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Direct multisearch for multiobjective optimization A. Ismael F. Vaz University of Minho - Portugal aivaz@dps.uminho.pt Joint work with A. L. Custdio, J. F. A. Madeira, and L. N. Vicente Southampton October 21, 2010 A.I.F. Vaz (UMinho) DMS

  1. Direct search for single objective Outline Introduction 1 Direct search 2 Direct search for single objective 3 Direct search for multiobjective 4 Numerical results 5 Conclusions and references 6 A.I.F. Vaz (UMinho) DMS October 21, 2010 8 / 64

  2. Direct search for single objective Derivative-free optimization Problem formulation (single objective) min x ∈ Ω f ( x ) where Ω = { x ∈ R n : ℓ ≤ x ≤ u } f : R n → R ∪ { + ∞} , ℓ ∈ ( R ∪ {−∞} ) n and u ∈ ( R ∪ { + ∞} ) n We aim at solving this problem without using derivatives of f . A.I.F. Vaz (UMinho) DMS October 21, 2010 9 / 64

  3. Direct search for single objective Some definitions Positive spanning set Is a set of vectors that spans R n with nonnegative coefficients. Examples D ⊕ = { e 1 , . . . , e n , − e 1 , . . . , − e n } D ⊗ = { e 1 , . . . , e n , − e 1 , . . . , − e n , e, − e } Extreme barrier function � f ( x ) if x ∈ Ω , f Ω ( x ) = + ∞ otherwise. A.I.F. Vaz (UMinho) DMS October 21, 2010 10 / 64

  4. Direct search for single objective Some definitions Positive spanning set Is a set of vectors that spans R n with nonnegative coefficients. Examples D ⊕ = { e 1 , . . . , e n , − e 1 , . . . , − e n } D ⊗ = { e 1 , . . . , e n , − e 1 , . . . , − e n , e, − e } Extreme barrier function � f ( x ) if x ∈ Ω , f Ω ( x ) = + ∞ otherwise. A.I.F. Vaz (UMinho) DMS October 21, 2010 10 / 64

  5. Direct search for single objective Some definitions Positive spanning set Is a set of vectors that spans R n with nonnegative coefficients. Examples D ⊕ = { e 1 , . . . , e n , − e 1 , . . . , − e n } D ⊗ = { e 1 , . . . , e n , − e 1 , . . . , − e n , e, − e } Extreme barrier function � f ( x ) if x ∈ Ω , f Ω ( x ) = + ∞ otherwise. A.I.F. Vaz (UMinho) DMS October 21, 2010 10 / 64

  6. Direct search for single objective A direct-search method (0) Initialization Choose x 0 ∈ Ω , α 0 > 0 . Let D be a (possibly infinite) set of positive spanning sets. For k = 0 , 1 , 2 , . . . (1) Search step (Optional) Try to compute a point x , using a finite number of trial points, in the grid � x k + α k D k z, z ∈ N | D k | � M k = 0 with D k ⊆ D and f Ω ( x ) < f ( x k ) . If f Ω ( x ) < f ( x k ) then set x k +1 = x , declare the iteration and the search step successful, and skip the poll step. A.I.F. Vaz (UMinho) DMS October 21, 2010 11 / 64

  7. Direct search for single objective A direct-search method (0) Initialization Choose x 0 ∈ Ω , α 0 > 0 . Let D be a (possibly infinite) set of positive spanning sets. For k = 0 , 1 , 2 , . . . (1) Search step (Optional) Try to compute a point x , using a finite number of trial points, in the grid � x k + α k D k z, z ∈ N | D k | � M k = 0 with D k ⊆ D and f Ω ( x ) < f ( x k ) . If f Ω ( x ) < f ( x k ) then set x k +1 = x , declare the iteration and the search step successful, and skip the poll step. A.I.F. Vaz (UMinho) DMS October 21, 2010 11 / 64

  8. Direct search for single objective A direct-search method (0) Initialization Choose x 0 ∈ Ω , α 0 > 0 . Let D be a (possibly infinite) set of positive spanning sets. For k = 0 , 1 , 2 , . . . (1) Search step (Optional) Try to compute a point x , using a finite number of trial points, in the grid � x k + α k D k z, z ∈ N | D k | � M k = 0 with D k ⊆ D and f Ω ( x ) < f ( x k ) . If f Ω ( x ) < f ( x k ) then set x k +1 = x , declare the iteration and the search step successful, and skip the poll step. A.I.F. Vaz (UMinho) DMS October 21, 2010 11 / 64

  9. Direct search for single objective A direct-search method (0) Initialization Choose x 0 ∈ Ω , α 0 > 0 . Let D be a (possibly infinite) set of positive spanning sets. For k = 0 , 1 , 2 , . . . (1) Search step (Optional) Try to compute a point x , using a finite number of trial points, in the grid � x k + α k D k z, z ∈ N | D k | � M k = 0 with D k ⊆ D and f Ω ( x ) < f ( x k ) . If f Ω ( x ) < f ( x k ) then set x k +1 = x , declare the iteration and the search step successful, and skip the poll step. A.I.F. Vaz (UMinho) DMS October 21, 2010 11 / 64

  10. Direct search for single objective A direct-search method (0) Initialization Choose x 0 ∈ Ω , α 0 > 0 . Let D be a (possibly infinite) set of positive spanning sets. For k = 0 , 1 , 2 , . . . (1) Search step (Optional) Try to compute a point x , using a finite number of trial points, in the grid � x k + α k D k z, z ∈ N | D k | � M k = 0 with D k ⊆ D and f Ω ( x ) < f ( x k ) . If f Ω ( x ) < f ( x k ) then set x k +1 = x , declare the iteration and the search step successful, and skip the poll step. A.I.F. Vaz (UMinho) DMS October 21, 2010 11 / 64

  11. Direct search for single objective A direct-search method (2) Poll step Optionally order the poll set P k = { x k + α k d, d ∈ D k } with D k ⊆ D . If a poll point x k + α k d k is found such that f Ω ( x k + α k d k ) < f ( x k ) then stop polling, set x k +1 = x k + α k d k , and declare the iteration and the poll step successful. Otherwise declare the iteration (and the poll step) unsuccessful and set x k +1 = x k . A.I.F. Vaz (UMinho) DMS October 21, 2010 12 / 64

  12. Direct search for single objective A direct-search method (2) Poll step Optionally order the poll set P k = { x k + α k d, d ∈ D k } with D k ⊆ D . If a poll point x k + α k d k is found such that f Ω ( x k + α k d k ) < f ( x k ) then stop polling, set x k +1 = x k + α k d k , and declare the iteration and the poll step successful. Otherwise declare the iteration (and the poll step) unsuccessful and set x k +1 = x k . A.I.F. Vaz (UMinho) DMS October 21, 2010 12 / 64

  13. Direct search for single objective A direct-search method (2) Poll step Optionally order the poll set P k = { x k + α k d, d ∈ D k } with D k ⊆ D . If a poll point x k + α k d k is found such that f Ω ( x k + α k d k ) < f ( x k ) then stop polling, set x k +1 = x k + α k d k , and declare the iteration and the poll step successful. Otherwise declare the iteration (and the poll step) unsuccessful and set x k +1 = x k . A.I.F. Vaz (UMinho) DMS October 21, 2010 12 / 64

  14. Direct search for single objective A direct-search method (3) Step size update: If the iteration was successful then maintain the step size parameter ( α k +1 = α k ) or double it ( α k +1 = 2 α k ) after two consecutive poll successes along the same direction. If the iteration was unsuccessful, halve the step size parameter ( α k +1 = α k / 2 ). A.I.F. Vaz (UMinho) DMS October 21, 2010 13 / 64

  15. Direct search for single objective A direct-search method (3) Step size update: If the iteration was successful then maintain the step size parameter ( α k +1 = α k ) or double it ( α k +1 = 2 α k ) after two consecutive poll successes along the same direction. If the iteration was unsuccessful, halve the step size parameter ( α k +1 = α k / 2 ). A.I.F. Vaz (UMinho) DMS October 21, 2010 13 / 64

  16. Direct search for single objective Some comments We could present the previous algorithm in a different form, namely by fixing the set D k ( D k = D, ∀ k ) not to change with the iteration number (problem with only bound constraints). allowing the set D k to be computed in a way to conform with possible linear constraints. to use a forcing function ρ ( · ) ( e.g. , ρ ( t ) = t 2 ) instead of a integer lattice (the mesh M k ). A forcing function ρ ( · ) is continuous, positive, and satisfies lim t − → 0 + ρ ( t ) /t = 0 and ρ ( t 1 ) ≤ ρ ( t 2 ) if t 1 < t 2 . A point x is accepted (successful) in the search step if f Ω ( x ) < f ( x k ) − ρ ( α k ) . A.I.F. Vaz (UMinho) DMS October 21, 2010 14 / 64

  17. Direct search for single objective Some comments We could present the previous algorithm in a different form, namely by fixing the set D k ( D k = D, ∀ k ) not to change with the iteration number (problem with only bound constraints). allowing the set D k to be computed in a way to conform with possible linear constraints. to use a forcing function ρ ( · ) ( e.g. , ρ ( t ) = t 2 ) instead of a integer lattice (the mesh M k ). A forcing function ρ ( · ) is continuous, positive, and satisfies lim t − → 0 + ρ ( t ) /t = 0 and ρ ( t 1 ) ≤ ρ ( t 2 ) if t 1 < t 2 . A point x is accepted (successful) in the search step if f Ω ( x ) < f ( x k ) − ρ ( α k ) . A.I.F. Vaz (UMinho) DMS October 21, 2010 14 / 64

  18. Direct search for single objective Some comments We could present the previous algorithm in a different form, namely by fixing the set D k ( D k = D, ∀ k ) not to change with the iteration number (problem with only bound constraints). allowing the set D k to be computed in a way to conform with possible linear constraints. to use a forcing function ρ ( · ) ( e.g. , ρ ( t ) = t 2 ) instead of a integer lattice (the mesh M k ). A forcing function ρ ( · ) is continuous, positive, and satisfies lim t − → 0 + ρ ( t ) /t = 0 and ρ ( t 1 ) ≤ ρ ( t 2 ) if t 1 < t 2 . A point x is accepted (successful) in the search step if f Ω ( x ) < f ( x k ) − ρ ( α k ) . A.I.F. Vaz (UMinho) DMS October 21, 2010 14 / 64

  19. Direct search for single objective Some comments We could present the previous algorithm in a different form, namely by fixing the set D k ( D k = D, ∀ k ) not to change with the iteration number (problem with only bound constraints). allowing the set D k to be computed in a way to conform with possible linear constraints. to use a forcing function ρ ( · ) ( e.g. , ρ ( t ) = t 2 ) instead of a integer lattice (the mesh M k ). A forcing function ρ ( · ) is continuous, positive, and satisfies lim t − → 0 + ρ ( t ) /t = 0 and ρ ( t 1 ) ≤ ρ ( t 2 ) if t 1 < t 2 . A point x is accepted (successful) in the search step if f Ω ( x ) < f ( x k ) − ρ ( α k ) . The presented algorithm just suits for the multiobjective version to be described. A.I.F. Vaz (UMinho) DMS October 21, 2010 14 / 64

  20. Direct search for multiobjective Outline Introduction 1 Direct search 2 Direct search for single objective 3 Direct search for multiobjective 4 Numerical results 5 Conclusions and references 6 A.I.F. Vaz (UMinho) DMS October 21, 2010 15 / 64

  21. Direct search for multiobjective Derivative-free multiobjective optimization MOO problem x ∈ Ω F ( x ) ≡ ( f 1 ( x ) , f 2 ( x ) , . . . , f m ( x )) ⊤ min where Ω = { x ∈ R n : ℓ ≤ x ≤ u } f j : R n → R ∪ { + ∞} , j = 1 , . . . , m , ℓ ∈ ( R ∪ {−∞} ) n and u ∈ ( R ∪ { + ∞} ) n Several objectives, often conflicting. Functions with unknown derivatives. Expensive function evaluations, possibly subject to noise. Impractical to compute approximations to derivatives. A.I.F. Vaz (UMinho) DMS October 21, 2010 16 / 64

  22. Direct search for multiobjective Derivative-free multiobjective optimization MOO problem x ∈ Ω F ( x ) ≡ ( f 1 ( x ) , f 2 ( x ) , . . . , f m ( x )) ⊤ min where Ω = { x ∈ R n : ℓ ≤ x ≤ u } f j : R n → R ∪ { + ∞} , j = 1 , . . . , m , ℓ ∈ ( R ∪ {−∞} ) n and u ∈ ( R ∪ { + ∞} ) n Several objectives, often conflicting. Functions with unknown derivatives. Expensive function evaluations, possibly subject to noise. Impractical to compute approximations to derivatives. A.I.F. Vaz (UMinho) DMS October 21, 2010 16 / 64

  23. Direct search for multiobjective Derivative-free multiobjective optimization MOO problem x ∈ Ω F ( x ) ≡ ( f 1 ( x ) , f 2 ( x ) , . . . , f m ( x )) ⊤ min where Ω = { x ∈ R n : ℓ ≤ x ≤ u } f j : R n → R ∪ { + ∞} , j = 1 , . . . , m , ℓ ∈ ( R ∪ {−∞} ) n and u ∈ ( R ∪ { + ∞} ) n Several objectives, often conflicting. Functions with unknown derivatives. Expensive function evaluations, possibly subject to noise. Impractical to compute approximations to derivatives. A.I.F. Vaz (UMinho) DMS October 21, 2010 16 / 64

  24. Direct search for multiobjective Derivative-free multiobjective optimization MOO problem x ∈ Ω F ( x ) ≡ ( f 1 ( x ) , f 2 ( x ) , . . . , f m ( x )) ⊤ min where Ω = { x ∈ R n : ℓ ≤ x ≤ u } f j : R n → R ∪ { + ∞} , j = 1 , . . . , m , ℓ ∈ ( R ∪ {−∞} ) n and u ∈ ( R ∪ { + ∞} ) n Several objectives, often conflicting. Functions with unknown derivatives. Expensive function evaluations, possibly subject to noise. Impractical to compute approximations to derivatives. A.I.F. Vaz (UMinho) DMS October 21, 2010 16 / 64

  25. Direct search for multiobjective DMS algorithmic main lines Does not aggregate any of the objective functions. Generalizes ALL direct-search methods of directional type to MOO. Makes use of search/poll paradigm. Implements an optional search step (only to disseminate the search). Tries to capture the whole Pareto front from the polling procedure. A.I.F. Vaz (UMinho) DMS October 21, 2010 17 / 64

  26. Direct search for multiobjective DMS algorithmic main lines Does not aggregate any of the objective functions. Generalizes ALL direct-search methods of directional type to MOO. Makes use of search/poll paradigm. Implements an optional search step (only to disseminate the search). Tries to capture the whole Pareto front from the polling procedure. A.I.F. Vaz (UMinho) DMS October 21, 2010 17 / 64

  27. Direct search for multiobjective DMS algorithmic main lines Does not aggregate any of the objective functions. Generalizes ALL direct-search methods of directional type to MOO. Makes use of search/poll paradigm. Implements an optional search step (only to disseminate the search). Tries to capture the whole Pareto front from the polling procedure. A.I.F. Vaz (UMinho) DMS October 21, 2010 17 / 64

  28. Direct search for multiobjective DMS algorithmic main lines Does not aggregate any of the objective functions. Generalizes ALL direct-search methods of directional type to MOO. Makes use of search/poll paradigm. Implements an optional search step (only to disseminate the search). Tries to capture the whole Pareto front from the polling procedure. A.I.F. Vaz (UMinho) DMS October 21, 2010 17 / 64

  29. Direct search for multiobjective DMS algorithmic main lines Does not aggregate any of the objective functions. Generalizes ALL direct-search methods of directional type to MOO. Makes use of search/poll paradigm. Implements an optional search step (only to disseminate the search). Tries to capture the whole Pareto front from the polling procedure. A.I.F. Vaz (UMinho) DMS October 21, 2010 17 / 64

  30. Direct search for multiobjective DMS algorithmic main lines Keeps a list of feasible nondominated points. Poll centers are chosen from the list. Successful iterations correspond to list changes. A.I.F. Vaz (UMinho) DMS October 21, 2010 18 / 64

  31. Direct search for multiobjective DMS algorithmic main lines Keeps a list of feasible nondominated points. Poll centers are chosen from the list. Successful iterations correspond to list changes. A.I.F. Vaz (UMinho) DMS October 21, 2010 18 / 64

  32. Direct search for multiobjective DMS algorithmic main lines Keeps a list of feasible nondominated points. Poll centers are chosen from the list. Successful iterations correspond to list changes. A.I.F. Vaz (UMinho) DMS October 21, 2010 18 / 64

  33. Direct search for multiobjective DMS example A.I.F. Vaz (UMinho) DMS October 21, 2010 19 / 64

  34. Direct search for multiobjective DMS example A.I.F. Vaz (UMinho) DMS October 21, 2010 20 / 64

  35. Direct search for multiobjective DMS example A.I.F. Vaz (UMinho) DMS October 21, 2010 21 / 64

  36. Direct search for multiobjective DMS example A.I.F. Vaz (UMinho) DMS October 21, 2010 22 / 64

  37. Direct search for multiobjective DMS example A.I.F. Vaz (UMinho) DMS October 21, 2010 23 / 64

  38. Direct search for multiobjective DMS example A.I.F. Vaz (UMinho) DMS October 21, 2010 24 / 64

  39. Direct search for multiobjective DMS example A.I.F. Vaz (UMinho) DMS October 21, 2010 25 / 64

  40. Direct search for multiobjective DMS example A.I.F. Vaz (UMinho) DMS October 21, 2010 26 / 64

  41. Direct search for multiobjective DMS example A.I.F. Vaz (UMinho) DMS October 21, 2010 27 / 64

  42. Direct search for multiobjective DMS search & poll steps Evaluate a finite set of feasible points ֒ → L add . Remove dominated points from L k ∪ L add ֒ → L filtered . Select list of feasible nondominated points ֒ → L trial . Compare L trial to L k (success if L trial � = L k , unsuccess otherwise). A.I.F. Vaz (UMinho) DMS October 21, 2010 28 / 64

  43. Direct search for multiobjective DMS search & poll steps Evaluate a finite set of feasible points ֒ → L add . Remove dominated points from L k ∪ L add ֒ → L filtered . Select list of feasible nondominated points ֒ → L trial . Compare L trial to L k (success if L trial � = L k , unsuccess otherwise). A.I.F. Vaz (UMinho) DMS October 21, 2010 28 / 64

  44. Direct search for multiobjective DMS search & poll steps Evaluate a finite set of feasible points ֒ → L add . Remove dominated points from L k ∪ L add ֒ → L filtered . Select list of feasible nondominated points ֒ → L trial . Compare L trial to L k (success if L trial � = L k , unsuccess otherwise). A.I.F. Vaz (UMinho) DMS October 21, 2010 28 / 64

  45. Direct search for multiobjective DMS search & poll steps Evaluate a finite set of feasible points ֒ → L add . Remove dominated points from L k ∪ L add ֒ → L filtered . Select list of feasible nondominated points ֒ → L trial . Compare L trial to L k (success if L trial � = L k , unsuccess otherwise). A.I.F. Vaz (UMinho) DMS October 21, 2010 28 / 64

  46. Direct search for multiobjective Direct MultiSearch for MOO (0) Initialization Choose x 0 ∈ Ω with F ( x 0 ) < + ∞ , α 0 > 0 . Set L 0 = { ( x 0 ; α 0 ) } . Let D be a (possibly infinite) set of positive spanning sets. For k = 0 , 1 , 2 , . . . (1) Selection of iterate point Order L k and select ( x k ; α k ) ∈ L k . A.I.F. Vaz (UMinho) DMS October 21, 2010 29 / 64

  47. Direct search for multiobjective Direct MultiSearch for MOO (0) Initialization Choose x 0 ∈ Ω with F ( x 0 ) < + ∞ , α 0 > 0 . Set L 0 = { ( x 0 ; α 0 ) } . Let D be a (possibly infinite) set of positive spanning sets. For k = 0 , 1 , 2 , . . . (1) Selection of iterate point Order L k and select ( x k ; α k ) ∈ L k . A.I.F. Vaz (UMinho) DMS October 21, 2010 29 / 64

  48. Direct search for multiobjective Direct MultiSearch for MOO (0) Initialization Choose x 0 ∈ Ω with F ( x 0 ) < + ∞ , α 0 > 0 . Set L 0 = { ( x 0 ; α 0 ) } . Let D be a (possibly infinite) set of positive spanning sets. For k = 0 , 1 , 2 , . . . (1) Selection of iterate point Order L k and select ( x k ; α k ) ∈ L k . A.I.F. Vaz (UMinho) DMS October 21, 2010 29 / 64

  49. Direct search for multiobjective Direct MultiSearch for MOO (0) Initialization Choose x 0 ∈ Ω with F ( x 0 ) < + ∞ , α 0 > 0 . Set L 0 = { ( x 0 ; α 0 ) } . Let D be a (possibly infinite) set of positive spanning sets. For k = 0 , 1 , 2 , . . . (1) Selection of iterate point Order L k and select ( x k ; α k ) ∈ L k . A.I.F. Vaz (UMinho) DMS October 21, 2010 29 / 64

  50. Direct search for multiobjective Direct MultiSearch for MOO (2) Search step (Optional) Evaluate a finite set of points L add = { ( z s ; α k ) } s ∈ S (in the mesh or using a forcing function). ( L k ; L add ) ֒ → L filtered ֒ → L trial If success is achieved then set L k +1 = L trial , declare the iteration and the search step successful, and skip the poll step. A.I.F. Vaz (UMinho) DMS October 21, 2010 30 / 64

  51. Direct search for multiobjective Direct MultiSearch for MOO (2) Search step (Optional) Evaluate a finite set of points L add = { ( z s ; α k ) } s ∈ S (in the mesh or using a forcing function). ( L k ; L add ) ֒ → L filtered ֒ → L trial If success is achieved then set L k +1 = L trial , declare the iteration and the search step successful, and skip the poll step. A.I.F. Vaz (UMinho) DMS October 21, 2010 30 / 64

  52. Direct search for multiobjective Direct MultiSearch for MOO (3) Poll step Evaluate L add = { ( x k + α k d ; α k ) , d ∈ D k }, with D k ⊆ D ( L k ; L add ) ֒ → L filtered ֒ → L trial If success is achieved then set L k +1 = L trial , declare the iteration and the poll step successful Otherwise declare the iteration (and the poll step) unsuccessful and set L k +1 = L trial A.I.F. Vaz (UMinho) DMS October 21, 2010 31 / 64

  53. Direct search for multiobjective Direct MultiSearch for MOO (3) Poll step Evaluate L add = { ( x k + α k d ; α k ) , d ∈ D k }, with D k ⊆ D ( L k ; L add ) ֒ → L filtered ֒ → L trial If success is achieved then set L k +1 = L trial , declare the iteration and the poll step successful Otherwise declare the iteration (and the poll step) unsuccessful and set L k +1 = L trial A.I.F. Vaz (UMinho) DMS October 21, 2010 31 / 64

  54. Direct search for multiobjective Direct MultiSearch for MOO (3) Poll step Evaluate L add = { ( x k + α k d ; α k ) , d ∈ D k }, with D k ⊆ D ( L k ; L add ) ֒ → L filtered ֒ → L trial If success is achieved then set L k +1 = L trial , declare the iteration and the poll step successful Otherwise declare the iteration (and the poll step) unsuccessful and set L k +1 = L trial A.I.F. Vaz (UMinho) DMS October 21, 2010 31 / 64

  55. Direct search for multiobjective Direct MultiSearch for MOO (4) Step size update: If the iteration was successful then maintain the step size parameter ( α k +1 = α k ) or double it ( α k +1 = 2 α k ) after two consecutive poll successes along the same direction. If the iteration was unsuccessful, halve the step size parameter ( α k +1 = α k / 2 ). A.I.F. Vaz (UMinho) DMS October 21, 2010 32 / 64

  56. Direct search for multiobjective Direct MultiSearch for MOO (4) Step size update: If the iteration was successful then maintain the step size parameter ( α k +1 = α k ) or double it ( α k +1 = 2 α k ) after two consecutive poll successes along the same direction. If the iteration was unsuccessful, halve the step size parameter ( α k +1 = α k / 2 ). A.I.F. Vaz (UMinho) DMS October 21, 2010 32 / 64

  57. Direct search for multiobjective Numerical Example — Problem SP1 [Huband et al.] � Evaluated points since beginning. � Current iterate list. A.I.F. Vaz (UMinho) DMS October 21, 2010 33 / 64

  58. Direct search for multiobjective Numerical example — problem SP1 [Huband et al.] � Evaluated poll points. � Evaluated points since beginning. A.I.F. Vaz (UMinho) DMS October 21, 2010 34 / 64

  59. Direct search for multiobjective Numerical example — problem SP1 [Huband et al.] • Nondominated evaluated poll points. A.I.F. Vaz (UMinho) DMS October 21, 2010 35 / 64

  60. Direct search for multiobjective Numerical example — problem SP1 [Huband et al.] � Evaluated poll points. � Evaluated points since beginning. � Current iterate list. A.I.F. Vaz (UMinho) DMS October 21, 2010 36 / 64

  61. Direct search for multiobjective Numerical example — problem SP1 [Huband et al.] � Evaluated poll points. � Evaluated points since beginning. A.I.F. Vaz (UMinho) DMS October 21, 2010 37 / 64

  62. Direct search for multiobjective Numerical example — problem SP1 [Huband et al.] • Nondominated evaluated poll points. A.I.F. Vaz (UMinho) DMS October 21, 2010 38 / 64

  63. Direct search for multiobjective Numerical example — problem SP1 [Huband et al.] � Evaluated poll points. � Evaluated points since beginning. � Current iterate list. A.I.F. Vaz (UMinho) DMS October 21, 2010 39 / 64

  64. Direct search for multiobjective Numerical example — problem SP1 [Huband et al.] � Evaluated poll points. � Evaluated points since beginning. � Current iterate list. A.I.F. Vaz (UMinho) DMS October 21, 2010 40 / 64

  65. Direct search for multiobjective Numerical example — problem SP1 [Huband et al.] � Evaluated poll points. � Evaluated points since beginning. � Current iterate list. A.I.F. Vaz (UMinho) DMS October 21, 2010 41 / 64

  66. Direct search for multiobjective Numerical example — problem SP1 [Huband et al.] � Evaluated poll points. � Evaluated points since beginning. � Current iterate list. A.I.F. Vaz (UMinho) DMS October 21, 2010 42 / 64

  67. Direct search for multiobjective Numerical example — problem SP1 [Huband et al.] � Evaluated poll points. � Evaluated points since beginning. � Current iterate list. A.I.F. Vaz (UMinho) DMS October 21, 2010 43 / 64

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