Artificial Intelligence Heuristic Search Methods

Transcription

1 Artificial Intelligence Heuristic Search Methods Chung-Ang University, Jaesung Lee The original version of this content is created by School of Mathematics, University of Birmingham professor Sandor Zoltan Nemeth. URL:

2 Hill-Climbing Search Why we use Heuristic? Circumvent combinatorial explosion Good approximations are usually sufficient The worst cases occur very infrequently More understanding the problem Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 2

3 Neighborhoods TSP is an optimization problem. Solutions which are close to a given solution Two or more solutions with similar representation Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 3

4 Neighborhoods, = 0 That is the distance of a point from itself is zero., =, This property is called symmetry. Our TSP problem is symmetric TSP.,, +, This property is called triangular inequality. Our TSP problem does not hold this. Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 4

5 Neighborhoods, = 0, =,,, +, = {, < } Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 5

6 Algorithm 2-opt for TSP Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 6

7 Neighborhood as a Mapping 2-swap mapping 2-interchange mapping Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 7

8 Local Optima Local optimum w.r.t. ( ) if, Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 8

9 Hill-climbing analogy Height: the quality of a node or solution based on evaluation function Peaks: (possibly local) optimal solutions Orientation: evaluating neighboring positions Hill-climbing: the problem of orientation and moving on a surface Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 9

10 Basic Hill-climbing algorithm Evaluate the initial point. If the objective is met, return. Otherwise set as the current point. Loop until a solution is found: Generate all the neighbors of. Select the best one (or randomly one of the best ones). If x is at least as good as, then return, else set as the current point. Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 10

11 Properties of hill-climbing Stuck in local minimum Plateaus No information about global optimum Affected by initial solution Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 11

12 Simulated Annealing Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 12

13 Motivation 1980 s Kirkpatrick, Gelatt, and Vecchi and independently Cerny Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 13

14 Analogy with physical annealing Physical system State Energy Ground state Temperature Careful annealing Optimization problem Feasible solution Evaluation function value Optimal solution Control parameter Simulated annealing Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 14

15 Hill-Climbing and Simulated Annealing Iterative Hill-Climbing 1. Choose a starting solution _ and initialize 2. := 0; := _ 3. Repeat until = : (a) Repeat until a local optimum is found: i. Select as the neighbor of with best value of the evaluation function ( ) ii. If ( ) is better than ( ) then := else a local optimum is found (b) := +1 (c) if is better than best then := (d) := new random solution Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 15

16 Hill-Climbing and Simulated Annealing Stochastic Hill-Climbing Modification of Iterative Hill-Climbing Selecting just one point from neighborhood Accept the new point with probability depending on the relative merit Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 16

17 Hill-Climbing and Simulated Annealing Stochastic Hill-Climbing 1. Choose a starting solution and evaluate it 2. = 0; = 3. Repeat until = : (a) select as a neighbor of (b) = with probability (c) = + 1 Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 17

18 Hill-Climbing and Simulated Annealing Stochastic Hill-Climbing (Example) For example, ( ) = 107, ( ) = 120 Acceptance Prob. = If = 1 ( is very small), acceptance probability is close to 1. As increases, the probability of acceptance decreases. If = 10 ( is very large) the probability of acceptance is 0.5. Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 18

19 Basic Structure of Simulated Annealing Pseudocode 1. Choose a starting solution 2. Initialize, 2. = 0; = 3. Repeat until halting criterion is satisfied: (a) Repeat times: i. Generate as a neighbor of ii. If ( ) h ( ) then = elseif exp > (0,1) then = (b) = + 1 (c) Calculate (d) Calculate the temperature Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 19

20 Properties of Simulated Annealing Simulated Annealing accepts some deterioration in the quality of solutions High temperature: Large deteriorations are accepted Low temperature: Small deteriorations are accepted 0 Temperature: Local optimization Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 20

21 SA for TSP Not so different The starting point: random solution or output of a local search should be proportional to the neighborhood size Post-processing may be helpful Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 21

22 Tabu Search 1986 s Glover and Hansen Tabu search is: a meta-heuristic superimposed on another heuristic. The overall approach is to avoid entrapment in cycles by forbidding or penalizing moves which take the solution, in the next iteration, to points in the solution space previously visited (hence tabu). Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 22

23 Basic features Accepts non-improving solutions deterministically Prevent the search from revisiting previously visited solutions Explore the unvisited areas of the solution space Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 23

24 Tabu search for TSP Example of Tabu List 6 7 Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 24

25 Tabu search for TSP Recency-based Memory Current tour: (7, 3, 5, 6, 1, 2, 4, 8) after 500 iterations Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 25

26 Tabu search for TSP Frequency-based Memory Current tour: (7, 3, 5, 6, 1, 2, 4, 8) after 500 iterations Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 26

27 Tabu search for TSP Neighborhood based on 2-interchange Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 27

28 Tabu search based on 2-interchange Sketch 1. Generate a tour 2. Repeat ITER times (a) identify a set of 2-interchange moves (b) select and make a 2-interchange (c) update tabu list (d) update local best tour info 3. Update global best tour info Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 28

29 Advantages and drawbacks of Tabu search Advantage Tabu search generally finds good solutions. Shortcomings Tabu list construction is problem specific. There is no guarantee of finding a global optimum. Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 29

30 Comparison to Simulated Annealing Both were designed to escape local optima and work on complete solutions. Tabu search only selects worse moves if it is stuck in a local optimum, whereas simulated annealing can do that all the time. Simulated annealing is stochastic, but Tabu search is deterministic. The parameters must be carefully chosen for both. Artificial Intelligence / Chung-Ang University / Professor Jaesung Lee 30