb (b 1 =0,b 5 =0) = 0+h 2 (b 5 ) +h 3 (b 5 ) = = 11 f mb (b 1 =0,b 5 =1) = r 5 +h 2 (b 5 ) +h 3 (b 5 ) = = 4 f mb (b 1 =0,b 5 =0,b 2 =1) = r 2 +h 4 (b 2 ) +h 3 (b 5 ) = = 11 f mb (b 1 =0,b 5 =0,b 2 =1,b 3 =1) = 6+r 3 +h 4 (b 2 ) = = 11 f mb (b 1 =1) = r 1 +h 5 = 8+11 = 19 f mb (b 1 =1,b 5 =1) = 8+r 5 +h 2 (b 5 ) +h 3 (b 5 ) = = 10 Example: Combinatorial Auction 3/25/2013Constraint Optimization h4(b2)h4(b2) b2b2 C h3(b5)h3(b5) b5b5 C h2(b5)h2(b5) b5b5 C b1b1 b2b2 b3b3 b4b4 b5b5 h5h5 Bound L= Lower bound is initialized to lower bound found during mini-bucket

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Mini-bucket Heuristics Properties 3/25/2013Constraint Optimization47 Definition 13.1: Given an ordered set of augmented buckets generated by the mini-bucket algorithm, the bounding function is the sum of all the new functions that are generated in buckets p+1 through n and reside in buckets 1 through p. That is: Theorem 13.5 (paraphrased): The mini-bucket heuristic never under-estimates the cost of the optimal extension of a given partial assignment The heuristic is monotonic, as search goes deeper the bound can only decrease, meaning it is more accurate.

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Example: Combinatorial Auction 3/25/2013Constraint Optimization48 In this case the bounding functions perform similarly, but mini-bucket is more accurate! First-choice Mini-bucket

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Outline 3/25/2013Constraint Optimization49 Motivation Constraint Optimization and Cost Networks Section 13.1 Branch-and-Bound Search Section 13.2 Bucket Elimination for Optimization Section 13.3 Mini-bucket Elimination Section 13.4 Search with Mini-bucket Heuristics Section 13.5 Summary

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Constraint optimization problems consist of a constraint network and a global cost function. A COP can be solved as a CSP, but search and inference methods more efficient Branch-and-bound search is a simple method which extends backtrack search with a bounding function Bucket Elimination is an inference method which can generate COP solutions in a backtrack-free manner Mini-bucket Elimination partitions buckets into subsets in order to reduce complexity, while still generating a bound for a COP BnB can be augmented with a mini-bucket based heuristic allowing inference and search methods to be combined. 3/25/2013Constraint Optimization50

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Thank You! Any Questions? 3/25/2013Constraint Optimization51

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