1. Single-solution based metaheuristics

2. Outline Local Search Simulated annealing Tabu search …

3. 3 local search LS() 1 choose an initial solution X randomly 2 while the stop criterion is not yet satisfied do 3 choose a neighbor X' ∈ N(X) 4 if f(X')<f(X) then X ← X' 5 return X

4. Neighbourhood For each solution X  S, N(X)  S is a neighborhoodIn some sense each X'  N(X) is in some sense “ close ” to S Defined in terms of some operation Very like the “action” in search Exchange two elements Change the value of elements

5. Local search Elements of Local Search Representation of the solution Evaluation function; Neighbourhood function: to define solutions which can be considered close to a given solution. Neighbourhood search strategy: random and systematic search; Acceptance criterion: first improvement, best improvement, best of non-improving solutions, random criteria;

6. Simulated annealing Combinatorial search technique inspired by the physical process of annealing A stochastic local search algorithm Simulated annealing is an approach for solving all sorts of optimization problems: Say you have a really huge search space. You want to find the global optimum for some function in that space.

7. Simulated annealing Basic ideas: like hill-climbing identify the quality of the local improvements instead of picking the best move, pick one randomly say the change in objective function is Δf if Δf is positive, then move to that state otherwise: move to this state with probability proportional to Δf thus: worse moves (very large negative Δf ) are executed less often

8. 8 SA() 1 choose an initial solution X 0 randomly 2 give an initial temperature T 0, X ← X 0, T ← T 0 3 while the stop criterion is not yet satisfied do 4 for i ← 1 to L do 5 pick a solution X' ∈ N(X) randomly 6 Δf ← f(X')-f(X) 7 if Δf<0 then X ← X' 8 else X ← X' with probability exp(- Δf/T) 9 T← g(T) //generally, T ← aT 10 return X

9. Generic choices for annealing schedule initial temperature T 0 (example: based on statistics of evaluation function) Cooling schedule-how to change temperature over time (example: geometric cooling, T ← aT) L: number of iterations at each temperature (example :multiple of the neighborhood size Stopping criterion (example: no improved solution found for a number of temperature values)

10. Simulated Annealing in Practice method proposed in 1983 by IBM researchers for solving VLSI layout problems (Kirkpatrick et al, Science, 220:671-680, 1983). theoretically will always find the global optimum (the best solution) useful for some problems, but can be very slow slowness comes about because T must be decreased very gradually to retain optimality In practice how do we decide the rate at which to decrease T? (this is a practical problem with this method)

11. 11 Tabu search Tabu Search (TS) is a metaheuristic which is concerned with imposing restrictions to guide a search process. These restrictions operate in several forms both by direct exclusion of search alternatives classed as tabu and by modifying evaluations and probabilities of selection of such alternatives Tabu search begins in the same way as ordinary local or neighborhood search, proceeding iteratively from one solution to another until a chosen termination criterion is satisfied. Each X ∈ S has an associated neighborhood N(X)  S, and each solution X' ∈ N(X) is reached from X by an operation called a move.

12. 12 Tabu search TS( ) 1 choose an initial solution X 2 X* ← X (record the best solution found so far) 3 while the stop criterion is not yet satisfied do 4 N*(X) ←{X' ∈ N(X) |the move from X to X' is not tabu OR X' satisfies the aspiration criterion} 5 choose the best neighbor X' ∈ N(X) 6 X ← X' //even if f(X') > f(X) 7 if f(X) < f(X*) then X* ← X 8 return X*

13. Elements of Tabu Search Neighborhood structure stop criterion Tabu List(short term memory) Tabu tenure Aspiration criteria Long term memory attributes

14. Elements of TS: Recency Memory related - recency (How recent the solution has been reached) Tabu List (short term memory): to record a limited number of attributes of solutions (moves, selections, assignments, etc) to be discouraged in order to prevent revisiting a visited solution; Tabu tenure (length of tabu list): number of iterations a tabu move is considered to remain tabu;

15. Elements of Tabu Search Memory related – recency (How recent the solution has been reached) Tabu tenure List of moves does not grow forever – restrict the search too much Restrict the size of list FIFO Other ways: dynamic

16. Elements of Tabu Search Long term memory: to record attributes of elite solutions to be used in: Intensification: giving priority to attributes of a set of elite solutions (usually in weighted probability manner) Diversification: Discouraging attributes of elite solutions in selection functions in order to diversify the search to other areas of solution space; trade-off

17. Memory related: frequency observe frequency of selected attributes Penalization of moves Making use of frequency can modify the evaluation function so that the attributes with less frequency are given the bigger evaluation value

18. Elements of TS: Aspiration If a move is good, but it’s tabu-ed, do we still reject it? Aspiration criteria: accepting an improving solution even if generated by a tabu move Similar to SA in always accepting improving solutions, but accepting non-improving ones when there is no improving solution in the neighbourhood;

19. Adaptive Tabu Search Tabu tenure denotes an attribute is tabu in recent t iteration If t (tenure) to small, we will return to the same local min Adaptively modify t If we see the same local min, increase t When we see evidence that local min escaped (e.g. improved sol), lower t

20. Path Relinking Basic idea: Given 2 good solutions, perhaps a better solution lies somewhere in-between Try to combine “ good features ” from two solutions Gradually convert one solution to the other

21. Applications Simulated annealing for optimization problem 3-CNF SAT, Strip packing, Vehicle Routing, … Tabu search for optimization problem 3-CNF SAT, Strip packing, Vehicle Routing, …

22. 22 Homework Experiments(1 or 2): 1. Implement TS or SA for strip packing problem 2. Implement TS or SA for 3-CNF SAT problem Download the following paper to read Qisen Cai, Defu Zhang, Wei Zheng, Stephen C.H. Leung. A new fuzzy time series forecasting model combined with ant colony optimization and auto-regression. Knowledge-Based Systems. 74(1) (2015) 61–68.