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Published byJanel Clark Modified over 8 years ago
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Escaping Local Optima
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Where are we? Optimization methods Complete solutions Partial solutions Exhaustive search Hill climbing Exhaustive search Hill climbing Exhaustive search Branch and bound Greedy Best first A* Divide and Conquer Dynamic programming Exhaustive search Branch and bound Greedy Best first A* Divide and Conquer Dynamic programming
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Where are we going? Optimization methods Complete solutions Partial solutions Exhaustive search Branch and bound Greedy Best first A* Divide and Conquer Dynamic programming Exhaustive search Branch and bound Greedy Best first A* Divide and Conquer Dynamic programming Improved methods based on complete solutions Exhaustive search Hill climbing Exhaustive search Hill climbing
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Escaping local optima Many strategies including Simulated annealing Tabu search and many others… some examples -->
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(0) Basic Hill climbing determine initial solution s while s is not a local optimum choose s’ in N(s) such that f(s’)>f(s) s = s’ return s
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(1) Randomized Hill climbing determine initial solution s; bestS = s while termination condition not satisfied with probability p choose neighbour s’ at random (uniform) else choose s’ with f(s’) > f(s) //climb if possible or s’ with max (f(s’)) over N(s) s = s’; if (f(s) > f(bestS)) bestS = s return bestS
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(2) Variable Neighbourhood determine initial solution s i = 1 repeat choose neighbour s’ in N i (s) with max(f(s’)) if ((f(s’) > f(s)) s = s’ i = 1// restart in first neighbourhood else i = i+1// go to next neighbourhood until i > iMax return s
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Stochastic local search many other important algorithms address the problem of avoiding the trap of local optima (possible source of project topics) M&F focus on two only simulated annealing tabu search
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Simulated annealing metaphor: slow cooling of liquid metals to allow crystal structure to align properly “temperature” T is slowly lowered to reduce random movement of solution s in solution space
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Simulated Annealing determine initial solution s; bestS = s T = T 0 while termination condition not satisfied choose s’ in N(s) probabilistically if (s’ is “acceptable”) // function of T s = s’ if (f(s) > f(sBest)) bestS = s update T return bestS
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Accepting a new solution - acceptance more likely if f(s’) > f(s) - as execution proceeds, probability of acceptance of s’ with f(s’) < f(s) decreases (becomes more like hillclimbing) determine initial solution s; bestS = s T = T 0 while termination condition not satisfied choose s’ in N(s) probabilistically if (s’ is “acceptable”) // function of T s = s’ if (f(s) > f(sBest)) bestS = s update T return bestS determine initial solution s; bestS = s T = T 0 while termination condition not satisfied choose s’ in N(s) probabilistically if (s’ is “acceptable”) // function of T s = s’ if (f(s) > f(sBest)) bestS = s update T return bestS
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the acceptance function T evolves *sometimes p=1 when f(s’)-f(s)> 0
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Simulated annealing with SAT algorithm p.123 SA-SAT propositions:P 1,… P n expression:F = D 1 D 2 …D k where clause D i is a disjunction of propositions and negative props e.g.,P x ~P y P z ~P w fitness function:number of true clauses
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Inner iteration assign random truth setTFFT repeat for i=1 to 4 flip truth of prop iFFFT evaluateFTFT decide to keep (or not)FFTT changed valueFFTF reduce TFFTT
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Tabu search (taboo) always looks for best solution but some choices (neighbours) are ineligible (tabu) ineligibility is based on recent moves: once a neighbour edge is used, it is tabu for a few iterations search does not stop at local optimum
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Symmetric TSP example set of 9 cities {A,B,C,D,E,F,G,H,I} neighbour definition based on 2-opt* (27 neighbours) current sequence: B - D - A - I - H - F - E - C - G - B move to 2-opt neighbour B - E - F - H - I - A - D - C - G - B edges B-E and D-C are now tabu i.e., next 2-opt swap cannot involve these edges *example in book uses 2-swap, p 131
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TSP example, algorithm p 133 how long will an edge be tabu?3 iterations how to track and restore eligibility? data structure to store tabu status of 9*8/2 = 36 edges B - D - A - I - H - F - E - C - G - B recency-based memory ABCDEFGH I00000002 H0000010 G010000 F00000 E0030 D230 C00 B0
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procedure tabu search begin tries <- 0 repeat generate a tour count <- 0 repeat identify a set T of 2-opt moves select best admissible move from T make appropriate 2-opt update tabu list and other vars if new tour is best-so-far for a given tries update local best tour information count <- count + 1 until count == ITER tries <- tries + 1 if best-so-far for given tries is best-so-far (for all ‘tries’) update global best information until tries == MAX-TRIES end
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applying 2-opt with tabu from the table, some edges are tabu: B - D - A - I - H - F - E - C - G - B 2-opt can only consider: AI and FE AI and CG FE and CG ABCDEFGH I00000002 H0000010 G010000 F00000 E0030 D230 C00 B0
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importance of parameters once algorithm is designed, it must be “tuned” to the problem selecting fitness function and neighbourhood definition setting values for parameters this is usually done experimentally
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procedure tabu search begin tries <- 0 repeat generate a tour count <- 0 repeat identify a set T of 2-opt moves select best admissible move from T make appropriate 2-opt update tabu list and other vars if new tour is best-so-far for a given tries update local best tour information count <- count + 1 until count == ITER tries <- tries + 1 if best-so-far for given tour is best-so-far for all tries update global best information until tries == MAX-TRIES end
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