Computationally Expensive Multi-objective Optimization. Juliane Müller

Transcription

1 Computationally Expensive Multi-objective Optimization Juliane Müller Lawrence Berkeley National Lab IMA Research Collaboration Workshop University of Minnesota, February 23, 2016 J. Müller Surrogate Models and MOP 1/30

2 Outline Multi-objective optimization problem J. Müller Surrogate Models and MOP 2/30

3 Outline Multi-objective optimization problem J. Müller Surrogate Models and MOP 3/30

4 Examples of multi-objective optimization Design optimization: airfoil design maximize lift minimize drag s 1 (0,0) (x 1, y 1 ) (x 1, y 1 ) (x 2, y 2 ) (x 2, y 2 ) s 2 (1,0) J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 4/30

5 Examples of multi-objective optimization Design optimization: airfoil design maximize lift minimize drag s 1 (0,0) (x 1, y 1 ) (x 1, y 1 ) (x 2, y 2 ) (x 2, y 2 ) s 2 (1,0) Flight choice minimize ticket price minimize number/time of layovers J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 4/30

6 Multi-objective optimization problem Minimize f(x) = [f 1 (x), f 2 (x),..., f k (x)] T < x l j x j x u j <, j = 1,..., d, (1a) (1b) where f i : R d R, i = 1,..., k, k 2 f i are generally in conflict (improving one objective will impair another objective) There does not exist one optimal solution Goal: find the best trade-off (Pareto-optimal) solutions J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 5/30

7 Pareto front (objective space), biobjective problem The points x A and x B that correspond to A and B, respectively are non-dominated. The point x C that corresponds to C is dominated. J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 6/30

8 Decision vector domination and Pareto optimality Decision vector domination Decision vector x 1 dominates x 2 (x 1 x 2 ) iff f i (x 1 ) f i (x 2 ) i = 1,..., k and m = 1,..., k : f m (x 1 ) < f m (x 2 ) Pareto-optimal set P = {x : x such that x x } Pareto-optimal front P F = {f = [f 1 (x ), f 2 (x ),..., f k (x )] T : x P } J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 7/30

9 Difficulties of solving problem (1) Evaluating the objective functions is computationally extremely expensive (one evaluation may take several hours) f i are computed by black-box simulations: we do not have an analytic description and no derivative information Traditional methods for solving (1) generally require thousands of function evaluations J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 8/30

10 Traditional solution methods using reformulation For computationally cheap problems Linear scalarization of (1a): min x ɛ-constraint method k w i f i (x), w i > 0 (2) i=1 min x f m (x) (3a) s.t. f i (x) ɛ i, i m Solve to optimality and obtain one Pareto-optimal point Not efficient for computationally expensive problems (3b) J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 9/30

11 Goal of this research Devise an efficient algorithm that finds a good approximation of the Pareto front and that only uses few hundred function evaluations J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 10/30

12 Outline Multi-objective optimization problem J. Müller Surrogate Models and MOP 11/30

13 (single objective) Use surrogate models to approximate the expensive objective functions f 1,..., f k : f i (x) = s i (x) + e i (x), i = 1,..., k, f i (x): the ith computationally-expensive objective function s i (x): the ith surrogate model output (cheap to evaluate) e i (x): the difference between both J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 12/30

14 Need some initial data to fit a surrogate model S = {x 1,..., x n } f i (x 1 ),..., f i (x n ) i = 1,..., k Fit a surrogate model s i to each of the data pairs (x l, f i (x l )), l = 1,..., n, i = 1,..., k J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 13/30

15 Need some initial data to fit a surrogate model S = {x 1,..., x n } f i (x 1 ),..., f i (x n ) i = 1,..., k Fit a surrogate model s i to each of the data pairs (x l, f i (x l )), l = 1,..., n, i = 1,..., k 15 s 1 (x) sample data f 1 (x) x J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 13/30

16 Need some initial data to fit a surrogate model S = {x 1,..., x n } f i (x 1 ),..., f i (x n ) i = 1,..., k Fit a surrogate model s i to each of the data pairs (x l, f i (x l )), l = 1,..., n, i = 1,..., k s 1 (x) s 2 (x) sample data f 1 (x) sample data f (x) x J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 13/30

17 Radial basis function (RBF) surrogate model where s(x) = n λ l φ( x x l 2 ) + p(x), (4) l=1 φ(r) = r 3 the cubic radial basis function p(x) = a + b T x the polynomial tail x l, l = 1,..., n, the already evaluated points determine parameters λ l R, l = 1,..., n, a R, and b = [b 1,..., b d ] T R d by solving a linear system of equations J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 14/30

18 Outline Multi-objective optimization problem J. Müller Surrogate Models and MOP 15/30

19 Overview Sampling strategies Algorithm steps 1: Initial experimental design, x 1,..., x n0. 2: Computationally expensive objective function evaluations, f i (x 1 ),..., f i (x n0 ), i = 1,..., k. 3: Identify the non-dominated points. 4: Fit the RBF surrogate models to the data. 5. Select the new sample point x new : Balance between local and global search 5: Evaluate f i (x new ), i = 1,..., k and go to Step 3. J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 16/30

20 Overview Sampling strategies Algorithm steps 1: Initial experimental design, x 1,..., x n0. 2: Computationally expensive objective function evaluations, f i (x 1 ),..., f i (x n0 ), i = 1,..., k. 3: Identify the non-dominated points. 4: Fit the RBF surrogate models to the data. 5. Select the new sample point x new : Balance between local and global search 5: Evaluate f i (x new ), i = 1,..., k and go to Step 3. J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 16/30

21 Overview Sampling strategies Sampling based on decision and objective space J. Müller Surrogate Models and MOP 17/30

22 Overview Sampling strategies Target value sampling objective space Approximate the Pareto front with a piecewise linear function Find the point x for which f 1 (x) = 2 and f 2 (x) = 20 J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 18/30

23 Overview Sampling strategies Target value sampling decision space Target values (t1, t2) = (2, 20) Find the point x for which f 1 (x) = 2 and f 2 (x) = 20 Use the computationally cheap surrogate models s 1 and s 2 and solve a computationally cheap multi-objective optimization problem min x [ s 1 (x) t1, s 2 (x) t2 ] T (5a) < x l j x j x u j <, j = 1,..., d. (5b) J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 19/30

24 Overview Sampling strategies Perturbation of non-dominated points 1 Decision space x Sample points Non dominated points Perturbation points x J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 20/30

25 Overview Sampling strategies Minimum point of each surrogate model Find the minimum point of each surrogate model (computationally cheap) i: min x s i (x) s.t. x l j x j x u j, j = 1,..., d s 1 (x) s (x) 2 sample data f 1 (x) sample data f 2 (x) x J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 21/30

26 Overview Sampling strategies Stochastic sampling and scoring Randomly generate points in the variable domain Score each random point and select the best one: Use the surrogate models s1,..., s k to predict the objective function values (surrogate model score) Compute the distance of each random point to the set of already evaluated points (distance score) Compute a weighted sum of both scores J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 22/30

27 Overview Sampling strategies Solve the surrogate multi-objective problem Use a multi-objective genetic algorithm to find the Pareto-optimal points of the approximation problem Minimize s(x) = [s 1 (x), s 2 (x),..., s k (x)] T < x l j x j x u j <, j = 1,..., d, May generate a large set of new sample points (6a) (6b) J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 23/30

28 Outline Multi-objective optimization problem J. Müller Surrogate Models and MOP 24/30

29 Example with d = 2, k = 2 min f(x) = [f 1 (x), f 2 (x)] T 2 f 1 (x 1, x 2 ) = 1 exp j=1 2 f 2 (x 1, x 2 ) = 1 exp j=1 4 x 1, x 2 4 [ x j 1 [ x j + 1 ] 2 2 ] 2 2 (7a) (7b) (7c) (7d) J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 25/30

30 Example with d = 2, k = 2 f 1 1 f x x x x 1 5 J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 26/30

31 Example with d = 2, k = 2 1 Objective space 0.8 minimize f minimize f 1 J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 26/30

32 Example with d = 2, k = 2 1 Objective space 0.8 minimize f minimize f 1 J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 26/30

33 Example with d = 2, k = 2 1 Objective space 0.8 minimize f minimize f 1 J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 26/30

34 Example with d = 2, k = 2 1 Objective space 0.8 minimize f minimize f 1 J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 26/30

35 Example with d = 2, k = 2 1 Objective space 0.8 minimize f minimize f 1 J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 26/30

36 Example with d = 2, k = 2 1 Objective space 0.8 minimize f minimize f 1 J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 26/30

37 Example with d = 2, k = 2 1 Objective space 0.8 minimize f minimize f 1 J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 26/30

38 Example with d = 2, k = 2 1 Objective space 0.8 minimize f minimize f 1 J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 26/30

39 Example with d = 2, k = 2 1 Objective space 0.8 minimize f minimize f 1 J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 26/30

40 Example with d = 2, k = 2 1 Objective space 0.8 minimize f minimize f 1 J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 26/30

41 Example with d = 2, k = 2 1 Objective space 0.8 minimize f minimize f 1 J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 26/30

42 Example with d = 2, k = 2 1 Objective space 0.8 minimize f minimize f 1 J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 26/30

43 Comparison to MO-Genetic Algorithm (MOGA) We compared the algorithm on 58 benchmark problems, one application from airfoil design, one application from structural optimization 1-35 variables, 2-10 objective functions Pareto fronts: convex connected, concave connected, disconnected, unknown J. Müller Surrogate Models and MOP 27/30

44 Comparison to MO-Genetic Algorithm (MOGA) Performance measures (a) Number of non-dominated solutions (b) Set coverage metric: the proportion of solutions from MOGA that are weakly dominated by solutions from SOCEMO and vice versa (c) hypervolume: the size of the space covered J. Müller Surrogate Models and MOP 28/30

45 Comparison to MO-Genetic Algorithm (MOGA) Large numbers are better SOCEMO MOGA SOCEMO MOGA SOCEMO MOGA (a) (b) (c) (a) Number of non-dominated solutions (b) Set coverage metric (c) Hypervolume J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 29/30

46 Conclusions We developed an algorithm for computationally expensive black-box multi-objective optimization problems We use computationally cheap surrogate models to approximate the expensive objective functions We use a combination of local and global search strategies showed that SOCEMO is an efficient and effective approach J. Müller (julianemueller@lbl.gov) Surrogate Models and MOP 30/30