Kyriakos C. Giannakoglou. Associate Professor, Lab. of Thermal Turbomachines (LTT), National Technical University of Athens (NTUA), GREECE

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1 EVOLUTIONARY ALGORITHMS: What are EAs? Mathematical Formulation & Computer Implementation Multi-objective Optimization, Constraints Computing cost reduction Kyriakos C. Giannakoglou Associate Professor, Lab. of Thermal Turbomachines (LTT), National Technical University of Athens (NTUA), GREECE

2 Outline From traditional problem solving techniques to EAs. Generalized EA: Basic and advanced operators. Mathematical foundations of EAs. EAs for multi-objective optimization. Distributed Evolutionary Algorithms (DGAs). Hierarchical Evolutionary Algorithms (HGAs). Constraints handling. Efficient ways for reducing the computing cost of EAs. Applications in the field of aeronautics, turbomachinery, energy production, logistics.

3 Effective/Efficient Problem Solving Techniques The number of possible solutions in the search space is so large as to forbid exhaustive search Seeking the best combination of approaches that addresses the purpose to be achieved Finding the solution using the available computing resources Finding the solution within the available time One or more (contradictory) targets Solving the problem under a number of (hard/soft) constraints

4 Basic Concepts of Problem Solving Techniques Representation: how to encode alternative candidate solutions for manipulation Objective: describes the purpose to be fulfilled Evaluation function: returns a value that indicates the (numeric of ordinal) quality of any particular solution, given the representation

5 Hill-Climbing: A Traditional (Deterministic) PST Useful Definitions: Neighborhood of a solution Local Optimum x 2 Requirements: A starting point Computation of gradient Termination criteria Ideas for creating Hill-Climbing Method variants How to select the new solution for comparison with the current solution (how to compute the gradient) To use more than one starting solutions, if necessary To be less deterministic x 1

6 Algorithms Relying on Analogies to Natural Processes Evolutionary Programming Genetic Algorithms Evolution Strategies Simulated Annealing Classifier Systems Neural Networks Traditional Problem Solving Techniques Algorithms with analogies to Natural Processes

7 The Subclass we are interested in Methods which are based on the principle of evolution (i.e. the survival of the fittest) Handling populations of candidate solutions Undergoing unary (mutation-type) operations Undergoing higher-order (crossover-type) operations Using a selection scheme biased towards fitter individuals Generation Generation Generation Generation

8 EA Schematic Presentation: g g = 0, S = S random g y = F( x), s S g, μ φ( y), s S g+ 1 g S = T3( T2( T1( S ))) g = g + 1 if( converge( g)) end

9 EA Prerequisites: Objective Function Fitness Function Parameterization & Representation y = F( x) φ( y) g+ 1 g S = T3( T2( T1( S ))) if( converge( g)) Evolution Operators Stopping Criterion

10 Encoding the free variables Binary Coding: b m, m=1,m xm x Gene Candidate solution Binary digits per variable Chromosome: bm Um Lm i 1 m = m + 2 b m, i m 2 1 i= 1 x L d x 1 x2 x M L m, U m user defined bounds

11 Schemata in binary strings: A schema is a similarity template describing a subset of strings with similarities at certain string positions (Holland, 1968) *10 * 1 Defining length d(s)=5-2=3 Order of schema o(s)=3 (fixed digits) m=length of string(=5) r=number of * s (=2) 2 m possible schemata 2 r strings matched by this schema

12 Why dealing with Schemata? *10 * 1 Defining length d(s)=5-2=3 Order of schema o(s)=3 (fixed digits) The order of a schema affects its survival probabilities during mutation The defining length of a schema affects its survival probabilities during crossover (Schema Theorem)

13 Encoding the free variable Real Coding: Representation: ( x, x,..., x,..., x ) 1 2 m M A step ahead: Representation including evolution parameters (the concept of ES): (,,...,,...,,,,...,,..., ) x1 x2 xm xm σ1 σ2 σm σ Μ

14 Schema Theorem (1/4) : The effect of selection: (m g ) examples of a particular schema (S), generation (g) F(S)=average fitness of the strings matching schema (S) m g+ 1 = FS ( ) mg F mean, g Reproductive Schema Growth Equation mg+ 1 = mg(1 + k) m g+1 >m g if F(S)>F mean,g m 1 0(1 ) g g m k + + = + 1 m g+1 <m g if F(S)<F mean,g Long term effect of selection (k=constant)

15 Schema Theorem (2/4) : The effect of crossover: Possibility of destructing the schema S during crossover: p p des main ds ( ) ****11010*01*1*11* = d(s) m 1 Possibility of maintaining the schema S: = 1 p Xover ds ( ) m 1 m-1 Schema Growth Equation (after selection & crossover): FS ( ) ds ( ) m + ( S) m ( S) (1 p ) g 1 g Xover Fmean, g( S) m 1

16 Schema Theorem (3/4) : The effect of mutation: Possibility of maintaining the schema S after mutation: p p p o S os ( ) surv = (1 mut ) 1 mut ( ) ****11010*01*1*11* Final Schema Growth Equation: FS ( ) ds ( ) m + ( S) m ( S) (1 p o( S) p ) g 1 g Xover mut Fmean, g( S) m 1 Short, low-order, above-average schemata should receive an (exponentially) increasing number of strings in the next generations

17 Schema Theorem (4/4) : Lessons Learned: Short, low-order, above-average schemata sould receive an (exponentially) increasing number of strings in the next generations (Schema Theorem). GA explore the search space by short, low-order schemata. GAs seek near-optimal performance through the juxtaposition of short, low-order, high-performance schemata (the so-called building blocks, Building Block Hypothesis).

18 Exploration vs. Exploitation : Exploration: seeking the global optimum in new and unknown areas in the search space. Exploitation: making use the knowledge gained from the previously examined points to guide the search towards new better points in the search space. Holland 1975: GAs but: Infinite population size Fitness function value accurately reflects the utility of a solution Genes in a chromosome do not interact significantly

19 EAs: Binary or Real Coding? Binary Coding: Creates lengthy binary strings if high accuracy is required. Offers the maximum number of schemata per bit of information, compared to any other coding. Facilitates theoretical analysis and the development of new genetic operators. Smallest alphabet that allows a natural expression of the problem (Goldberg, 1989). Real Coding: Problem-tailored genetic operators can readily be devised. High-cardinality alphabets contain more schemata (Antonisse, 1989).

20 Generalized Evolutionary Algorithm (EA) Offspring Parent Elite λ g, S λ Τ μ κ μ g, S μ e, S ge y = F ( x) Τ e Τ m φ Τ r K : 1 ρ xm, m = 1, M Fk, k = 1, K

21 ΕΑ Schematic Presentation: g = 0, S =, S =, S = S ge, g, μ g, λ random g, λ y = F( x), s S g+ 1, e g, λ ge, S = Te ( S S ) φ( y), s S S S g, λ g, λ g+ 1, e S = Te ( S S ) 2 g+ 1, μ g, μ g, λ S = Tμ ( S S ) + λ + + S =Τm Tr S S g = g + 1 if( converge( g)) end g, μ g, λ g+ 1, e ( ) μ ( ( )) g 1, g 1, g 1, e

22 Parent Selection Operator Τ μ (1/5) (Neglecting Elitism) Phase 1: Phase 2: ( ) prov, μ g, g, =Τμ,1 μ λ S S S S g+ 1, μ prov, μ =Τμ,2 ( S ) Life-span κ Fitness φ Selective Pressure (Fitness φ) (ES) (GA)

23 Parent Selection Operator Τ μ (2/5) Proportional selection Linear ranking Roulette wheel Probabilistic Tournament selection Indirect Selection (μ<λ) ( ) g+ 1, μ g, μ g, λ =Τμ,1 S S S S g+ 1, μ prov, μ =Τμ,2 ( S )

24 Parent Selection Operator Τ μ (3/5) Proportional selection: Select (λρ) individuals out of (μ) preselected ones μφ μ i= 1 () s φ () ι 1 () s number _ of _ copies = int Premature convergence due to the presence of a super-fit individual. λρμφ μ φ i= 1 () s () ι

25 Parent Selection Operator Τ μ (4/5) Roulette Wheel: Select (λρ) individuals out of (μ) preselected ones i= 1 () s φ angle _ slot = μ 360 () ι φ o Premature convergence due to the presence of a super-fit individual.

26 Parent Selection Operator Τ μ (5/5) Fitness Ranking: Individuals are sorted in the order of fitness values Reproductive trials are assigned according to rank Linear Ranking Exponential Ranking, etc Overcomes the problem of the presence of an extreme individual Fitness ranking performs better than fitness scaling.

27 Recombination Operation T r Binary Coding One-point ρ= } ρ= Two-point One- or two-point per variable Discrete (variables interchanging) Uniform (with parent-depending probability)

28 Recombination Operation T r Real Coding One-point, ρ=2 x 1 x 2 x 3 x 4 x 5 x 6 x 1 x 2 x 3 x 4 x 5 x 6 } x 1 x 2 x 3 x 4 x 5 x 6 Two-point M-point Discrete, ρ=2 x 4 =x 4 +r (x 4 -x 4 ), r [0,1] x m =x m +r m (x m -x m ), m=1,m, r m [0,1] Discrete Panmictic ρ=μ (50% selection probability from each parent {1,2}, {1,3},..., {1,Μ} ) Generalized Intermediate Panmictic x m =x m +r m (x m,ρ -x m ), m=1,m, r m [0,1] Blend Xover (BLX-a), c=(1+2a)r-a, a=0.5

29 Mutation Operator T m - Binary Coding P m M m= 1 b m Dynamic Adjustment of P m, depending on the number of generations without any improvement the number of evaluations without any improvement

30 Mutation Operator T m Real Coding Dynamic mutation probability per variable P m ή x m xm + D(, gum xm), r1 < 0.5 = xm D(, g xm Lm), r1 0.5 p g = 2 g max g p (1 ) g = max 2 Dga (, ) a r 1 Dga (, ) a 1 r or, using number of evaluations, instead of number of generations p~0.2

31 Mutation Operator T m Real Coding σ = σ exp ( τ Ν (0,1) + τ Ν (0,1) ) m m m x = x + σ N m m m m (0,1) τ ( 2 ) = Μ ( ) τ ' = 2Μ 1 1

32 Genetic Algorithms or Evolution Strategies or GA Binary Coding μ=λ, parents=offspring [Holland, 1970] ρ=2, two-parent recombination[goldberg, 1989] κ=0, zero life-span [Michalewicz, 1994] [Fogel] Ρ r <1, recombination probability K=1, one target ES Real Coding, including the evolution parameters Without parent Selection Operator (μ<λ) μ<<λ κ=0 (μ,0,λ) =(μ,λ) or κ= (μ,,λ) = (μ+λ) ρ=2 P r =1 [Schwefel, Rechenberg, 1965] Κ=1 [Bäck, 1996]

33 Multi-objective Optimization Non-Pareto Front Techniques Pareto Front Techniques (K = number of objectives)

34 Pareto Front F 2 Front 0 (Pareto), S ge F 1 Front 1 Front 2

35 Pareto Front Definition (Minimization Problems): Dominant Solution: ( p) ( q) ( p) ( q) x < x s < s ( p) ( q) k k k {1,..., K} : F F ( p) ( q) k k k {1,..., K} : F < F Pareto Optimal Solution: ' M ' x / x : x > x

36 Multi-Objective Optimization Computation of φ φ K = wk k = 1 F k φ = Fk, k [1, K] For parent selection, VEGA [Schaffer, 1984] φ = Pareto_Rank( F ), k = 1, K k [Goldberg, 1989]

37 Vector Evaluated Genetic Algorithm (VEGA) K=3 Selection based on φ=f 1 Selection based on φ=f 2 Selection based on φ=f 3 Recombination, Mutation

38 VEGA: Guess the final solutions F 2 Front 0 (Pareto), S ge F 1 Front 1 Front 2

39 φ computation using the Pareto front: Front Ranking [Goldberg, 1989] Niched Pareto GA (NPGA) [Horn, Nafpliotis, 1993] Nondominated Sorting GA (NSGA) [Srinivas, Deb, 1994] Strength Pareto EA (SPEA) [Zitzler, Thiele, 1998] Pareto Envelope-based Selection Algorithm (PESA) [Corne, Knowles, Oates, 2000] NSGA-II [Deb, Agrawal, Pratap, Meyarivan, 2000] Strength Pareto EA ΙΙ (SPEA II) [Zitzler, Laumanns, Thiele, 2001]

40 φ computation Sorting (NSGA) F 2 F 1

41 φ computation Niching (NSGA) F 2 σ share 0, dij (, ) σ Sh = dij (, ) j = 1 1, dij (, ) < σ σshare φ = φ Sh i N f share dummy share i φ dummy = d(i,j) 1 j σ share φ = φ + ε dummy prev _ front F 1

42 φ computation SPEA F 2 F () i e φ ( F ) = 1 + φ( ) [0,1) μ + λ + 1 > 1 S g,e Dominated Solutions Other Solutions F 1

43 Ways to reduce the number of evaluations: Improved evolution operators Hybridization with other optimization methods Distributed EAs (island model) Hierarchical EAs (faster solvers) Use of surrogate models (metamodels, fast approximate model)

44 Hybridization with other Optimization Methods GA Log 10 (score) AM evaluations

45 Use of Surrogate Evaluation Models Polynomial Interpolation Artificial Neural Networks Multilayer Perceptron Radial Basis Function Networks Kriging Ways of using the surrogate evaluation model: Decoupled from the exact evaluation tool (+Design of Experiments, DoE) In combination with the exact evaluation tool Regular Training (depending on the number of new entries in the DB) Dynamical Training (separately, for each new individual)

46 Use of Surrogate Models (with Off-Line Training) Design of Experiments Inner Loop Exact Evaluations Outer Loop Evaluations using the Surrogate Model Genetic Operators Surrogate model Optimal Solution

47 Use of Surrogate Models (with On-Line Training) Build the Surrogate Model(s) New Population Evaluations using the Surrogate Model(s) Genetic Operators best Evaluations using the Exact Model Fitness Value Homogenization

48 Inexact Pre-Evaluation (IPE) The Concept: Exact Evaluation of the most promising solutions Inexact Evaluations λ evaluations L<λ evaluations Evolution (to the next generation)

49 Local Surrogate Models - Training Set: T [ T, T ] x 2 min max x 1

50 Surrogate Models What else do they tell us??? Fitness Function Approximation Confidence Intervals Hessian Matrix Approximation Sensitivity Derivatives (Importance Factors) F err Optimal err

51 Optimization and Multi-processing Reduction of the Optimization Wall-Clock Time Parallelization Level EA CFD CPUs λ (ή L ) Loading Distribution: Heterogeneous Platforms Loading per processor Variable evaluation cost Synchronization (in each generation) Master: ΕΑ Module - waiting list for evaluations Slave: discrete (remote) evaluation process

52 Distributed ΕΑ Why? L (<< λ) < CPUs Persistant diversity in populations Straightforward parallelization Additional Parameters: Number of islands Communication topology Communication frequency Migration algorithm EA parameters per island?

53 Distributed ΕΑ on a Multi-Processor System Islands EA EA EA Thread 3 Thread 4 Thread 5 Migrations Thread 2 Evaluations Server Thread 1 Asynchronous Requests for Evaluations MASTER SLAVE1 SLAVE2 SLAVE3 SLAVE4

54 Applications The Evolutionary Algorithms SYstem v1.3 Developed by the National Technical University of Athens, (NTUA), Greece

55 Problem: Rastrigin s Function 1 2 Fx ( ) = 20 + e 20 exp( 0.2 xm ) M M 1 exp( cos(2 πxm )) M m = 1 M m = 1 M=30 30 x m 30

56 Results: Rastrigin s Function Cost Function Value (Rastrigin) GA DGA GA-IPE-IF D(GA-IPE-IF) Cost Measured in Exact Evaluations

57 Problem: Three-Element Airfoil, Lift Maximization M = 0.12, a = o F = C L Rotation Angle Initial 28.1 ο Δx Δy Rotation Angle -37 ο Δx Δy Initial Optimal

58 Results: Three-Element Airfoil, Lift Maximization Initial Optimal Rotation Angle 28.1 ο ο Δx Δy Rotation Angle -37 ο ο Δx Δy Importance slat flap Δy IFs Design Variable

59 Results: Three-Element Airfoil, Lift Maximization GA DGA GA-IPE-IF D(GA-IPE-IF) Cost Function Value (-Lift) Cost Measured in Exact Evaluations

60 Problem: 3D Compressor Blade Design NURBS Evaluations ω ref =0.09 Hub Section Πόδι Tip Section Ακροπτερύγιο

61 Problem: Design of a Compressor Cascade s=0.68c γ=30 ο c=0.07m Minimum Total Pressure Losses Constraint on the minimum (maximum thickness) Desirable Flow turning F = ω P P t t P t t t t max thres 1 = exp( ), thres = 0.9 max, ref, max < thres tthres P Δα Δα ref 2 = exp( max(1, )), Δ α = α1 α2 > 0 Δαref

62 Results: Design of a Compressor Cascade GA DGA GA-IPE-IF D(GA-IPE-IF) Cost Function Value Cost Measured in Exact Evaluations

63 Problem: Compressor Multi-Point Design ASME 90-GT-140 s=0.68c γ=30 ο c=0.07m α 1 43 o 45 o 47 o 49 o 52 o Μ Re 8.61E5 8.50E5 8.41E5 8.20E5 7.63E5 1/AVDR ω (exp) ω MISES α 2 (πexp) α 2 MISES MISES 2.53 ω = p p p p t2, is t2 t

64 Compressor Multi-Point Design (1 OP 1 target) (20,2,100) ω L= ω REF Evaluations

65 Compressor Multi-Point Design (3 OP 2 targets) F = ω P P F = ( ω + ω 2 ω ) P P > Evaluations

66 Compressor Multi-Point Design (5 OP 3 targets) Evaluations

67 Compressor Multi-Point Design (5 OP 5 targets) Fi = ωi P1 P2, i = 1,...,5 Evaluations

68 Problem: High-Lift, Low-Drag Optimization o C1 : M = 0.20, a = 10.8, Re = F 1 o C2 : M = 0.77, a = 1.0, Re = F 2 NS, k-ε WF min 200

69 Results: High-Lift, Low-Drag Optimization C 1 C evaluations 1500 evaluations 2500 evaluations

70 Optimization of Combined Cycle GT Power Plants 6 C T G HP LP WATER TANK ST Natural gas fired, dual-pressure CCGTPP configuration GT: 260 MWe, 38% efficiency, exhaust gas mass flow 615 kg/sec at 600C. ST G2 Design variables HP steam pressure LP steam pressure superheated HP steam temperature feedwater temperature at the inlet to the HP evaporator feedwater temperature at the outlet from the first HP economizer feedwater temperature at the inlet to the LP evaporator superheated LP steam temperature steam pressure fed to the water tank exhaust gas mass flow ratio (percentage of mass flowrate traversing the LP economizer) exhaust gas temperature at the HRSG outlet steam extraction pressure from the LP steam turbine exhaust gas temperature at the inlet to the condensate preheater

71 Optimization of Combined Cycle GT Power Plants 100 Temperature (Celcius) Flue Gas HP Water/Steam LP Water/Steam Preheater Exchanged Heat (MW) Constrained Optimization Problem Capital Cost (MEuro) Candidate Solutions Optimal Solution Pareto front Pressure Value (HP, bar) Efficiency Efficiency HP Pressure Values, over the Pareto Front