Computational Intelligence

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1 Computational Intelligence Winter Term 207/8 Prof Dr Günter Rudolph Lehrstuhl für Algorithm Engineering (LS ) Fakultät für Informatik TU Dortmund Slides prepared by Dr Nicola Beume (202) enriched with slides by Prof Dr Boris Naujoks, TH Cologne

2 The regular optimisation problem Multiobjective Optimization Minimize f : X Ă IR n ÝÑ Y Ă IR taste Subject to Equality constraints hpxq P X costs cooking time Inequality constraints gpxq ď P X Definitions x P X is (valid) solution X search, parameter, or decision space Y objective space nutrients Realworld problems: various demands on problem solution multiple conflictive objective functions B Naujoks MultiObjective Evolutionary Optimisation 2 November / 39 Nicola Beume (LS) CI / 28 Laptop Selection Comparing Apples and Oranges Name Display Battery Weight Price CPU RAM Graphic Disk Interfaces Dell Vostro h 2 kg 689 I57 8 GB DDR4 HD Graphics SSD VGA, HDMI, USB HP 4bs007ng 4 25 h,7 kg 699 I57 8 GB DDR4 HD Graphics SSD VGA, HDMI, USB HP 250 2HG7ES 56 2 h,86 kg 649 I57 8 GB DDR4 Radeon SSD VGA, HDMI, USB Lenovo ThinkPad L h,87 kg 699 I57 8 GB DDR4 HD Graphics SSD VGA, Disp, USB Fuijitsu Lifebook A h 2,4 kg 650 I57 8 GB DDR4 HD Graphics SSD VGA, HDMI, USB Levono ThinkPad E h 87 kg 699 I57 8 GB DDR4 GeForce 940MX 256 SSD HDMI, USB Levono ThinkPad E h 87 kg 849 I77 6 GB DDR4 GeForce 940MX 256 SSD HDMI, USB Lenovo ThinkPad L h 238 kg 849 I57 8 GB DDR4 HD Graphics SSD VGA; Disp, USB Lenovo ThinkPad h 44 kg 869 I57 8 GB DDR4 HD Graphics SSD HDMI, USB HP Power Pavilion 4cb03ng h 22 kg 39 I77 6 GB DDR4 GeForce CTX 050 Ti 256 SSD + T HDD HDMI, USB HP Power Pavilion 5cb03ng h 22 kg 39 I77 6 GB DDR4 GeForce GtX 050 Ti 256 SSD + T HDD HDMI, USB Asus X556UQDM885T 56 decent 23 kg 79 I57 8 GB DDR4 GeForce 940MX 256 SSD + T HDD VGA; HDMI; USB Acer Aspire 5 A555G5 RL 56 9 h 2 kg 849 I57 8 GB DDR4 Geforce MX50 28 SSD + T HDD HDMI; USB HP Pavilion 4bf007ng h 53 kg 666 I57 8 GB DDR4 HD Graphics SSD HDMI; USB Acer Swift 3 (SF34577W2) 4 0 h 65 kg 774 I77 8 GB DDR4 HD Graphics SSD HDMI; USB Lenovo ThinkPad X Carbon 4 2 h 3 kg 879 I74 8 GB DDR3L HD Graphics SSD HDMI; DISP; USB Fujitsu Lifebook A h 24 kg 63 I57 6 GB DDR4 HD Graphics SSD VGA; HDMI; USB Acer TravelMate P459G2M56T h 2 kg 694 I57 8 GB DDR4 HD Graphics SSD VGA; HDMI; USB AsusZenbook UX340UQGV999T 4 85 h 4 kg 999 I57 8 GB DDR4 GeForce 940MX 256 SSD + T HDD HDMI; USB Von: modified B Naujoks MultiObjective Evolutionary Optimisation 2 November / 39 B Naujoks MultiObjective Evolutionary Optimisation 2 November / 39

3 Multiobjective Optimization Pareto Dominance Multiobjective Problem f : S R n Z R d, min x R n f(x) = (f (x),, f d (x)) partial order among vectors in R d and thus in R n (, ) (5, 5) (8, 8) (, 8) (5, 5) (8, ) How to relate vectors? a b, a weakly dominates b : i {,, d} : a i b i a b, a dominates b : a b and a b, ie, i {,, d} : a i < b i a b, a and b are incomparable: neither a b nor b a Nicola Beume (LS) CI / 28 Nicola Beume (LS) CI / 28 Laptop Selection Aim of Optimization Name Display Battery Weight Price CPU RAM Graphic Disk Interfaces Dell Vostro h 2 kg 689 I57 8 GB DDR4 HD Graphics SSD VGA, HDMI, USB HP 4bs007ng 4 25 h,7 kg 699 I57 8 GB DDR4 HD Graphics SSD VGA, HDMI, USB HP 250 2HG7ES 56 2 h,86 kg 649 I57 8 GB DDR4 Radeon SSD VGA, HDMI, USB Lenovo ThinkPad L h,87 kg 699 I57 8 GB DDR4 HD Graphics SSD VGA, Disp, USB Fuijitsu Lifebook A h 2,4 kg 650 I57 8 GB DDR4 HD Graphics SSD VGA, HDMI, USB Levono ThinkPad E h 87 kg 699 I57 8 GB DDR4 GeForce 940MX 256 SSD HDMI, USB Levono ThinkPad E h 87 kg 849 I77 6 GB DDR4 GeForce 940MX 256 SSD HDMI, USB Lenovo ThinkPad L h 238 kg 849 I57 8 GB DDR4 HD Graphics SSD VGA; Disp, USB Lenovo ThinkPad h 44 kg 869 I57 8 GB DDR4 HD Graphics SSD HDMI, USB HP Power Pavilion 4cb03ng h 22 kg 39 I77 6 GB DDR4 GeForce CTX 050 Ti 256 SSD + T HDD HDMI, USB HP Power Pavilion 5cb03ng h 22 kg 39 I77 6 GB DDR4 GeForce GtX 050 Ti 256 SSD + T HDD HDMI, USB Asus X556UQDM885T 56 decent 23 kg 79 I57 8 GB DDR4 GeForce 940MX 256 SSD + T HDD VGA; HDMI; USB Acer Aspire 5 A555G5 RL 56 9 h 2 kg 849 I57 8 GB DDR4 Geforce MX50 28 SSD + T HDD HDMI; USB HP Pavilion 4bf007ng h 53 kg 666 I57 8 GB DDR4 HD Graphics SSD HDMI; USB Acer Swift 3 (SF34577W2) 4 0 h 65 kg 774 I77 8 GB DDR4 HD Graphics SSD HDMI; USB Lenovo ThinkPad X Carbon 4 2 h 3 kg 879 I74 8 GB DDR3L HD Graphics SSD HDMI; DISP; USB Fujitsu Lifebook A h 24 kg 63 I57 6 GB DDR4 HD Graphics SSD VGA; HDMI; USB Acer TravelMate P459G2M56T h 2 kg 694 I57 8 GB DDR4 HD Graphics SSD VGA; HDMI; USB AsusZenbook UX340UQGV999T 4 85 h 4 kg 999 I57 8 GB DDR4 GeForce 940MX 256 SSD + T HDD HDMI; USB Pareto front: set of optimal solution vectors in R d, ie, PF = {x Z x Z with x x} Aim of optimization: find Pareto front? PF maybe infinitively large PF hard to hit exactly in continuous space too ambitious! Aim of optimization: approximate Pareto front! B Naujoks MultiObjective Evolutionary Optimisation 2 November / 39 Nicola Beume (LS) CI / 28

4 Laptop Selection Scalarization Isn t there an easier way? Name Display Battery Weight Price CPU RAM Graphic Disk Interfaces Dell Vostro h 2 kg 689 I57 8 GB DDR4 HD Graphics SSD VGA, HDMI, USB HP 4bs007ng 4 25 h,7 kg 699 I57 8 GB DDR4 HD Graphics SSD VGA, HDMI, USB HP 250 2HG7ES 56 2 h,86 kg 649 I57 8 GB DDR4 Radeon SSD VGA, HDMI, USB Lenovo ThinkPad L h,87 kg 699 I57 8 GB DDR4 HD Graphics SSD VGA, Disp, USB Fuijitsu Lifebook A h 2,4 kg 650 I57 8 GB DDR4 HD Graphics SSD VGA, HDMI, USB Levono ThinkPad E h 87 kg 699 I57 8 GB DDR4 GeForce 940MX 256 SSD HDMI, USB Levono ThinkPad E h 87 kg 849 I77 6 GB DDR4 GeForce 940MX 256 SSD HDMI, USB Lenovo ThinkPad L h 238 kg 849 I57 8 GB DDR4 HD Graphics SSD VGA; Disp, USB Lenovo ThinkPad h 44 kg 869 I57 8 GB DDR4 HD Graphics SSD HDMI, USB HP Power Pavilion 4cb03ng h 22 kg 39 I77 6 GB DDR4 GeForce CTX 050 Ti 256 SSD + T HDD HDMI, USB HP Power Pavilion 5cb03ng h 22 kg 39 I77 6 GB DDR4 GeForce GtX 050 Ti 256 SSD + T HDD HDMI, USB Asus X556UQDM885T 56 decent 23 kg 79 I57 8 GB DDR4 GeForce 940MX 256 SSD + T HDD VGA; HDMI; USB Acer Aspire 5 A555G5 RL 56 9 h 2 kg 849 I57 8 GB DDR4 Geforce MX50 28 SSD + T HDD HDMI; USB HP Pavilion 4bf007ng h 53 kg 666 I57 8 GB DDR4 HD Graphics SSD HDMI; USB Acer Swift 3 (SF34577W2) 4 0 h 65 kg 774 I77 8 GB DDR4 HD Graphics SSD HDMI; USB Lenovo ThinkPad X Carbon 4 2 h 3 kg 879 I74 8 GB DDR3L HD Graphics SSD HDMI; DISP; USB Fujitsu Lifebook A h 24 kg 63 I57 6 GB DDR4 HD Graphics SSD VGA; HDMI; USB Acer TravelMate P459G2M56T h 2 kg 694 I57 8 GB DDR4 HD Graphics SSD VGA; HDMI; USB AsusZenbook UX340UQGV999T 4 85 h 4 kg 999 I57 8 GB DDR4 GeForce 940MX 256 SSD + T HDD HDMI; USB Scalarize objectives to singleobjective function: f : S R n Z R 2 f scal = w f (x) + w 2 f 2 (x) Result: single solution Specify desired solution by choice of w, w B Naujoks MultiObjective Evolutionary Optimisation 2 November / 39 Nicola Beume (LS) CI / 28 Scalarization Previous example: convex Pareto front Consider concave Pareto front only boundary solutions are optimal scalarization by simple weighting is not a good idea Classification apriori approach first specify preferences, then optimize more advanced scalarization techniques (eg Tschebyscheff) allow to access all elements of PF 2 2 remaining difficulty: how to express your desires through parameter values!? aposteriori approach first optimize (approximate Pareto front), then choose solution back to aposteriori approach stateoftheart methods: evolutionary algorithms Nicola Beume (LS) CI / 28 Nicola Beume (LS) CI / 28

5 Evolutionary Algorithms Selection in EMOA Evolutionary Multiobjective Optimization Algorithms (EMOA) Multiobjective Optimization Evolutionary Algorithms (MOEA) initialization evaluation of population parent selection for reproduction variation (recombination/crossover, mutation) evolution evaluation of offspring Selection requires sortable population to choose best individuals How to sort ddimensional objective vectors? Primary selection criterion: use Pareto dominance relation to sort comparable individuals selection of succeeding population termination condition fulfilled? stop Secondary selection criterion: apply additional measure to incomparable individuals to enforce order What to change in case of multiobjective optimization? Selection! Remaining operators may work on search space only Nicola Beume (LS) CI / 28 Nicola Beume (LS) CI / 28 Nondominated Sorting Example for primary selection criterion partition population into sets of mutually incomparable solutions (antichains) nondominated set: best elements of set NDS(M) = {x M x M with x x} Simple algorithm: iteratively remove nondominated set until population empty Nondominated Sorting Example for primary selection criterion partition population into sets of mutually incomparable solutions (antichains) nondominated set: best elements of set NDS(M) = {x M x M with x x} Simple algorithm: iteratively remove nondominated set until population empty Nicola Beume (LS) CI / 28 Nicola Beume (LS) CI / 28

6 Nondominated Sorting Example for primary selection criterion partition population into sets of mutually incomparable solutions (antichains) nondominated set: best elements of set NDS(M) = {x M x M with x x} Simple algorithm: iteratively remove nondominated set until population empty NSGAII Popular EMOA: Nondominated Sorting Genetic Algorithm II (µ + µ)selection: perform nondominated sorting on all µ + µ individuals 2 take best subsets as long as they can be included completely 3 if population size µ not reached but next subset does not fit in completely: apply secondary selection criterion crowding distance to that subset 4 fill up population with best ones wrt the crowding distance Nicola Beume (LS) CI / 28 Nicola Beume (LS) CI / 28 NSGAII Crowding distance: /2 perimeter of empty bounding box around point value of infinity for boundary points large values good Difficulties of Selection imagine point in the middle of the search space d = 2: /4 better, /4 worse, /2 incomparable d = 3: /8 better, /8 worse, 3/4 incomparable general: fraction 2 d+ comparable, decreases exponentially typical case: all individuals incomparable mainly secondary selection criterion in operation Drawback of crowding distance: rewards spreading of points, does not reward approaching the Pareto front NSGAII diverges for large d, difficulties already for d = 3 Nicola Beume (LS) CI / 28 Nicola Beume (LS) CI / 28

7 Difficulties of Selection Hypervolumen (Smetric) as Quality Measure dominated hypervolume: size of dominated space bounded by reference point Secondary selection criterion has to be meaningful! Desired: choose best subset of size µ from individuals How to compare sets of partially incomparable points? use quality indicators for sets One approach for selection for each point: determine contribution to quality value of set sort points according to contribution f 2 v () v (2) v (3) r ( m ) H(M, r) := Leb [v (i), r] i= M = {v (), v (2),, v (m) } r reference point to be maximized v (4) v (5) f Nicola Beume (LS) CI / 28 Nicola Beume (LS) CI / 28 SMS(SMetric Selection)EMOA Stateoftheart EMOA (µ + )selection nondominated sorting SMS(SMetric Selection)EMOA Stateoftheart EMOA (µ + )selection nondominated sorting Nicola Beume (LS) CI / 28 Nicola Beume (LS) CI / 28

8 SMS(SMetric Selection)EMOA Stateoftheart EMOA (µ + )selection nondominated sorting SMS(SMetric Selection)EMOA Stateoftheart EMOA (µ + )selection nondominated sorting Nicola Beume (LS) CI / 28 Nicola Beume (LS) CI / 28 SMS(SMetric Selection)EMOA Stateoftheart EMOA (µ + )selection nondominated sorting SMS(SMetric Selection)EMOA Stateoftheart EMOA (µ + )selection nondominated sorting Nicola Beume (LS) CI / 28 Nicola Beume (LS) CI / 28

9 SMS(SMetric Selection)EMOA Stateoftheart EMOA (µ + )selection nondominated sorting Computational complexity of hypervolume Lower Bound Ω(m log m) Upper Bound O(m d/2 2 O(log m) ) proof: hypervolume as special case of Klee s measure problem f 2 r f 2 f f Nicola Beume (LS) CI / 28 Nicola Beume (LS) CI / 28 Conclusions on EMOA Conclusions NSGAII only suitable in case of d=2 objective functions otherwise no convergence to Pareto front SMSEMOA also effective for d > 2 due to hypervolume hypervolume calculation timeconsuming use approximation of hypervolume Other stateoftheart EMOA, eg MOCMAES: CMAES + hypervolume selection ɛmoea: objective space partitioned into grid, only point per cell MSOPS: selection acc to ranks of different scalarizations realworld problems are often multiobjective Pareto dominance only a partial order a priory: parameterization difficult a posteriori: choose solution after knowing possible compromises stateoftheart a posteriori methods: EMOA, MOEA EMOA require sortable population for selection use quality measures as secondary selection criterion hypervolume: excellent quality measure, but computationally intensive use stateoftheart EMOA, other may fail completely Nicola Beume (LS) CI / 28 Nicola Beume (LS) CI / 28

10 Exercise Exercise Solution, Given the following table Car Consumption (l/00 km) Price (T Euro) Draw the cars in objective space Calculate the hypervolume of the set wrt reference point p68; 6q ^ F a t tf If rb xnf I : I : tint : 3, ; tae# = lf gohn d Li i, i / i n4 :, : * i i ' M 5 t 6, f Hypuvotume : 2 08 t 06ft 03 B Naujoks MultiObjective Evolutionary Optimisation 2 November / 39 a 6 t 06 it 03=25 B Naujoks MultiObjective Evolutionary Optimisation 2 November / 39

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