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
The regular optimisation problem Multiobjective Optimization Minimize f : X Ă IR n ÝÑ Y Ă IR taste Subject to Equality constraints hpxq 0 @x P X costs cooking time Inequality constraints gpxq ď 0 @x 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 207 4 / 39 Nicola Beume (LS) CI 202 250202 2 / 28 Laptop Selection Comparing Apples and Oranges Name Display Battery Weight Price CPU RAM Graphic Disk Interfaces Dell Vostro 5 5568 56 2 h 2 kg 689 I57 8 GB DDR4 HD Graphics 620 256 SSD VGA, HDMI, USB HP 4bs007ng 4 25 h,7 kg 699 I57 8 GB DDR4 HD Graphics 620 256 SSD VGA, HDMI, USB HP 250 2HG7ES 56 2 h,86 kg 649 I57 8 GB DDR4 Radeon 520 256 SSD VGA, HDMI, USB Lenovo ThinkPad L470 4 0 h,87 kg 699 I57 8 GB DDR4 HD Graphics 620 256 SSD VGA, Disp, USB Fuijitsu Lifebook A557 56 8 h 2,4 kg 650 I57 8 GB DDR4 HD Graphics 620 256 SSD VGA, HDMI, USB Levono ThinkPad E470 4 8 h 87 kg 699 I57 8 GB DDR4 GeForce 940MX 256 SSD HDMI, USB Levono ThinkPad E470 4 8 h 87 kg 849 I77 6 GB DDR4 GeForce 940MX 256 SSD HDMI, USB Lenovo ThinkPad L570 56 0 h 238 kg 849 I57 8 GB DDR4 HD Graphics 620 256 SSD VGA; Disp, USB Lenovo ThinkPad 3 33 2 h 44 kg 869 I57 8 GB DDR4 HD Graphics 620 256 SSD HDMI, USB HP Power Pavilion 4cb03ng 56 45 h 22 kg 39 I77 6 GB DDR4 GeForce CTX 050 Ti 256 SSD + T HDD HDMI, USB HP Power Pavilion 5cb03ng 56 45 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 4 025 h 53 kg 666 I57 8 GB DDR4 HD Graphics 620 256 SSD HDMI; USB Acer Swift 3 (SF34577W2) 4 0 h 65 kg 774 I77 8 GB DDR4 HD Graphics 620 256 SSD HDMI; USB Lenovo ThinkPad X Carbon 4 2 h 3 kg 879 I74 8 GB DDR3L HD Graphics 5000 256 SSD HDMI; DISP; USB Fujitsu Lifebook A557 56 8 h 24 kg 63 I57 6 GB DDR4 HD Graphics 620 52 SSD VGA; HDMI; USB Acer TravelMate P459G2M56T4 56 8 h 2 kg 694 I57 8 GB DDR4 HD Graphics 620 256 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: http://xkcdcom/388/, modified B Naujoks MultiObjective Evolutionary Optimisation 2 November 207 5 / 39 B Naujoks MultiObjective Evolutionary Optimisation 2 November 207 8 / 39
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 202 250202 3 / 28 Nicola Beume (LS) CI 202 250202 4 / 28 Laptop Selection Aim of Optimization Name Display Battery Weight Price CPU RAM Graphic Disk Interfaces Dell Vostro 5 5568 56 2 h 2 kg 689 I57 8 GB DDR4 HD Graphics 620 256 SSD VGA, HDMI, USB HP 4bs007ng 4 25 h,7 kg 699 I57 8 GB DDR4 HD Graphics 620 256 SSD VGA, HDMI, USB HP 250 2HG7ES 56 2 h,86 kg 649 I57 8 GB DDR4 Radeon 520 256 SSD VGA, HDMI, USB Lenovo ThinkPad L470 4 0 h,87 kg 699 I57 8 GB DDR4 HD Graphics 620 256 SSD VGA, Disp, USB Fuijitsu Lifebook A557 56 8 h 2,4 kg 650 I57 8 GB DDR4 HD Graphics 620 256 SSD VGA, HDMI, USB Levono ThinkPad E470 4 8 h 87 kg 699 I57 8 GB DDR4 GeForce 940MX 256 SSD HDMI, USB Levono ThinkPad E470 4 8 h 87 kg 849 I77 6 GB DDR4 GeForce 940MX 256 SSD HDMI, USB Lenovo ThinkPad L570 56 0 h 238 kg 849 I57 8 GB DDR4 HD Graphics 620 256 SSD VGA; Disp, USB Lenovo ThinkPad 3 33 2 h 44 kg 869 I57 8 GB DDR4 HD Graphics 620 256 SSD HDMI, USB HP Power Pavilion 4cb03ng 56 45 h 22 kg 39 I77 6 GB DDR4 GeForce CTX 050 Ti 256 SSD + T HDD HDMI, USB HP Power Pavilion 5cb03ng 56 45 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 4 025 h 53 kg 666 I57 8 GB DDR4 HD Graphics 620 256 SSD HDMI; USB Acer Swift 3 (SF34577W2) 4 0 h 65 kg 774 I77 8 GB DDR4 HD Graphics 620 256 SSD HDMI; USB Lenovo ThinkPad X Carbon 4 2 h 3 kg 879 I74 8 GB DDR3L HD Graphics 5000 256 SSD HDMI; DISP; USB Fujitsu Lifebook A557 56 8 h 24 kg 63 I57 6 GB DDR4 HD Graphics 620 52 SSD VGA; HDMI; USB Acer TravelMate P459G2M56T4 56 8 h 2 kg 694 I57 8 GB DDR4 HD Graphics 620 256 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 207 5 / 39 Nicola Beume (LS) CI 202 250202 5 / 28
Laptop Selection Scalarization Isn t there an easier way? Name Display Battery Weight Price CPU RAM Graphic Disk Interfaces Dell Vostro 5 5568 56 2 h 2 kg 689 I57 8 GB DDR4 HD Graphics 620 256 SSD VGA, HDMI, USB HP 4bs007ng 4 25 h,7 kg 699 I57 8 GB DDR4 HD Graphics 620 256 SSD VGA, HDMI, USB HP 250 2HG7ES 56 2 h,86 kg 649 I57 8 GB DDR4 Radeon 520 256 SSD VGA, HDMI, USB Lenovo ThinkPad L470 4 0 h,87 kg 699 I57 8 GB DDR4 HD Graphics 620 256 SSD VGA, Disp, USB Fuijitsu Lifebook A557 56 8 h 2,4 kg 650 I57 8 GB DDR4 HD Graphics 620 256 SSD VGA, HDMI, USB Levono ThinkPad E470 4 8 h 87 kg 699 I57 8 GB DDR4 GeForce 940MX 256 SSD HDMI, USB Levono ThinkPad E470 4 8 h 87 kg 849 I77 6 GB DDR4 GeForce 940MX 256 SSD HDMI, USB Lenovo ThinkPad L570 56 0 h 238 kg 849 I57 8 GB DDR4 HD Graphics 620 256 SSD VGA; Disp, USB Lenovo ThinkPad 3 33 2 h 44 kg 869 I57 8 GB DDR4 HD Graphics 620 256 SSD HDMI, USB HP Power Pavilion 4cb03ng 56 45 h 22 kg 39 I77 6 GB DDR4 GeForce CTX 050 Ti 256 SSD + T HDD HDMI, USB HP Power Pavilion 5cb03ng 56 45 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 4 025 h 53 kg 666 I57 8 GB DDR4 HD Graphics 620 256 SSD HDMI; USB Acer Swift 3 (SF34577W2) 4 0 h 65 kg 774 I77 8 GB DDR4 HD Graphics 620 256 SSD HDMI; USB Lenovo ThinkPad X Carbon 4 2 h 3 kg 879 I74 8 GB DDR3L HD Graphics 5000 256 SSD HDMI; DISP; USB Fujitsu Lifebook A557 56 8 h 24 kg 63 I57 6 GB DDR4 HD Graphics 620 52 SSD VGA; HDMI; USB Acer TravelMate P459G2M56T4 56 8 h 2 kg 694 I57 8 GB DDR4 HD Graphics 620 256 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 2 2 2 B Naujoks MultiObjective Evolutionary Optimisation 2 November 207 5 / 39 Nicola Beume (LS) CI 202 250202 6 / 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 202 250202 7 / 28 Nicola Beume (LS) CI 202 250202 8 / 28
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 202 250202 9 / 28 Nicola Beume (LS) CI 202 250202 0 / 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 202 250202 / 28 Nicola Beume (LS) CI 202 250202 2 / 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 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 202 250202 3 / 28 Nicola Beume (LS) CI 202 250202 4 / 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 202 250202 5 / 28 Nicola Beume (LS) CI 202 250202 6 / 28
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 202 250202 7 / 28 Nicola Beume (LS) CI 202 250202 8 / 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 202 250202 9 / 28 Nicola Beume (LS) CI 202 250202 20 / 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 202 250202 2 / 28 Nicola Beume (LS) CI 202 250202 22 / 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 202 250202 23 / 28 Nicola Beume (LS) CI 202 250202 24 / 28
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 202 250202 25 / 28 Nicola Beume (LS) CI 202 250202 26 / 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 202 250202 27 / 28 Nicola Beume (LS) CI 202 250202 28 / 28
Exercise Exercise Solution, Given the following table Car 2 3 4 6 7 Consumption (l/00 km) 62 60 65 62 65 67 Price (T Euro) 6 4 5 3 2 4 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 6,0 6 62 63 64 65 6 f 67 68 Hypuvotume : 2 08 t 06ft 03 B Naujoks MultiObjective Evolutionary Optimisation 2 November 207 37 / 39 a 6 t 06 it 03=25 B Naujoks MultiObjective Evolutionary Optimisation 2 November 207 38 / 39