Heuristic Optimisation

Transcription

1 Heuristic Optimisation Part 2: Basic concepts Sándor Zoltán Németh University of Birmingham S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 1 / 20

2 Overview Representation Objective Evaluation function Optimisation problem definition Neighbourhoods and local optima The hill-climbing method S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 2 / 20

3 Introduction Problem Model Solution For any algorithmic approach: The representation - the encoding of alternative solutions The objective - the purpose (may or may not be minimising one function) The evaluation function - how good a solution (given the representation) is S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 3 / 20

4 Representation SAT problem: n binary variables a bit string of length n Search space has size 2 n, each point corresponds to a feasible solution TSP: A permutation of natural numbers 1, 2,..., n (each number corresponds to one city) Search space has size (n 1)!/2 for symmetric TSP NLP: All real numbers in n dimensions Infinite search space; for approximate representation with precision of six digits size 10 7n S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 4 / 20

5 Representation cont d The size of the search space is determined by the representation and its corresponding interpretation The problem itself does not determine the search space size Choosing the right representation is of utmost importance! S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 5 / 20

6 Objective The mathematical statement of the task SAT: find the bit vector for which the Boolean statement evaluates to TRUE TSP: minimise the total distance, when visiting each city exactly once and returning to the starting city min dist(x, y) NLP: minimise or maximise a nonlinear function S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 6 / 20

7 Evaluation function Generally it is not the same thing as the objective The evaluation function is a mapping from the space of (feasible) solutions to a set of numbers (reals). The numeric value indicates quality of solution TSP: tour the sum of distances along the route - exact Sometimes it is sufficient to know whether one solution is better than the other, without knowing how much better S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 7 / 20

8 How to choose the evaluation function? The evaluation function is not given with the problem, only the objective is A solution that meets the objective should also have the best evaluation A solution that fails to meet the objective cannot be evaluated to be better than a solution that meets the objective The objective may suggest the evaluation function (TSP, NLP) or may not (SAT) The feasible region of the search space must be considered S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 8 / 20

9 Defining the optimisation problem Search problem = optimisation problem Given a search space S together with its feasible part F S, find x F such that eval(x) eval(y), y F (minimisation) x is a global solution S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 9 / 20

10 Reasons for using heuristics We can avoid combinatorial explosion We rarely need the optimal solution, good approximations are usually sufficient The approximations may not be very good in the worst case, but in reality worst cases occur very rarely Trying to understand why a heuristic works/does not work leads to a better understanding of the problem S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 10 / 20

11 Neighbourhoods dist : S S R N(x) = {y S : dist(x, y) < ε}, given ε 0 NLP: Euclidean distance dist(x, y) = n i=1(x i y i ) 2 SAT: Hamming distance (number of bit positions with different truth assignments) S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 11 / 20

12 Neighbourhood as mapping Alternatively the neighbourhood can be defined as a mapping m : S 2 S TSP: 2-swap mapping generates for any potential solution x the set of potential solutions obtained by swapping two cities in x SAT: 1-flip mapping (flipping one bit) S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 12 / 20

13 Local optima A potential solution x F is a local optimum with respect to neighbourhood N(x) iff eval(x) eval(y), y N(x) Local search methods operate on neighbourhoods: They try to modify the current solution x into a better one within its neighbourhood N(x) S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 13 / 20

14 Hill climbing S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 14 / 20

15 Hill climbing analogy Orientation and moving on a surface Height: the quality of a node (the evaluation function) Peaks: optimal solutions Orientation: evaluating neighbouring positions S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 15 / 20

16 Basic hill climbing algorithm 1. Evaluate the initial point x. If the objective is met, return x. Otherwise set x as the current point. 2. Loop until a solution is found: (a) Generate all the neighbours of x. Select the best one y. (b) If y is not better than x return x else set y as the current point. For better efficiency, the algorithm can be restarted a predefined number of times (iterative hill climbing) S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 16 / 20

17 Example of hill climbing S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 17 / 20

18 Example of hill climbing (5,1) 7 Hill climbing 6 (4,1) (5,2) 6 S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 17 / 20

19 Example of hill climbing (5,1) 7 Hill climbing (5,2) 6 S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 17 / 20

20 Example of hill climbing (5,1) 7 Hill climbing (5,2) 6 (4,2) 5 S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 17 / 20

21 Example of hill climbing (5,1) 7 Hill climbing (5,2) 6 (4,2) 5 6 (4,1) (4,3) 4 S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 17 / 20

22 Example of hill climbing (5,1) 7 Hill climbing (5,2) 6 (4,2) 5 (4,3) 4 S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 17 / 20

23 Example of hill climbing (5,1) 7 Hill climbing (5,2) 6 (4,2) 5 (4,3) 4 3 (3,3) (4,4) 3 S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 17 / 20

24 Example of hill climbing (5,1) 7 Hill climbing (5,2) 6 (4,2) 5 (4,3) 4 (4,4) 3 S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 17 / 20

25 Example of hill climbing (5,1) 7 Hill climbing (5,2) 6 (4,2) 5 (4,3) 4 (4,4) 3 2 (3,4) (5,4) 4 S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 17 / 20

26 Example of hill climbing (5,1) 7 Hill climbing (5,2) 6 (4,2) 5 (4,3) 4 (4,4) 3 2 (3,4) S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 17 / 20

27 Properties of hill climbing Can get stuck in local maxima backtrack Plateaus (areas of search space where the evaluation function is flat) are a problem big jumps There is no information about how far the found local optimum from the global optimum is The solution depends on the initial configuration S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 18 / 20

28 Summary Choosing the right representation is essential for success The evaluation function must also be carefully chosen Effective search balances exploitation of the best solutions so far and exploration of the search space Hill climbing is an easy local search method that is good at exploitation but neglects exploration Random search is good at exploration, but does no exploitation S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 19 / 20

29 Recommended reading Z. Michalewicz & D.B. Fogel How to Solve It: Modern Heuristics Chapter 2. Basic concepts S Z Németh (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 20 / 20