1. Heuristic Optimization Methods Scatter Search

2. 2 Agenda Scatter Search (SS) –For Local Search based Metaheuristics: SA based on ideas from nature TS based on problem-solving and learning –For population based Metaheuristics: GA based on ideas from nature SS based on problem-solving and learning –Nature works, but usually very slowly –Being clever is better than emulating nature?

3. Scatter Search: Methodology and Applications Manuel Laguna University of Colorado Rafael Martí University of Valencia The following is a presentation previously held at the conference ICS 2003...

4. 4 Based on … Laguna, M. and R. Martí (2003) Scatter Search: Methodology and Implementations in C, Kluwer Academic Publishers, Boston.

5. Scatter Search Methodology

6. 6 Metaheuristic A metaheuristic refers to a master strategy that guides and modifies other heuristics to produce solutions beyond those that are normally generated in a quest for local optimality. A metaheuristic is a procedure that has the ability to escape local optimality

7. 7 Typical Search Trajectory

8. 8 Metaheuristic Classification x/y/z Classification –x = A (adaptive memory) or M (memoryless) –y = N (systematic neighborhood search) or S (random sampling) –Z = 1 (one current solution) or P (population of solutions) Some Classifications –Tabu search (A/N/1) –Genetic Algorithms (M/S/P) –Scatter Search (M/N/P)

9. 9 Scatter Search P Diversification Generation Method Repeat until |P| = PSize Subset Generation Method Improvement Method Solution Combination Method Improvement Method Stop if no more new solutions Reference Set Update Method RefSet

10. 10 Repeat until |P| = PSize Scatter Search with Rebuilding P Diversification Generation Method Subset Generation Method Improvement Method Solution Combination Method Improvement Method No more new solutions Reference Set Update Method RefSet Diversification Generation Method Improvement Method Stop if MaxIter reached

11. 11 Tutorial Unconstrained Nonlinear Optimization Problem

12. 12 Diversification Generation Method -10+10-50+5 Subrange 1Subrange 2Subrange 3Subrange 4 Probability of selecting a subrange is proportional to a frequency count

13. 13 Diverse Solutions

14. 14 Improvement Method Nelder and Mead (1965)

15. 15 Reference Set Update Method (Initial RefSet) RefSet of size b b 1 high-quality solutions b 2 diverse solutions Min-max criterion and Euclidean distances to measure diversity Objective function value to measure quality

16. 16 Initial RefSet High-Quality Solutions Diverse Solutions

17. 17 Subset Generation Method All pairs of reference solutions that include at least one new solution The method generates (b 2 -b)/2 pairs from the initial RefSet

18. 18 Combination Method

19. 19 Alternative Combination Method

20. 20 Reference Set Update Method RefSet of size b Worst BestQuality1 b 2...... New trial solution Updated RefSet Worst Best1 b 2......

21. 21 Static Update RefSet of size b Worst BestQuality1 b 2...... Pool of new trial solutions Updated RefSet = Best b from RefSet  Pool

22. 22 RefSet after Update x1x1 x2x2 x3x3 x4x4 f(x)f(x) 1.13831.29650.83060.7150.14 0.70160.52971.20781.46330.36 0.52690.2871.26451.60770.59 1.19631.39680.68010.4460.62 0.33260.10311.36321.83110.99 0.33680.10991.38181.93891.02 0.31270.09491.35121.85891.03 0.75920.5231.31391.71951.18 0.20040.03441.40371.94381.24 1.38921.93050.1252-0.01521.45

23. 23 Additional Strategies Reference Set –Rebuilding –Multi-tier Subset Generation –Subsets of size > 2 Combination Method –Variable number of solutions

24. 24 Rebuilding RefSet Rebuilt RefSet b1b1 b2b2 Diversification Generation Method Reference Set Update Method

25. 25 2-Tier RefSet RefSet b1b1 b2b2 Solution Combination Method Improvement Method Try here first If it fails, then try here

26. 26 3-Tier RefSet RefSet b1b1 b2b2 Solution Combination Method Improvement Method Try here first If it fails, then try here b3b3 Try departing solution here

27. 27 Subset Generation Subset Type 1: all 2-element subsets. Subset Type 2: 3-element subsets derived from the 2- element subsets by augmenting each 2-element subset to include the best solution not in this subset. Subset Type 3: 4-element subsets derived from the 3- element subsets by augmenting each 3-element subset to include the best solutions not in this subset. Subset Type 4: the subsets consisting of the best i elements, for i = 5 to b.

28. 28 Subsets of Size > 2

29. 29 Variable Number of Solutions RefSet of size b Worst BestQuality1 b 2...... Generate 5 solutions Generate 3 solutions Generate 1 solution

30. 30 Hybrid Approaches Use of Memory –Tabu Search mechanisms for intensification and diversification GRASP Constructions Combination Methods –GA Operators –Path Relinking

31. 31 Multiobjective Scatter Search This is a fruitful research area Many multiobjective evolutionary approaches exist (Coello, et al. 2002) SS can use similar techniques developed for MOEA (multiobjective evolutionary approches)

32. 32 Multiobjective EA Techniques Independent Sampling –Search on f(x) =  w i f i (x) –Change weights and rerun Criterion Selection –Divide reference set into k subsets –Admission to i th subset is according to f i (x)

33. 33 Advanced Designs Reference Set Update –Dynamic / Static –2 Tier / 3 Tier Subset Generation Use of Memory –Explicit Memory –Attributive Memory Path Relinking

34. 34 An Example The Linear Ordering Problem Given a matrix of weights E = { e ij } mxm, the LOP consists of finding a permutation p of the columns (and rows) in order to maximize the sum of the weights in the upper triangle Applications Triangulation for Input-Output Economic Tables. Aggregation of individual preferences Classifications in Sports Maximize

35. 35 An Instance 12341234 1 2 3 4 34123412 3 4 1 2 p=(1,2,3,4) c E (p)=12+5+3+2+6+9=37 p*=(3,4,1,2) c E (p*)=9+8+3+11+4+12=47

36. 36 Diversification Generator Use of problem structure to create methods in order to achieve a good balance between quality and diversity. Quality –Deterministic constructive method Diversity –Random Generator –Systematic Generators (Glover, 1998) GRASP constructions. –The method randomly selects from a short list of the most attractive sectors. Use of Memory –Modifying a measure of attractiveness proposed by Becker with a frequency-based memory measure that discourages sectors from occupying positions that they have frequently occupied.

37. 37 Diversity vs. Quality  d = Standardized Diversity  C = Standardized Quality Compare the different generators Create a set of 100 solutions with each one

38. 38 Improvement Method INSERT_MOVE ( p j, i ) consist of deleting p j from its current position j to be inserted in position i Apply a first strategy –scans the list of sectors in search for the first sector whose movement results in an improvement MoveValue = C E (p’) - C E (p) CE (p’) = 78 + (1 - 4) + (6 - 0) + (2 - 6) + (13 - 4) = 78 + 8 = 86

39. 39 Solution Combination Method The method scans (from left to right) each reference permutation. –Each reference permutation votes for its first element that is still not included in the combined permutation (“incipient element”). –The voting determines the next element to enter the first still unassigned position of the combined permutation. –The vote of a given reference solution is weighted according to the incipient element’s position. Incipient element (3,1,4,2,5)votes for 4Solution under construction: (1,4,3,5,2)votes for 4(3,1,2,4,_ ) (2,1,3,5,4)votes for 5

40. 40 Experiments with LOLIB GDCKCK10TSSS Optima deviation 0.15 % 0.02%0.04%0.01% Number of optima 11 273342 Run time (seconds) 0.010.101.060.492.35 49 Input-Output Economic Tables

41. 41 Another Example A commercial SS implementation OptQuest Callable Library (by OptTek) As other context-independent methods separates the method and the evaluation.

42. 42 OptQuest based Applications Solution Generator Solution Evaluator

43. 43 Feasibility and Evaluation The OptQuest engine generates a new solution User Implementation Returns to OptQuest

44. 44 Comparison with Genocop Average on 28 hard nonlinear instances

45. 45 Conclusions The development of metaheuristics usually entails a fair amount of experimentation (“skill comes from practice”). Code objectives: –Quick Start –Benchmark –Advanced Designs Scatter Search provides a flexible “framework” to develop solving methods.

46. 46 Metaheuristic Classification x/y/z Classification –x = A (adaptive memory) or M (memoryless) –y = N (systematic neighborhood search) or S (random sampling) –Z = 1 (one current solution) or P (population of solutions) Some Classifications –Tabu search (A/N/1) –Genetic Algorithms (M/S/P) –Scatter Search (M/N/P)

47. 47 Some Classifications Simulated Annealing M/S/1 Tabu Search A/N/1 Genetic Algorithm M/S/P Scatter Search M/N/P (randomized) (systematic) (local search) (population)

48. 48 About the Classifications Our four main methods (SA, TS, GA, SS) all belong far from the center (they are very randomized or very systematic) Other methods have both some element of randomized and some element of systematic behaviour Most implementations will mix the ingredients, and we have an element of local search in population based methods (e.g., Memetic Algorithms), or an element of randomness in systematic approaches (such as random tabu tenure in TS) The classifications highlight the differences between methods, but there are also many similarities

49. 49 GA vs. SS (1) GA has a ”long” history: proposed in the 1970s, and immediately becoming popular –Not initially used for optimization –Gradually morphed into a metodology whose major concern is the solution of optimization problems The concepts and principles of SS was also proposed early (1970s), but was not popularized until the 1990s –The SS template most often used is from 1998 –Propsed to solve Integer Programming problems

50. 50 GA vs. SS (2) GA is based on natural processes (genetics, the ”survival of the fittest”, and imitation of the nature) SS is based on strategic ideas for how to use adaptive memory –Some TS concepts are critically linked with SS

51. 51 GA vs. SS (3) Diversification –GA: mutation –SS: favoring diverse solutions in the reference set, and generating diverse solutions in the initialization Intensification –GA: probabilistic selection of parents, favoring the fittest parents (but this is not really very intensifying) –SS: the improvement method

52. 52 GA vs. SS (4) GA has a population that is usually 10x larger than the reference set in SS GA applies operators (mutation, crossover) to random solutions, SS applies operators (combination, improvement) non-randomly Evolution in GA follows random ”survival of the fittest”, in SS there are deterministic rules in the reference set update method

53. 53 GA vs. SS (5) Use of Local Search (improvement) is an integral part of SS, but added to GA only to create hybrid/improved approaches GA usually limited to combine only a pair of solutions (parents), while SS allows combination of any number of solutions GA uses full randomization to create initial population, while SS balances diversity and quality (diversification generation method)

54. 54 Summary of Todays’s Lecture Scatter Search –M/N/P (memoryless, systematic ”neighborhood”, population of solutions) Components of Scatter Search: –Diversification Generation –Improvement –Reference Set Update –Subset Generation –Solution Combination