its search jim, but not as we know it. local search (aka neighbourhood search) we start off with a...
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It’s search Jim, but not as we know it
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Local Search (aka neighbourhood search)
We start off with a complete solution and improve it
or
We gradually construct a solution, make our best move as we go
We need:• a (number of) move operator(s)
• take a state S and produce a new state S’• an evaluation function
• so we can tell if we appear to be moving in a good direction• let’s assume we want to minimise this function, i.e. cost.
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Wooooooosh! Let’s scream down hill.
Hill climbing/descending
Find the lowest cost solution
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Trapped at a local minima
How can we escape?
Find the lowest cost solution
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tsp
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But first, an example
TSP• given n cities with x/y coordinates• select any city as a starting and ending point• arrange the n-1 cities into a tour of minimum cost
Representation• a permutation of the n-1 cities
A move operator• swap two positions (let’s say)• 2-opt?
• Take a sub-tour and reverse it• how big is the neighbourhood of a state?• how big is the state space?• What dimensional space are we moving through?
Evaluation Function• cost/distance of the tour
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How might we construct initial tour?
Nearest neighbour
Furthest Insertion
Random
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But first, an exampleA move operator• 2-opt?
• Take a sub-tour and reverse it
A tour, starting and ending at city 99 1 4 2 7 3 5 6 8 9
9
1
8
7
5
4
3
26
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But first, an exampleA move operator• 2-opt?
• Take a sub-tour and reverse it
9 1 4 2 7 3 5 6 8 9
9
1
8
7
5
4
3
26
reverse
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But first, an exampleA move operator• 2-opt?
• Take a sub-tour and reverse it
9 1 4 2 7 3 5 6 8 9
9 1 6 5 3 7 2 4 8 9
9
1
8
7
5
4
3
26
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Steepest descent
S := construct(n)improvement := truewhile improvementdo let N := neighbourhood(S), S’ := bestOf(N) in if cost(S’) <= cost(S) then S := S’ improvement := true else improvement := false
But … it gets stuck at a local minima
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Consider 1-d Bin Packing
• how might we construct initial solution?• how might we locally improve solution
• what moves might we have?• what is size of neighbourhood?• what cost function (to drive steepest/first descent)?
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Warning: Local search does not guarantee optimality
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Simulated Annealing (SA) Kirkpatrick, Gelatt, & Vecci Science 220, 1983
Annealing, to produce a flawless crystal, a structure in a minimum energy state
• At high temperatures, parts of the structure can be freely re-arranged• we can get localised increases in temperature
• At low temperatures it is hard to re-arrange into anything other than a lower energy state• Given a slow cooling, we settle into low energy states
Apply this to local search, with following control parameters• initial temperature T• cooling rate R• time at temperature E (time to equilibrium)
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Simulated Annealing (SA) Kirkpatrick, Gelatt, & Vecci Science 220, 1983
Apply this to local search, with following control parameters• initial temperature T (whatever)• cooling rate R (typically R = 0.9)• time at temperature E (time to equilibrium, number of moves examined)• Δ change in cost (+ve means non-improving)
Accept a non-improving move with probability
Te
Throw a dice (a uniformly random real in range 0.0 to 1.0), and if it delivers a value less than above then acceptthe non-improving move.
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62.01035
72.01025
85.01015
95.010015
85.01015
2.0115
tktK
Replaced e with kAs we increase temp t we increase probability of acceptAs delta increases (cost is worse) acceptance decreases
SA
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Simulated Annealing Sketch (SA) Kirkpatrick, Gelatt, & Vecci Science 220, 1983
S := construct(n) while T > limitdo begin for in (1 .. E) do let N := neighbourhood(S), S’ := bestOf(N) delta := cost(S’) - cost(S) in if delta < 0 or (random(1.0) < exp(-delta/T)) then S := S’ if S’ is best so far then save it T := T * R end
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Tabu Search (TS) Fred Glover
SA can cycle! • Escape a local minima• Next move, fall back in!
• Maintain a list of local moves that we have made• the tabu list!• Not states, but moves made (e.g. 2-opt with positions j and k)
• Don’t accept a move that is tabu• unless it is the best found so far
• To encourage exploration
• Consider• size of tabu-list• what to put into the list• representation of entries in list• consider tsp and 1-d bp
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Guided Local Search (GLS) Tsang & Voudouris
• (1) Construct a solution, going down hill, with steepest or 1st descent• (2) analyse solution at local minima
• determine most costly component of solution• in tsp this might be longest arc
• (3) penalise the most costly feature• giving a new cost function
• (4) loop back to (1) if time left
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Genetic Algorithms (GA) John Holland, 1981
• Represent solution as a chromosome• Have a population of these (solutions)• Select the fittest, a champion
• note, evaluation function considered measure of fitness• Allow that champion to reproduce with others
• using crossover primarily• mutation, as a secondary low lever operator
• Go from generation to generation• Eventually population becomes homogenised
• Attempts to balance exploration and optimisation
• Analogy is Evolution, and survival of the fittest
It didn’t work for me. I want a 3d hand, eyes on the back of my head, good looks , ...
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• Arrange population in non-decreasing order of fitness• P[1] is weakest and P[n] is fittest in population
• generate a random integer x in the range 1 to n-1• generate a random integer y in the range x to n• Pnew := crossover(P[x],P[y])• mutate(Pnew,pMutation)• insert(Pnew,P)• delete(P[1])• loop until no time left
GA sketch
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HC, SA, TS, GLS are point based
GA is population based
All of them retain the best solution found so far(of course!)
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Local Search: a summary
• cannot guarantee finding optimum (i.e. incomplete)• these are “meta heuristics” and need insight/inventiveness to use• they have parameters that must be tuned• tricks may be needed for evaluation functions to smooth out landscape• genetic operators need to be invented (for GA)
• example in TSP, with PMX or order-based chromosome• this may result in loss of the spirit of the meta heuristic
• challenge to use in CP environment (see next slides)
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Local search for a csp (V,C,D)
let cost be the number of conflicts
Therefore we try to move to a state with less conflicts
WarningLocal search cannot prove that there is no solutionneither can it be guaranteed to find a solution
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Problems with local search and CP?
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Problems with local search and csp?
• how do we do a local move?• how do we undo the effects of propagation?• can we use propagation?• maybe use 2 models
• one active, conventional• one for representing state
• used to compute neighbourhood• estimate cost of local moves
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Dynadec / Solution Videos
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Local Search for CP, some recent work
• min-conflicts • Minton @ nasa
• WalkSat• reformulate CSP as SAT
• GENET• Tsang & Borrett
• Weak-commitment search• Yokoo AAAI-94
• Steve Prestwich• Cork
• Large Neighbourhood Search• Paul Shaw, ILOG
• Incorporating Local Search in CP (for VRP)• deBaker, Furnon, Kilby, Prosser, Shaw
• LOCALIZER• COMET
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Is local search used?(aka “who cares”)
You bet!(aka industry)
Early work on SA was Aart’s work on scheduling
BT use SA for Workforce management (claim $120M saving per year)
ILOG Dispatcher uses TS & GLS (TLS?)
COMET Dynadec
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Limited Discrepancy Search (LDS) Ginsberg & Harvey
In complete systematic search (BT (still using that old greasy stuff?))where are errors usually made?
At the top of search, where our heuristics are least informed
When we reach a dead end, go back to the root and start againbut our 1st decision is anti-heuristic.
If that fails, go back and this time let our 2nd decision be anti-heuristic
And so on, for 1 discrepancy
Then increase the discrepancy count
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LDS is quasi-complete
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Limited Discrepancy Search (LDS) Ginsberg & Harvey
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Limited Discrepancy Search (LDS) Ginsberg & Harvey
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Limited Discrepancy Search (LDS) Ginsberg & Harvey
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Limited Discrepancy Search (LDS) Ginsberg & Harvey
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Limited Discrepancy Search (LDS) Ginsberg & Harvey
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Limited Discrepancy Search (LDS) Ginsberg & Harvey
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Limited Discrepancy Search (LDS) Ginsberg & Harvey
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Limited Discrepancy Search (LDS) Ginsberg & Harvey
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Limited Discrepancy Search (LDS) Ginsberg & Harvey
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Limited Discrepancy Search (LDS) Ginsberg & Harvey
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Limited Discrepancy Search (LDS) Ginsberg & Harvey
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Limited Discrepancy Search (LDS) Ginsberg & Harvey
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Limited Discrepancy Search (LDS) Ginsberg & Harvey
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Limited Discrepancy Search (LDS) Ginsberg & Harvey
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Limited Discrepancy Search (LDS) Ginsberg & Harvey
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Limited Discrepancy Search (LDS) Ginsberg & Harvey
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Now take 2 discrepancies
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Limited Discrepancy Search (LDS) Ginsberg & Harvey
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Limited Discrepancy Search (LDS) Ginsberg & Harvey
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Limited Discrepancy Search (LDS) Ginsberg & Harvey
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Variants of LDS
• Depth bounded discrepancy search• Toby Walsh
• Iterative LDS• Pedro Meseguer
Who cares?
See ILOG Solver
Guess!
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Barbara’s view, min-conflicts, …
AR33, pages 56,57,58
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Local Search: where’s it going?
The physicists have returned
We are trying to understand the landscape of problemsand the behaviour of the algorithms
We are looking at the emergence of backbonesand other phenomena
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That’s all for now folks
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