1. Optimization Techniques Gang Quan Van Laarhoven, Aarts Sources used

2. Scheduling using Simulated Annealing Reference: Devadas, S.; Newton, A.R. Algorithms for hardware allocation in data path synthesis. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, July 1989, Vol.8, (no.7):768-81.

3. Iterative Improvement 1 General method to solve combinatorial optimization problems Principles: 1.Start with initial configuration 2.Repeatedly search neighborhood and select a neighbor as candidate 3.Evaluate some cost function (or fitness function) and accept candidate if "better"; if not, select another neighbor 4.Stop if quality is sufficiently high, if no improvement can be found or after some fixed time

4. Iterative Improvement 2 Needed are: 1.A method to generate initial configuration 2.A transition or generation function to find a neighbor as next candidate 3.A cost function 4.An Evaluation Criterion 5.A Stop Criterion

5. Iterative Improvement 3 Simple Iterative Improvement or Hill Climbing: Candidate is always and only accepted if cost is lower (or fitness is higher) than current configuration Stop when no neighbor with lower cost (higher fitness) can be found Disadvantages: Local optimum as best result Local optimum depends on initial configuration Generally, no upper bound can be established on the number of iterations

6. Hill climbing

7. Simulated Annealing Local Search Solution space Cost function ?

8. How to cope with disadvantages 1.Repeatdifferent initial 1.Repeat algorithm many times with different initial configurations gathered in previous runs 2.Use information gathered in previous runs 3.Use a more complex Generation Function to jump out of local optimum Evaluation Criterion 4.Use a more complex Evaluation Criterion that accepts sometimes (randomly) also solutions away from the (local) optimum

9. Simulated Annealing Use a more complex Evaluation Function: Do sometimes accept candidates with higher cost to escape from local optimum Adapt the parameters of this Evaluation Function during execution analogyBased upon the analogy with the simulation of the annealing of solids

10. Simulated Annealing

11. Other Names Monte Carlo Annealing Statistical Cooling Probabilistic Hill Climbing Stochastic Relaxation Probabilistic Exchange Algorithm

12. Optimization Techniques Mathematical Programming Network Analysis Branch & Bound Genetic Algorithm Simulated Annealing Algorithm Tabu Search

13. Simulated Annealing What –Exploits an analogy between the annealing process and the search for the optimum in a more general system.

14. Annealing Process –Raising the temperature up to a very high level (melting temperature, for example), the atoms have a higher energy state and a high possibility to re-arrange the crystalline structure. –Cooling down slowly, the atoms have a lower and lower energy state and a smaller and smaller possibility to re-arrange the crystalline structure.

15. Statistical Mechanics Combinatorial Optimization State {r:}(configuration -- a set of atomic position ) weight e -E({r:])/K B T -- Boltzmann distribution E({r:]): energy of configuration K B : Boltzmann constant T: temperature Low temperature limit ??

16. Analogy Physical System State (configuration) Energy Ground State Rapid Quenching Careful Annealing Optimization Problem Solution Cost function Optimal solution Iteration improvement Simulated annealing

17. Simulated Annealing Analogy –Metal  Problem –Energy State  Cost Function –Temperature  Control Parameter –A completely ordered crystalline structure  the optimal solution for the problem Global optimal solution can be achieved as long as the cooling process is slow enough.

18. Other issues related to simulated annealing 1.Global optimal 1.Global optimal solution is possible, but near optimal is practical Tuning 2.Parameter Tuning 1.Aarts, E. and Korst, J. (1989). Simulated Annealing and Boltzmann Machines. John Wiley & Sons. 3.Not easy for parallel implementation, but was implemented. 4.Random generator quality is important

19. Analogy Slowly cool down a heated solid, so that all particles arrange in the ground energy state At each temperature wait until the solid reaches its thermal equilibrium Probability of being in a state with energy E : Z(T) Pr { E = E } = 1 / Z(T). exp (-E / k B.T) EEnergy TTemperature k B Boltzmann constant Z(T) Z(T) Normalization factor (temperature dependant)

20. Simulation of cooling (Metropolis 1953) At a fixed temperature T : Perturb (randomly) the current state to a new state  E is the difference in energy between current and new state If  E < 0 (new state is lower), accept new state as current state If  E  0, accept new state with probability Pr (accepted) = exp (-  E / k B.T) Eventually the systems evolves into thermal equilibrium at temperature T ; then the formula mentioned before holds When equilibrium is reached, temperature T can be lowered and the process can be repeated

21. Simulated Annealing Same algorithm can be used for combinatorial optimization problems: Energy ECost function CEnergy E corresponds to the Cost function C Temperature T corresponds to control parameter c Pr { configuration = i } = 1/Q(c). exp (-C(i) / c) CCost cControl parameter Q(c) Normalization factor (not important)

22. Metropolis Loop Metropolis Loop is the essential characteristic of simulated annealing Determining how to: – randomly explore new solution, –reject or accept the new solution at a constant temperature T. Finished until equilibrium is achieved.

23. Metropolis Criterion Let : –X be the current solution and X ’ be the new solution –C(x) be the energy state (cost) of x –C(x’) be the energy state of x’ Probability P accept = exp [(C(x)-C(x’))/ T] Let N = Random(0,1) Unconditional accepted if –C(x’) < C(x), the new solution is better Probably accepted if –C(x’) >= C(x), the new solution is worse. –Accepted only when N < P accept

24. Simulated Annealing Algorithm Initialize: –initial solution x, –highest temperature T h, –and coolest temperature T l T= T h When the temperature is higher than T l While not in equilibrium Search for the new solution X’ Accept or reject X’ according to Metropolis Criterion End Decrease the temperature T End

25. Components of Simulated Annealing Definition of solution Search mechanism, i.e. the definition of a neighborhood Cost-function

26. Control Parameters 1.Definition of equilibrium 1.Definition is reached when we cannot yield any significant improvement after certain number of loops 2.A constant number of loops is assumed to reach the equilibrium 2.Annealing schedule (i.e. How to reduce the temperature ) 1.A constant value is subtracted to get new temperature, T’ = T - T d 2.A constant scale factor is used to get new temperature, T’= T * R d A scale factor usually can achieve better performance 1.How to define equilibrium? 2.How to calculate new temperature for next step?

27. Control Parameters: Temperature Temperature determination: –Artificial, without physical significant –Initial temperature 1.Selected so high that leads to 80-90% acceptance rate –Final –Final temperature 1.Final temperature is a constant value, i.e., based on the total number of solutions searched. No improvement during the entire Metropolis loop 2.Final temperature when acceptance rate is falling below a given (small) value need to be tunedProblem specific and may need to be tuned

28. Example of Simulated Annealing Traveling Salesman ProblemTraveling Salesman Problem (TSP) –Given 6 cities and the traveling cost between any two cities –A salesman need to start from city 1 and travel all other cities then back to city 1 –Minimize the total traveling cost

29. Example: SA for traveling salesman Solution representation –An integer list, i.e., (1,4,2,3,6,5) Search mechanism –Swap any two integers (except for the first one) (1,4,2,3,6,5)  (1,4,3,2,6,5) Cost function

30. Temperature 1.Initial temperature determination 1.Initial temperature is set at such value that there is around 80% acceptation rate for “bad move” 2.Determine acceptable value for (C new – C old ) 2.Final temperature determination Stop criteria Solution space coverage rate Example: SA for traveling salesman Annealing schedule (i.e. How to reduce the temperature ) –A constant value is subtracted to get new temperature, T’ = T – T d –For instance new value is 90% of previous value. Depending on solution space coverage rate

31. Homogeneous Algorithm of Simulated Annealing initialize; REPEAT perturb ( config.i  config.j,  C ij ); perturb ( config.i  config.j,  C ij ); IF  C ij < 0 THEN accept IF  C ij < 0 THEN accept ELSE IF exp(-  C ij /c) > random[0,1) THEN accept; ELSE IF exp(-  C ij /c) > random[0,1) THEN accept; IF accept THEN update(config.j); IF accept THEN update(config.j); UNTIL equilibrium is approached sufficient closely; UNTIL equilibrium is approached sufficient closely; c := next_lower(c); UNTIL system is frozen or stop criterion is reached In homogeneous algorithm the value of c is kept constant in the inner loop and is only decreased in the outer loop

32. Inhomogeneous Algorithm Previous algorithm is the homogeneous variant: c is kept constant in the inner loop and is only decreased in the outer loop Alternative is the inhomogeneous variant: 1.There is only one loop; 2. c 2. c is decreased each time in the loop, 3.but only very slightly

33. Selection of Parameters for Inhomogeneous variants 1.Choose the start value of c so that in the beginning nearly all perturbations are accepted (exploration), but not too big to avoid long run times 2.The function next_lower in the homogeneous variant is generally a simple function to decrease c, e.g. a fixed part (80%) of current c 3.At the end c is so small that only a very small number of the perturbations is accepted (exploitation) 4.If possible, always try to remember explicitly the best solution found so far; the algorithm itself can leave its best solution and not find it again

34. Markov Chains for use in Simulation Annealing Markov Chain: Sequence of trials where the outcome of each trial depends only on the outcome of the previous one conditional probabilitiesMarkov Chain is a set of conditional probabilities: P ij (k-1,k) Probability that the outcome of the k-th trial is j, when trial k-1 is i optimal solution circuit algorithm Stage k-1 Stage k 1/4 1/2 This example is just a particular application in natural language analysis and generation

35. Markov Chains for use in Simulation Annealing Markov Chain: Sequence of trials where the outcome of each trial depends only on the outcome of the previous one conditional probabilitiesMarkov Chain is a set of conditional probabilities: P ij (k-1,k) Probability that the outcome of the k-th trial is j, when trial k-1 is i Markov Chain is homogeneous when the probabilities do not depend on k

36. Homogeneous and inhomogeneous Markov Chains in Simulated Annealing one homogeneous Markov ChainWhen c is kept constant (homogeneous variant), the probabilities do not depend on k and for each c there is one homogeneous Markov Chain one inhomogeneous Markov ChainWhen c is not constant (inhomogeneous variant), the probabilities do depend on k and there is one inhomogeneous Markov Chain

37. Performance of Simulated Annealing SA is a general solution method that is easily applicable to a large number of problems "Tuning" of the parameters (initial c, decrement of c, stop criterion) is relatively easy Generally the quality of the results of SA is good, although it can take a lot of time

38. Performance of Simulated Annealing Results are generally not reproducible: another run can give a different result SA can leave an optimal solution and not find it again (so try to remember the best solution found so far) Proven to find the optimum under certain conditions; one of these conditions is that you must run forever

39. Basic Ingredients for S.A. 1. Solution space 2. Neighborhood Structure 3. Cost function 4. Annealing Schedule

40. Optimization Techniques Mathematical Programming Network Analysis Branch & Bond Genetic Algorithm Simulated Annealing Tabu Search

42. What –Neighborhood search + memory Neighborhood search MemoryMemory –Record the search history – the “tabu list” –Forbid cycling search Main idea of tabu

43. Algorithm of Tabu Search 1.Choose an initial solution X 2.Find a subset of N(x) the neighbors of X which are not in the tabu list. 3.Find the best one (x’) in set N(x). 4.If F(x’) > F(x) then set x=x’. 5.Modify the tabu list. 6.If a stopping condition is met then stop, else go to the second step.

44. Effective Tabu Search Effective Modeling –Neighborhood structure –Objective function (fitness or cost) Example: 1.Graph coloring problem: –Find the minimum number of colors needed such that no two connected nodes share the same color. Aspiration criteria –The criteria for overruling the tabu constraints and differentiating the preference of among the neighbors

45. Effective Tabu Search Effective Computing –“Move” may be easier to be stored and computed than a completed solution move: the process of constructing of x’ from x –Computing and storing the fitness difference may be easier than that of the fitness function.

46. Effective Tabu Search Memory UseEffective Memory Use –Variable tabu list size For a constant size tabu list –Too long: –Too long: deteriorate the search results –Too short: –Too short: cannot effectively prevent from cycling –Intensification of the search Decrease the tabu list size –Diversification of the search Increase the tabu list size Penalize the frequent move or unsatisfied constraints

47. Eample of Tabu Search A hybrid approach for graph coloring problem –R. Dorne and J.K. Hao, A New Genetic Local Search Algorithm for Graph Coloring, 1998

48. Problem Given an undirected graph G=(V,E) –V={v 1,v 2,…,v n } –E={e ij } Determine a partition of V in a minimum number of color classes C 1,C 2,…,C k such that for each edge e ij, v i and v j are not in the same color class. NP-hard

49. General Approach Transform an optimization problem into a decision problem Genetic Algorithm + Tabu SearchGenetic Algorithm + Tabu Search –Meaningful crossover –Using Tabu search for efficient local search

50. Encoding Individual –(C i1, C i2, …, C ik ) Cost function –Number of total conflicting nodes Conflicting nodeConflicting node –having same color with at least one of its adjacent nodes Neighborhood (move) definition –Changing the color of a conflicting node Cost evaluation –Special data structures and techniques to improve the efficiency

51. Implementation Parent Selection –Random Reproduction/Survivor Crossover Operator –Unify independent set (UIS) crossover Independent set –Conflict-free nodes set with the same color Try to increase the size of the independent set to improve the performance of the solutions

52. Unify independent set (UIS) crossover It can be made very similar to Simulated Annealing or Genetic Algorithm

53. Implementation of Tabu Search MutationMutation –With Probability P w, randomly pick neighbor –With Probability 1 – P w, Tabu search Tabu search –Tabu list List of {V i, c j } –Tabu tenure (the length of the tabu list) L = a * N c + Random(g) N c : Number of conflicted nodes a,g: empirical parameters

54. Summary on Tabu Search 1.Neighbor Search 2.TS prevent being trapped in the local minimum with tabu list 3.TS directs the selection of neighbor 4.TS cannot guarantee the optimal result 5.Sequential 6.Adaptive