A coverage-based Box-Algorithm to compute a representation for optimization problems with three objective functions

Transcription

1 A coverage-based Box-Algorithm to compute a representation for optimization problems with three objective functions Tobias Kuhn (Coauthor: Stefan Ruzika) Department of Mathematics University of Kaiserslautern Recent Advances in Multi-Objective Optimization Lancaster, June 24, 2016 Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

2 Thanks to... We gratefully acknowledge support by Federal Ministry of Education and Research, Germany, BMBF, project Robuste Evakuierung und zivile Sicherheitsplanung (RobEZiS), grant number 13N Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

3 Multiple Objective Programming Problem (Multiple Objective Programming Problem) Let f i : X R n R, i {1,..., p} =: [p]. (MOP) min f (x) := (f 1(x),..., f p (x)) x X f 2 (x) f 1 (x) Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

4 Optimality Concept Definition x X efficient : x X : f (x) f (x ) : x X i [p] : f i (x) f i (x ) f (x) f (x ) y Y = f (X ) nondominated : y = f (x ), x efficient X E : Set of efficient solutions Y N : Set of nondominated points f 2 (x) y I : Ideal point y N : Nadir point f 1 (x) Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

5 Optimality Concept Definition x X efficient : x X : f (x) f (x ) : x X i [p] : f i (x) f i (x ) f (x) f (x ) y Y = f (X ) nondominated : y = f (x ), x efficient X E : Set of efficient solutions Y N : Set of nondominated points f 2 (x) y I : Ideal point y N : Nadir point y N y I f 1 (x) Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

6 Motivation General goal: Compute the nondominated set Y N (and present it to the decision maker)! Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

7 Motivation General goal: Compute the nondominated set Y N (and present it to the decision maker)! But: Y N may... be very large in practice, Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

8 Motivation General goal: Compute the nondominated set Y N (and present it to the decision maker)! But: Y N may... be very large in practice,... consist of infinitely many points, Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

9 Motivation General goal: Compute the nondominated set Y N (and present it to the decision maker)! But: Y N may... be very large in practice,... consist of infinitely many points,... grow exponentially in the input size of the problem. Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

10 Motivation General goal: Compute the nondominated set Y N (and present it to the decision maker)! But: Y N may... be very large in practice,... consist of infinitely many points,... grow exponentially in the input size of the problem. compute representation of Y N, i. e., an appropriate substitute for the nondominated set. Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

11 Representative System Definition (Representative System) For some MOP with outcome set Y, we call a finite approximation Rep Y representative system and its elements representative points. Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

12 Representative System Definition (Representative System) For some MOP with outcome set Y, we call a finite approximation Rep Y representative system and its elements representative points. What is a good representative system? Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

13 Quality Measures Definition (Sayin 2000, Ruzika 2007) a) Coverage error: max min z y. y Y N z Rep Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

14 Quality Measures Definition (Sayin 2000, Ruzika 2007) a) Coverage error: max min z y. y Y N z Rep b) Uniformity: min z ẑ. z,ẑ Rep z ẑ Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

15 Quality Measures Definition (Sayin 2000, Ruzika 2007) a) Coverage error: max min z y. y Y N z Rep b) Uniformity: min z ẑ. z,ẑ Rep z ẑ c) Cardinality: Rep. Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

16 Quality Measures Definition (Sayin 2000, Ruzika 2007) a) Coverage error: max min z y. y Y N z Rep b) Uniformity: min z ẑ. z,ẑ Rep z ẑ c) Cardinality: Rep. d) Representation error: max z Rep min y Y N y z. Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

17 Some Literature using Boxes for MOPs Boxes / rectangles / cuboids... are frequently used in multiple objective programming: Laumans et al. (2006), Dhaenens et al. (2010), Kirlik and Sayin (2014): exact nondominated set by fixing one objective and projecting the others; grid-based structure; ε-constraint method. Dächert and Klamroth (2013): Improvement of splitting of boxes + generic algorithm; ε-constraint or Tchebycheff method Boland et al. (2014): Partition of projected search space by L-shapes and rectangles; 3-objective integer problems; experimental quality assessment. + several other approaches, e. g. in evolutionary algorithms. Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

18 Our Contribution Goal: Simple algorithm for computing a representative system with desired coverage error for MOPs with p = 3 objectives with the capability of relating the run time of the algorithm to the quality of the representative system. Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

19 Our Contribution Goal: Simple algorithm for computing a representative system with desired coverage error for MOPs with p = 3 objectives with the capability of relating the run time of the algorithm to the quality of the representative system. Idea: Extending the Box-Algorithm of [Hamacher et al. 2007] to the case of three objective functions Hamacher, Pedersen, Ruzika Finding representative sytems for discrete bicriterion optimization problems Operations Research Letters 35(3): , 2007 Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

20 Short Review of the Box-Algorithm (2D) f2 f 1 Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

21 Short Review of the Box-Algorithm (2D) f2 f 1 Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

22 Short Review of the Box-Algorithm (2D) y N f2 y I f 1 Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

23 Short Review of the Box-Algorithm (2D) y N f2 y I ε f 1 Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

24 Short Review of the Box-Algorithm (2D) y N f2 y I ε f 1 Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

25 Short Review of the Box-Algorithm (2D) f2 f 1 Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

26 Box-Algorithm for MOPs with Three Objectives Initialization (MOP) min (f 1 (x), f 2 (x), f 3 (x)) s. t. x X R n Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

27 Box-Algorithm for MOPs with Three Objectives Initialization (MOP) min (f 1 (x), f 2 (x), f 3 (x)) s. t. x X R n Definition Let l, u R 3 with l u. We refer to the cuboid B(l, u) := l + R 3 u R3 = {y R3 l y u} as the box defined by l and u. Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

28 Box-Algorithm for MOPs with Three Objectives Initialization (MOP) min (f 1 (x), f 2 (x), f 3 (x)) s. t. x X R n Definition Let l, u R 3 with l u. We refer to the cuboid B(l, u) := l + R 3 u R3 = {y R3 l y u} as the box defined by l and u. Lemma Let l 0 y I R 3 and let u0 y N + R 3. Then Y N B(l 0, u 0 ). Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

29 Update Step Definition Let l, u R 3, l u and let ε i = l i + u i l i 2 for i = 1, 2, 3. Then, we define the lexicographic ε-constraint scalarizations with lower bounds (P 1 ε 1,ε 2 ), (P 2 ε 1,ε 3 ), and (P 3 ε 2,ε 3 ) as (P 1 ε 1,ε 2 ) lex min (f 3 (x), f 2 (x), f 1 (x)) s. t. x X l 1 f 1 (x) ε 1 l 2 f 2 (x) ε 2 l 3 f 3 (x) ( u 3 ) =: f (x) B(l, u)(ε 1,ε 2,u 3 ) Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

30 Update Step Definition Let l, u R 3, l u and let ε i = l i + u i l i 2 for i = 1, 2, 3. Then, we define the lexicographic ε-constraint scalarizations with lower bounds (P 1 ε 1,ε 2 ), (P 2 ε 1,ε 3 ), and (P 3 ε 2,ε 3 ) as (P 1 ε 1,ε 2 ) lex min (f 3 (x), f 2 (x), f 1 (x)) and, analogously, s. t. x X l 1 f 1 (x) ε 1 l 2 f 2 (x) ε 2 l 3 f 3 (x) ( u 3 ) =: f (x) B(l, u)(ε 1,ε 2,u 3 ) (P 2 ε 1,ε 3 ) lex min (f 2(x), f 1(x), f 3(x)) s. t. x X f (x) B(l, u) (ε 1,u 2,ε 3 ) (P 3 ε 2,ε 3 ) lex min (f 1(x), f 3(x), f 2(x)) s. t. x X f (x) B(l, u) (u 1,ε 2,ε 3 ). Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

31 Update Step Proposition Let z be the image under f of an optimal solution of (Pε 1 1,ε 2 ). Then, there does not exist a y Y N \ {z } such that ( y B(z, u) B l, (ε 1, ε 2, z3 ) ) \ B ((l 1, z2, z3 ), (z1, ε 2, z3 ) ) Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

32 Update Step Proposition Let z be the image under f of an optimal solution of (Pε 1 1,ε 2 ). Then, there does not exist a y Y N \ {z } such that ( y B(z, u) B l, (ε 1, ε 2, z3 ) ) \ B ((l 1, z2, z3 ), (z1, ε 2, z3 ) ) Proof. Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

33 Update Step Proposition Let z be the image under f of an optimal solution of (Pε 1 1,ε 2 ). Then, there does not exist a y Y N \ {z } such that ( y B(z, u) B l, (ε 1, ε 2, z3 ) ) \ B ((l 1, z2, z3 ), (z1, ε 2, z3 ) ) Proof. B(z, u) is dominated by z Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

34 Update Step Proposition Let z be the image under f of an optimal solution of (Pε 1 1,ε 2 ). Then, there does not exist a y Y N \ {z } such that ( y B(z, u) B l, (ε 1, ε 2, z3 ) ) \ B ((l 1, z2, z3 ), (z1, ε 2, z3 ) ) Proof. B(z, u) is dominated by z A nondominated point in B ( l, (ε 1, ε 2, z 3 ) ) \ B ( (l 1, z 2, z 3 ), (z 1, ε 2, z 3 ) ) contradicts optimality of z for (P 1 ε 1,ε 2 ) Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

35 Update Step Example: Consider an initial box with l = 0 and u = (12, 10, 10). Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

36 Update Step Example: Solve (P 1 ε 1,ε 2 ) with ε 1 = 6 and ε 2 = 5. Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

37 Update Step Example: Optimal solution with image z = (2, 3, 4) Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

38 Update Step Example: Optimal solution with image z = (2, 3, 4) Cut-off Regions (see first proposition): B ( (2, 3, 4), (12, 10, 10) ) und B ( 0, (6, 5, 4) ) \ B ( (0, 3, 4), (2, 5, 4) ) Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

39 Update Step Example: Optimal solution with image z = (2, 3, 4) Cut-off Regions (see first proposition): B ( (2, 3, 4), (12, 10, 10) ) und B ( 0, (6, 5, 4) ) \ B ( (0, 3, 4), (2, 5, 4) ) Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

40 Update Step Definition Let l, u R 3, l u and let ε i = l i + u i l i 2 for i = 1, 2, 3. Suppose (P 1 ε 1,ε 2 ) is solved. Then, the four quarters of the current box B(l, u) are defined as ε 1 Q 1,1 := B l, ε 2, u 3 ε 1 u 1 Q 1,2 := B l 2, ε 2, l 3 u 3 ε 1 Q 1,3 := B ε 2, u, l 3 l 1 ε 1 Q 1,4 := B ε 2, u 2. l 3 u 3 Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

41 Definition Let l, u R 3, l u and let ε i = l i + u i l i 2 for i = 1, 2, 3. Suppose (Pε 1 1,ε 2 ) is solved and let z R 3 denote the image under f of an optimal solution for (Pε 1 1,ε 2 ). Then, the subdivision of the current box B(l, u) consists of the boxes l 1 z 1 B 1,1 := B z2, ε 2, z3 u 3 ε 1 u 1 B 1,3 := B l 2, ε 2, l 3 z3 ε 1 u 1 B 1,5 := B ε 2, u 2, l 3 z3 l 1 l 1 ε 1 B 1,2 := B l 2, z z3 2 u 3 ε 1 u 1 B 1,4 := B l 2, z z3 2 u 3 B 1,7 := B ε 2, u 2 z3 u 3 l 1 ε 1 B 1,6 := B ε 2, u 2 l 3 z3 z 1 Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

42 Properties of the Subdivision Lemma 7 Y N B(l, u) i=1 B 1,i Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

43 Properties of the Subdivision Lemma 7 Y N B(l, u) i=1 B 1,i Proof. Follows directly from previous proposition. Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

44 Properties of the Subdivision Lemma 7 Y N B(l, u) i=1 B 1,i Proof. Follows directly from previous proposition. Observation Vol(B 1,i ) 1 Vol(B(l, u)) 4 Moreover, for each of the quarters Q 1,1, Q 1,2 and Q 1,3, we can find two pairs of boxes for which the combined volume fulfills this formula. Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

45 Properties of the Subdivision Lemma 7 Vol(B 1,i ) 3 Vol(B(l, u)) 4 i=1 Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

46 Properties of the Subdivision Lemma 7 Vol(B 1,i ) 3 Vol(B(l, u)) 4 i=1 Proof. B lex := B ( l, (ε 1, ε 2, z 3 ) ) and B dom := B (z, u) are cut off with Vol(B lex ) + Vol(B dom ) Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

47 Properties of the Subdivision Lemma 7 Vol(B 1,i ) 3 Vol(B(l, u)) 4 i=1 Proof. B lex := B ( l, (ε 1, ε 2, z 3 ) ) and B dom := B (z, u) are cut off with Vol(B lex ) + Vol(B dom ) Vol(B lex ) + Vol ( B ( (ε 1, ε 2, z 3 ), u )) Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

48 Properties of the Subdivision Lemma 7 Vol(B 1,i ) 3 Vol(B(l, u)) 4 i=1 Proof. B lex := B ( l, (ε 1, ε 2, z 3 ) ) and B dom := B (z, u) are cut off with Vol(B lex ) + Vol(B dom ) Vol(B lex ) + Vol ( B ( (ε 1, ε 2, z 3 ), u )) = Vol ( B lex + (ε 1 l 1, ε 2 l 2, 0) ) + Vol ( B ( (ε 1, ε 2, z 3 ), u )) Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

49 Properties of the Subdivision Lemma 7 Vol(B 1,i ) 3 Vol(B(l, u)) 4 i=1 Proof. B lex := B ( l, (ε 1, ε 2, z 3 ) ) and B dom := B (z, u) are cut off with Vol(B lex ) + Vol(B dom ) Vol(B lex ) + Vol ( B ( (ε 1, ε 2, z 3 ), u )) = Vol ( B lex + (ε 1 l 1, ε 2 l 2, 0) ) + Vol ( B ( (ε 1, ε 2, z 3 ), u )) = Vol ( B ( (ε 1, ε 2, l 3 ), (u 1, u 2, z 3 ) )) + Vol ( B ( (ε 1, ε 2, z 3 ), u )) Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

50 Properties of the Subdivision Lemma 7 Vol(B 1,i ) 3 Vol(B(l, u)) 4 i=1 Proof. B lex := B ( l, (ε 1, ε 2, z 3 ) ) and B dom := B (z, u) are cut off with Vol(B lex ) + Vol(B dom ) Vol(B lex ) + Vol ( B ( (ε 1, ε 2, z 3 ), u )) = Vol ( B lex + (ε 1 l 1, ε 2 l 2, 0) ) + Vol ( B ( (ε 1, ε 2, z 3 ), u )) = Vol ( B ( (ε 1, ε 2, l 3 ), (u 1, u 2, z 3 ) )) + Vol ( B ( (ε 1, ε 2, z 3 ), u )) = Vol ( B ( (ε 1, ε 2, l 3 ), u )) = Vol(Q 1,3 ) = 1 Vol(B(l, u)) 4 Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

51 Correctness Algorithm 1 Box-Algorithm for three objectives Require: MOP with three objectives, δ C > 0 Ensure: Rep representative system with coverage error at most δ C 1: S := {InitialBox()} 2: while S do 3: B := B(l, u) := SelectBox(S) 4: if l u δ C then 5: Use (Pu 1 1,u 2 ) to search for a representative point in B 6: else 7: Determine the 2 longest edges of B 8: Solve (P.,.), j j {1, 2, 3}, dividing these 2 edges 9: Add optimal outcome z to Rep 10: for i {1,..., 7} do 11: Add B j,i to S Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

52 Correctness Algorithm 1 Box-Algorithm for three objectives Require: MOP with three objectives, δ C > 0 Ensure: Rep representative system with coverage error at most δ C 1: S := {InitialBox()} 2: while S do 3: B := B(l, u) := SelectBox(S) 4: if l u δ C then 5: Use (Pu 1 1,u 2 ) to search for a representative point in B 6: else 7: Determine the 2 longest edges of B 8: Solve (P.,.), j j {1, 2, 3}, dividing these 2 edges 9: Add optimal outcome z to Rep 10: for i {1,..., 7} do 11: Add B j,i to S Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

53 Correctness Theorem Algorithm 1 terminates in finitely many steps. It outputs a collection of boxes containing all nondominated points. The representative system Rep has a coverage error of at most( δ C (w. r. t. ). More precisely, the ( ) ) L 2 log2 (7) algorithm performs at most O δ C many iterations, where L is the distance of the corner points of the initial box B(l 0, u 0 ), i.e., L := l 0 u 0. Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

54 Correctness Theorem Algorithm 1 terminates in finitely many steps. It outputs a collection of boxes containing all nondominated points. The representative system Rep has a coverage error of at most( δ C (w. r. t. ). More precisely, the ( ) ) L 2 log2 (7) algorithm performs at most O δ C many iterations, where L is the distance of the corner points of the initial box B(l 0, u 0 ), i.e., L := l 0 u 0. Proof. see our paper... Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

55 Correctness Corollary Let Rep be a representative system with coverage error less than or equal to δ C (w. r. t. ) and let Rep N denote all points of Rep which are not dominated by any other point in this set. Then it is ( Y N Rep N (δ C, δ C, δ C ) ) + R 3. Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

56 Correctness Corollary Let Rep be a representative system with coverage error less than or equal to δ C (w. r. t. ) and let Rep N denote all points of Rep which are not dominated by any other point in this set. Then it is ( Y N Rep N (δ C, δ C, δ C ) ) + R 3. Proof. Let y Y N. Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

57 Correctness Corollary Let Rep be a representative system with coverage error less than or equal to δ C (w. r. t. ) and let Rep N denote all points of Rep which are not dominated by any other point in this set. Then it is ( Y N Rep N (δ C, δ C, δ C ) ) + R 3. Proof. Let y Y N. Coverage property: z Rep with y z δ C which implies z (δ C, δ C, δ C ) y. Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

58 Correctness Corollary Let Rep be a representative system with coverage error less than or equal to δ C (w. r. t. ) and let Rep N denote all points of Rep which are not dominated by any other point in this set. Then it is ( Y N Rep N (δ C, δ C, δ C ) ) + R 3. Proof. Let y Y N. Coverage property: z Rep with y z δ C which implies z (δ C, δ C, δ C ) y. If z Rep N, then there exists ẑ Rep N with ẑ z. Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

59 Selection Rule max-dist selection rule Corollary Let SelectBox() always select the box with largest corner point distance. Suppose the algorithm is aborted prematurely after Γ 1 iterations and for all remaining boxes B S, we additionally execute a completion step. Then, the representative system Rep has a coverage error of at most L 2 (log 7 (6Γ+1))/2, where L equals the corner point distance l 0 u 0 of the initial box B(l 0, u 0 ). Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

60 Selection Rule nondominated selection rule Problem: Generation of (in a later step) dominated solutions. Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

61 Selection Rule nondominated selection rule Problem: Generation of (in a later step) dominated solutions. Idea: Do not select the box with maximal distance, but the box which cannot be dominated in a future iteration. Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

62 Selection Rule nondominated selection rule Problem: Generation of (in a later step) dominated solutions. Idea: Do not select the box with maximal distance, but the box which cannot be dominated in a future iteration. Store the list of unexplored boxes in analogy to a tree, where we perform a depth first search (DFS). Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

63 Selection Rule nondominated selection rule Problem: Generation of (in a later step) dominated solutions. Idea: Do not select the box with maximal distance, but the box which cannot be dominated in a future iteration. Store the list of unexplored boxes in analogy to a tree, where we perform a depth first search (DFS). Result: By appropriately cutting off dominated parts of (partial) dominated boxes during the algorithm, we obtain a representative system Rep = Rep N fulfilling the desired accuracy. Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

64 Bounding the Representation Error Rep Y N is not guaranteed in the algorithm Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

65 Bounding the Representation Error Rep Y N is not guaranteed in the algorithm The representation error max z Rep min y Y N z y can be positive. Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

66 Bounding the Representation Error Rep Y N is not guaranteed in the algorithm The representation error max z Rep min y Y N z y can be positive. Suppose representative system Rep has coverage error at most δ C w. r. t. some norm. Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

67 Bounding the Representation Error Rep Y N is not guaranteed in the algorithm The representation error max z Rep min y Y N z y can be positive. Suppose representative system Rep has coverage error at most δ C w. r. t. some norm. Definition Let B be the last subdivision (covering the whole set Y N ) obtained from Algorithm 1. Let z Rep and B(z) B be the set containing either the box for which z was computed or the corresponding child boxes contained in the corresponding quarter of the box for which z was computed. Then, we define B z := { B B : B ( z R 3 ) B B(z) }. If Bz, we call z a critical representative point. Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

68 Bounding the Representation Error Lemma For MOP, let (Rep, B) be the output of Algorithm 1 and z Rep be some representative point. Then, it holds (B z = ) = (z Y N ) Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

69 Bounding the Representation Error Lemma For MOP, let (Rep, B) be the output of Algorithm 1 and z Rep be some representative point. Then, it holds Proposition min max max z Rep B(l,u) B (B z = ) = (z Y N ) max min z y z Rep y Y N z Rep B z z l z, max z ẑ + δ C δ C (z R3 ) max ẑ B Rep where B Rep (z R 3 δ C ) := { y Rep : ỹ z R 3 y ỹ } δc and max l z returns the value 0 if B(l,u) B Bz =. z Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

70 Bounding the Representation Error Lemma For MOP, let (Rep, B) be the output of Algorithm 1 and z Rep be some representative point. Then, it holds Proposition min max max z Rep B(l,u) B (B z = ) = (z Y N ) max min z y z Rep y Y N z Rep B z z l z, max z ẑ + δ C δ C (z R3 ) max ẑ B Rep where B Rep (z R 3 δ C ) := { y Rep : ỹ z R 3 y ỹ } δc and max l z returns the value 0 if B(l,u) B Bz =. z Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

71 Bounding the Representation Error Lemma For MOP, let (Rep, B) be the output of Algorithm 1 and z Rep be some representative point. Then, it holds Proposition min max max z Rep B(l,u) B (B z = ) = (z Y N ) max min z y z Rep y Y N z Rep B z z l z, max z ẑ + δ C δ C (z R3 ) max ẑ B Rep where B Rep (z R 3 δ C ) := { y Rep : ỹ z R 3 y ỹ } δc and max l z returns the value 0 if B(l,u) B Bz =. z Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

72 Conclusion Box algorithm for computing representative sets for MOPs with p = 3 objectives easy to implement with desired coverage error with the capability of relating the run time of the algorithm to the quality of the representative system with different selection rules representative points are not necessarily nondominated but bound on representation error Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26

73 Conclusion Box algorithm for computing representative sets for MOPs with p = 3 objectives easy to implement with desired coverage error with the capability of relating the run time of the algorithm to the quality of the representative system with different selection rules representative points are not necessarily nondominated but bound on representation error Thank you for your attention! Tobias Kuhn (TU Kaiserslautern) Triobjective Box-Algorithm Lancaster, June 24, / 26