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Heuristic Optimization Methods Scatter Search 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? 2 The following is a presentation previously held at the conference ICS 2003... Scatter Search: Methodology and Applications Manuel Laguna University of Colorado Rafael Martí University of Valencia Based on … Laguna, M. and R. Martí (2003) Scatter Search: Methodology and Implementations in C, Kluwer Academic Publishers, Boston. 4 Scatter Search Methodology 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 6 Typical Search Trajectory 100 90 Objective Function 80 70 60 Value 50 Best Value 40 30 20 10 0 1 4 7 10 13 16 19 22 25 28 31 34 37 40 Iteration 7 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) 8 Scatter Search P Diversification Generation Method Repeat until |P| = PSize Improvement Method Improvement Method Reference Set Update Method Solution Combination Method Subset Generation Method RefSet Stop if no more new solutions 9 Scatter Search with Rebuilding P Diversification Generation Method Repeat until |P| = PSize Improvement Method Improvement Method Solution Combination Method Reference Set Update Method Stop if MaxIter reached Improvement Method Subset Generation Method RefSet No more new solutions Diversification Generation Method 10 Tutorial • Unconstrained Nonlinear Optimization Problem Minimize 100 x 2 2 2 x1 10.1 x 2 1 x 4 1 Subject to 2 10 x i 10 1 x1 90 x 4 2 2 19.8x 2 2 2 x3 1 x3 2 1x 4 1 for i 1, ,4 11 Diversification Generation Method Subrange 1 -10 Subrange 2 -5 Subrange 3 0 Subrange 4 +5 +10 Probability of selecting a subrange is proportional to a frequency count 12 Diverse Solutions Solution x1 x2 x3 x2 f(x) 1 2 3 4 5 6 7 8 9 10 1.11 -9.58 7.42 8.83 -6.23 1.64 0.2 -3.09 -6.08 -1.97 0.85 -6.57 -1.71 -8.45 7.48 3.34 -3.64 6.62 0.67 8.13 9.48 -8.81 9.28 4.52 6 -8.32 -5.3 -2.33 -6.48 -5.63 -6.35 -2.27 5.92 3.18 7.8 -8.66 -7.03 -3.12 1.48 8.02 835546.2 1542078.9 901878.0 775470.7 171450.5 546349.8 114023.8 7469.1 279099.9 54537.2 13 Improvement Method Solution 1 2 3 4 5 6 7 8 9 10 x1 x2 x3 x2 f(x) 2.5 -0.52 -2.6 0.49 -3.04 -1.4 -0.36 -1.63 -0.8 -2.47 5.4 -0.5 5.9 0.53 9.45 2.46 -0.31 2.51 0.69 5.32 2.59 0.35 4.23 2.47 1.14 0.37 0.8 0.73 -1.16 -2.92 5.67 -0.14 10 5.89 0.41 -3.94 1.14 0.56 1.5 8.15 1002.7 138.5 7653.7 213.7 720.1 1646.7 57.1 21.5 11.2 1416.7 Nelder and Mead (1965) 14 Reference Set Update Method (Initial RefSet) b1 high-quality solutions Objective function value to measure quality b2 diverse solutions Min-max criterion and Euclidean distances to measure diversity RefSet of size b 15 Initial RefSet High-Quality Solutions Solution number in P 35 46 34 49 38 x1 x2 x3 x4 -0.0444 1.133 -0.0075 1.1803 1.0323 0.0424 1.2739 0.0438 1.4606 0.9719 1.3577 -0.6999 1.4783 -0.344 -0.8251 1.8047 0.5087 2.2693 0.2669 0.695 f(x) 2.1 3.5 3.5 5.2 5.3 Diverse Solutions Solution 37 30 45 83 16 x1 -3.4331 3.8599 -4.4942 -0.2414 6.1626 x2 10 10 10 -6.5307 10 x3 1.0756 -4.0468 3.0653 -0.9449 0.1003 f(x) x4 0.3657 10 10 -9.4168 0.1103 1104.1 9332.4 13706.1 17134.8 78973.2 16 Subset Generation Method • All pairs of reference solutions that include at least one new solution • The method generates (b2-b)/2 pairs from the initial RefSet 17 Combination Method y x3 = x1 - r(x2 - x1) x3 = (9,7) r = 2/3 10 x4 = x1 + r(x2 - x1) x5 = x2 + r(x2 - x1) 9 x1 = (5,7) 8 7 6 5 x4 = (6.5,5.5) r = 1/2 x2 = (8,4) 4 3 x5 = (11,1) r=1 2 1 1 2 3 4 5 6 7 8 9 10 11 12 13 x 18 Alternative Combination Method y x3 = x1 - r(x2 - x1) 10 x4 = x1 + r(x2 - x1) x5 = x2 + r(x2 - x1) 9 x3 x1 = (5,7) 8 7 6 x4 5 x2 = (8,4) 4 3 x5 2 1 1 2 3 4 5 6 7 8 9 10 11 12 13 x 19 Reference Set Update Method Quality 1 2 . . . Best 1 2 . . . Best b Worst New trial solution b Worst RefSet of size b Updated RefSet 20 Static Update Pool of new trial solutions Quality 1 2 . . . Best Updated RefSet = Best b from RefSet Pool b Worst RefSet of size b 21 RefSet after Update x1 x2 x3 x4 f(x) 1.1383 1.2965 0.8306 0.715 0.14 0.7016 0.5297 1.2078 1.4633 0.36 0.5269 0.287 1.2645 1.6077 0.59 1.1963 1.3968 0.6801 0.446 0.62 0.3326 0.1031 1.3632 1.8311 0.99 0.3368 0.1099 1.3818 1.9389 1.02 0.3127 0.0949 1.3512 1.8589 1.03 0.7592 0.523 1.3139 1.7195 1.18 0.2004 0.0344 1.4037 1.9438 1.24 1.3892 1.9305 0.1252 -0.0152 1.45 22 Additional Strategies • Reference Set – Rebuilding – Multi-tier • Subset Generation – Subsets of size > 2 • Combination Method – Variable number of solutions 23 Rebuilding RefSet Rebuilt RefSet b1 b2 Diversification Generation Method Reference Set Update Method 24 2-Tier RefSet Solution Combination Method Improvement Method RefSet Try here first b1 If it fails, then try here b2 25 3-Tier RefSet Solution Combination Method Improvement Method RefSet Try here first b1 If it fails, then try here b2 Try departing solution here b3 26 Subset Generation • • • • Subset Type 1: all 2-element subsets. Subset Type 2: 3-element subsets derived from the 2element 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 3element 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. 27 Subsets of Size > 2 90% 80% 70% 60% 50% 40% 30% 20% 10% Random 0% Type 1 Type 2 LOLIB Type 3 Type 4 28 Variable Number of Solutions Quality 1 2 . . . Best Generate 5 solutions Generate 3 solutions Generate 1 solution b Worst RefSet of size b 29 Hybrid Approaches • Use of Memory – Tabu Search mechanisms for intensification and diversification • GRASP Constructions • Combination Methods – GA Operators – Path Relinking 30 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) 31 Multiobjective EA Techniques • Independent Sampling – Search on f(x) = wi fi(x) – Change weights and rerun • Criterion Selection – Divide reference set into k subsets – Admission to ith subset is according to fi(x) 32 Advanced Designs • Reference Set Update – Dynamic / Static – 2 Tier / 3 Tier • Subset Generation • Use of Memory – Explicit Memory – Attributive Memory • Path Relinking 33 An Example The Linear Ordering Problem Given a matrix of weights E = {eij}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 m1 m CE ( p) e pi p j Maximize i 1 j i 1 Applications Triangulation for Input-Output Economic Tables. Aggregation of individual preferences Classifications in Sports 34 An Instance 1 1 2 3 4 2 0 12 4 0 8 3 11 4 3 4 5 3 2 6 0 9 2 0 p=(1,2,3,4) cE(p)=12+5+3+2+6+9=37 3 3 4 1 2 4 1 2 0 9 8 3 2 0 11 4 5 3 0 12 2 6 4 0 p*=(3,4,1,2) cE(p*)=9+8+3+11+4+12=47 35 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. 36 Diversity vs. Quality Compare the different generators Create a set of 100 solutions with each one 1.6 d 1.4= Standardized Diversity C 1.2 = Standardized Quality C+ d 1.0 0.8 0.6 d 0.4 C 0.2 0.0 DG05 DG04 DG02 DG01 DG09 DG08 DG06 DG03 DG07 Procedure 37 DG10 Improvement Method • INSERT_MOVE (pj, i) consist of deleting pj 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 1 0 12 5 3 6 0 3 6 4 8 5 0 5 7 E ( p) 2 7 2 0 3 8 0 3 1 0 1 6 2 13 9 2 9 4 5 8 8 4 0 6 4 0 1 3 2 3 0 1 4 0 MoveValue = CE(p’) - CE(p) CE (p’) = 78 + (1 - 4) + (6 - 0) + (2 - 6) + (13 - 4) = 78 + 8 = 86 38 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 4 (1,4,3,5,2) votes for 4 (2,1,3,5,4) votes for 5 Solution under construction: (3,1,2,4,_ ) 39 Experiments with LOLIB 49 Input-Output Economic Tables GD CK CK10 TS SS Optima deviation 0.15 % 0.15% 0.02% 0.04% 0.01% Number of optima 11 11 27 33 42 Run time (seconds) 0.01 0.10 1.06 0.49 2.35 40 Another Example A commercial SS implementation OptQuest Callable Library (by OptTek) As other context-independent methods separates the method and the evaluation. Output Optimization Procedure Input System Evaluator 41 OptQuest based Applications Solution Generator OptQuest Callable Library User-written Application Solution Evaluator System Evaluator 42 Feasibility and Evaluation User Implementation The OptQuest engine generates a new solution Returns to OptQuest 43 Comparison with Genocop • Average on 28 hard nonlinear instances 1.0E+13 Average objective function value (Logarithmic scale) 1.0E+12 Genocop OCL 1.0E+11 1.0E+10 1.0E+09 1.0E+08 1.0E+07 1.0E+06 1.0E+05 1.0E+04 1.0E+03 0 1000 2000 3000 4000 5000 6000 Evaluations 7000 8000 9000 10000 44 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. 45 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) 46 Some Classifications (local search) Tabu Search A/N/1 Simulated Annealing M/S/1 (systematic) (randomized) Genetic Algorithm M/S/P Scatter Search M/N/P (population) 47 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 48 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 49 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 50 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 51 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 52 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) 53 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 54