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A New Method for Bound-Constrained Derivative-Free Global Optimization and its Application to Parameter Estimation in Astrophysics A. Ismael F. Vaz 1 Lus Nunes Vicente 2 Joo Manuel Fernandes 3 2 Production an Systems Department, University of

  1. Particle swarm New position and velocity The new particle position is updated by Update particle x p ( t + 1) = x p ( t ) + v p ( t + 1) , where v p ( t + 1) is the new velocity given by Update velocity � � � � v p j ( t + 1) = ι ( t ) v p y p j ( t ) − x p y j ( t ) − x p j ( t ) + µω 1 j ( t ) j ( t ) + νω 2 j ( t ) ˆ j ( t ) , for j = 1 , . . . , n . ι ( t ) is the inertial factor µ is the cognitive parameter and ν is the social parameter ω 1 j ( t ) and ω 2 j ( t ) are random numbers drawn from the uniform (0 , 1) distribution. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 7 / 57

  2. Particle swarm New position and velocity The new particle position is updated by Update particle x p ( t + 1) = x p ( t ) + v p ( t + 1) , where v p ( t + 1) is the new velocity given by Update velocity � � � � v p j ( t + 1) = ι ( t ) v p y p j ( t ) − x p y j ( t ) − x p j ( t ) + µω 1 j ( t ) j ( t ) + νω 2 j ( t ) ˆ j ( t ) , for j = 1 , . . . , n . ι ( t ) is the inertial factor µ is the cognitive parameter and ν is the social parameter ω 1 j ( t ) and ω 2 j ( t ) are random numbers drawn from the uniform (0 , 1) distribution. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 7 / 57

  3. Particle swarm New position and velocity The new particle position is updated by Update particle x p ( t + 1) = x p ( t ) + v p ( t + 1) , where v p ( t + 1) is the new velocity given by Update velocity � � � � v p j ( t + 1) = ι ( t ) v p y p j ( t ) − x p y j ( t ) − x p j ( t ) + µω 1 j ( t ) j ( t ) + νω 2 j ( t ) ˆ j ( t ) , for j = 1 , . . . , n . ι ( t ) is the inertial factor µ is the cognitive parameter and ν is the social parameter ω 1 j ( t ) and ω 2 j ( t ) are random numbers drawn from the uniform (0 , 1) distribution. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 7 / 57

  4. Particle swarm Handling bound constraints In particle swarm, simple bound constraints are handled by a projection onto Ω = { x ∈ R n : ℓ ≤ x ≤ u } , for all particles i = 1 , . . . , s . Projection if x i  ℓ j j ( t ) < ℓ j ,  proj Ω ( x i if x i j ( t )) = u j j ( t ) > u j , x i j ( t ) otherwise,  for j = 1 , . . . , n . Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 8 / 57

  5. Particle swarm Example iter=1, best fx=−0.6836, nfx=36 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 9 / 57

  6. Particle swarm Example iter=11, best fx=−0.0131, nfx=396 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 10 / 57

  7. Particle swarm Example iter=21, best fx=−0.0131, nfx=756 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 11 / 57

  8. Particle swarm Example iter=31, best fx=−0.0074, nfx=1116 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 12 / 57

  9. Particle swarm Example iter=41, best fx=−0.0040, nfx=1476 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 13 / 57

  10. Particle swarm Example iter=51, best fx=−0.0040, nfx=1836 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 14 / 57

  11. Particle swarm Example iter=271, best fx=−0.0000, nfx=9756 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 15 / 57

  12. Particle swarm Example iter=871, best fx=−0.0000, nfx=31356 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 16 / 57

  13. Particle swarm Example iter=1181, best fx=−0.0000, nfx=42516 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 17 / 57

  14. Particle swarm Some properties Easy to implement. Easy to deal with discrete variables. Easy to parallelize. For a correct choice of parameters the algorithm terminates ( lim t → + ∞ v ( t ) = 0 ). Uses only objective function values. Convergence for a global optimum under strong assumptions (unpractical). High number of function evaluations. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 18 / 57

  15. Particle swarm Some properties Easy to implement. Easy to deal with discrete variables. Easy to parallelize. For a correct choice of parameters the algorithm terminates ( lim t → + ∞ v ( t ) = 0 ). Uses only objective function values. Convergence for a global optimum under strong assumptions (unpractical). High number of function evaluations. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 18 / 57

  16. Particle swarm Some properties Easy to implement. Easy to deal with discrete variables. Easy to parallelize. For a correct choice of parameters the algorithm terminates ( lim t → + ∞ v ( t ) = 0 ). Uses only objective function values. Convergence for a global optimum under strong assumptions (unpractical). High number of function evaluations. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 18 / 57

  17. Particle swarm Some properties Easy to implement. Easy to deal with discrete variables. Easy to parallelize. For a correct choice of parameters the algorithm terminates ( lim t → + ∞ v ( t ) = 0 ). Uses only objective function values. Convergence for a global optimum under strong assumptions (unpractical). High number of function evaluations. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 18 / 57

  18. Particle swarm Some properties Easy to implement. Easy to deal with discrete variables. Easy to parallelize. For a correct choice of parameters the algorithm terminates ( lim t → + ∞ v ( t ) = 0 ). Uses only objective function values. Convergence for a global optimum under strong assumptions (unpractical). High number of function evaluations. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 18 / 57

  19. Particle swarm Some properties Easy to implement. Easy to deal with discrete variables. Easy to parallelize. For a correct choice of parameters the algorithm terminates ( lim t → + ∞ v ( t ) = 0 ). Uses only objective function values. Convergence for a global optimum under strong assumptions (unpractical). High number of function evaluations. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 18 / 57

  20. Particle swarm Some properties Easy to implement. Easy to deal with discrete variables. Easy to parallelize. For a correct choice of parameters the algorithm terminates ( lim t → + ∞ v ( t ) = 0 ). Uses only objective function values. Convergence for a global optimum under strong assumptions (unpractical). High number of function evaluations. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 18 / 57

  21. Coordinate search Outline Introduction 1 Particle swarm 2 Coordinate search 3 The hybrid algorithm 4 Numerical results with a set of test problems 5 Parameter estimation in Astrophysics 6 Conclusions and future work 7 Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 19 / 57

  22. Coordinate search Introduction to direct search methods Direct search methods are an important class of optimization methods that try to minimize a function by comparing objective function values at a finite number of points. Direct search methods do not use derivative information of the objective function nor try to approximate it. Coordinate search is a simple direct search method. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 20 / 57

  23. Coordinate search Introduction to direct search methods Direct search methods are an important class of optimization methods that try to minimize a function by comparing objective function values at a finite number of points. Direct search methods do not use derivative information of the objective function nor try to approximate it. Coordinate search is a simple direct search method. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 20 / 57

  24. Coordinate search Introduction to direct search methods Direct search methods are an important class of optimization methods that try to minimize a function by comparing objective function values at a finite number of points. Direct search methods do not use derivative information of the objective function nor try to approximate it. Coordinate search is a simple direct search method. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 20 / 57

  25. Coordinate search Some definitions Positive maximal basis Formed by the coordinate vectors and their negative counterparts: D ⊕ = { e 1 , . . . , e n , − e 1 , . . . , − e n } . D ⊕ spans R n with nonnegative coefficients. Coordinate search The direct search method based on D ⊕ is known as coordinate or compass search. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 21 / 57

  26. Coordinate search Some definitions Positive maximal basis Formed by the coordinate vectors and their negative counterparts: D ⊕ = { e 1 , . . . , e n , − e 1 , . . . , − e n } . D ⊕ spans R n with nonnegative coefficients. Coordinate search The direct search method based on D ⊕ is known as coordinate or compass search. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 21 / 57

  27. Coordinate search Some definitions Sets Given D ⊕ and the current point y ( t ) , two sets of points are defined: a grid M t and the poll set P t . The grid M t is given by � � y ( t ) + α ( t ) D ⊕ z, z ∈ N | D ⊕ | M t = , 0 where α ( t ) > 0 is the grid size parameter. The poll set is given by P t = { y ( t ) + α ( t ) d, d ∈ D ⊕ } . Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 22 / 57

  28. Coordinate search Example of M t and P t y(t)+ α (t)e 2 y(t)− α (t)e 1 y(t)+ α (t)e 1 y(t) y(t)− α (t)e 2 The grid M t and the set P t when D ⊕ = { e 1 , e 2 , − e 1 , − e 2 } Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 23 / 57

  29. Coordinate search Coordinate search The search step conducts a finite search on the grid M t . If no success is obtained in the search step then a poll step follows. The poll step evaluates the objective function at the elements of P t , searching for points which have a lower objective function value. If success is attained, the value of α ( t ) may be increased, otherwise it is reduced. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 24 / 57

  30. Coordinate search Coordinate search The search step conducts a finite search on the grid M t . If no success is obtained in the search step then a poll step follows. The poll step evaluates the objective function at the elements of P t , searching for points which have a lower objective function value. If success is attained, the value of α ( t ) may be increased, otherwise it is reduced. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 24 / 57

  31. Coordinate search Coordinate search The search step conducts a finite search on the grid M t . If no success is obtained in the search step then a poll step follows. The poll step evaluates the objective function at the elements of P t , searching for points which have a lower objective function value. If success is attained, the value of α ( t ) may be increased, otherwise it is reduced. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 24 / 57

  32. Coordinate search Coordinate search The search step conducts a finite search on the grid M t . If no success is obtained in the search step then a poll step follows. The poll step evaluates the objective function at the elements of P t , searching for points which have a lower objective function value. If success is attained, the value of α ( t ) may be increased, otherwise it is reduced. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 24 / 57

  33. Coordinate search Handling bound constraints For the coordinate search method it is sufficient to initialize the algorithm with a feasible initial guess ( y (0) ∈ Ω ) and to use ˆ f as the objective function. Penalty/Barrier function � f ( z ) if z ∈ Ω , ˆ f ( z ) = + ∞ otherwise. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 25 / 57

  34. The hybrid algorithm Outline Introduction 1 Particle swarm 2 Coordinate search 3 The hybrid algorithm 4 Numerical results with a set of test problems 5 Parameter estimation in Astrophysics 6 Conclusions and future work 7 Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 26 / 57

  35. The hybrid algorithm Motivation for PSwarm Hybrid algorithm The hybrid algorithm tries to combine the best of both algorithms. From particle swarm The particle swarm ability of searching for the global optimum. From coordinate search The guarantee to obtain at least a stationary point. Some robustness. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 27 / 57

  36. The hybrid algorithm Motivation for PSwarm Hybrid algorithm The hybrid algorithm tries to combine the best of both algorithms. From particle swarm The particle swarm ability of searching for the global optimum. From coordinate search The guarantee to obtain at least a stationary point. Some robustness. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 27 / 57

  37. The hybrid algorithm Motivation for PSwarm Hybrid algorithm The hybrid algorithm tries to combine the best of both algorithms. From particle swarm The particle swarm ability of searching for the global optimum. From coordinate search The guarantee to obtain at least a stationary point. Some robustness. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 27 / 57

  38. The hybrid algorithm Motivation for PSwarm Central idea A particle swarm iteration is performed in the search step and if no progress is attained a poll step is taken. Key points In the first iterations the algorithm takes advantage of the particle swarm ability to find a global optimum (exploiting the search space), while in the last iterations the algorithm takes advantage of the pattern search robustness to find a stationary point. The number of particles in the swarm search can be decreased along the iterations (no need to have a large number of particles around a local optimum). Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 28 / 57

  39. The hybrid algorithm Motivation for PSwarm Central idea A particle swarm iteration is performed in the search step and if no progress is attained a poll step is taken. Key points In the first iterations the algorithm takes advantage of the particle swarm ability to find a global optimum (exploiting the search space), while in the last iterations the algorithm takes advantage of the pattern search robustness to find a stationary point. The number of particles in the swarm search can be decreased along the iterations (no need to have a large number of particles around a local optimum). Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 28 / 57

  40. The hybrid algorithm Motivation for PSwarm Central idea A particle swarm iteration is performed in the search step and if no progress is attained a poll step is taken. Key points In the first iterations the algorithm takes advantage of the particle swarm ability to find a global optimum (exploiting the search space), while in the last iterations the algorithm takes advantage of the pattern search robustness to find a stationary point. The number of particles in the swarm search can be decreased along the iterations (no need to have a large number of particles around a local optimum). Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 28 / 57

  41. The hybrid algorithm Motivation for PSwarm Central idea A particle swarm iteration is performed in the search step and if no progress is attained a poll step is taken. Key points In the first iterations the algorithm takes advantage of the particle swarm ability to find a global optimum (exploiting the search space), while in the last iterations the algorithm takes advantage of the pattern search robustness to find a stationary point. The number of particles in the swarm search can be decreased along the iterations (no need to have a large number of particles around a local optimum). Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 28 / 57

  42. The hybrid algorithm Example iter=1, best fx=12.3615, pollsteps=0, suc=0, delta=0.81920000 nfx=20 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 29 / 57

  43. The hybrid algorithm Example iter=2, best fx=2.2032, pollsteps=1, suc=1, delta=0.81920000 nfx=43 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 30 / 57

  44. The hybrid algorithm Example iter=3, best fx=1.2456, pollsteps=1, suc=1, delta=0.81920000 nfx=60 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 31 / 57

  45. The hybrid algorithm Example iter=4, best fx=0.3038, pollsteps=1, suc=1, delta=0.81920000 nfx=70 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 32 / 57

  46. The hybrid algorithm Example iter=5, best fx=0.3038, pollsteps=2, suc=1, delta=0.40960000 nfx=81 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 33 / 57

  47. The hybrid algorithm Example iter=6, best fx=0.3038, pollsteps=3, suc=1, delta=0.20480000 nfx=90 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 34 / 57

  48. The hybrid algorithm Example iter=61, best fx=0.0543, pollsteps=58, suc=42, delta=0.00160000 nfx=400 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 35 / 57

  49. The hybrid algorithm Example iter=511, best fx=0.0264, pollsteps=211, suc=149, delta=0.40960000 nfx=1206 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 36 / 57

  50. The hybrid algorithm Example iter=6506, best fx=0.0018, pollsteps=1120, suc=499, delta=0.81920000 nfx=9997 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 37 / 57

  51. Numerical results with a set of test problems Outline Introduction 1 Particle swarm 2 Coordinate search 3 The hybrid algorithm 4 Numerical results with a set of test problems 5 Parameter estimation in Astrophysics 6 Conclusions and future work 7 Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 38 / 57

  52. Numerical results with a set of test problems Test problems 122 problems were collected from the global optimization literature. 12 problems of large dimension (between 100 and 300 variables). The others are small ( < 10 ) and medium size ( < 30 ). Majority of objective functions are differentiable, but non-convex. All problems have simple bounds on the variables (needed for the search step — particle swarm). The test problems were coded in AMPL ( A Modeling Language for Mathematical Programming ). Test problems available on http://www.norg.uminho.pt/aivaz (under software ). Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 39 / 57

  53. Numerical results with a set of test problems Test problems 122 problems were collected from the global optimization literature. 12 problems of large dimension (between 100 and 300 variables). The others are small ( < 10 ) and medium size ( < 30 ). Majority of objective functions are differentiable, but non-convex. All problems have simple bounds on the variables (needed for the search step — particle swarm). The test problems were coded in AMPL ( A Modeling Language for Mathematical Programming ). Test problems available on http://www.norg.uminho.pt/aivaz (under software ). Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 39 / 57

  54. Numerical results with a set of test problems Test problems 122 problems were collected from the global optimization literature. 12 problems of large dimension (between 100 and 300 variables). The others are small ( < 10 ) and medium size ( < 30 ). Majority of objective functions are differentiable, but non-convex. All problems have simple bounds on the variables (needed for the search step — particle swarm). The test problems were coded in AMPL ( A Modeling Language for Mathematical Programming ). Test problems available on http://www.norg.uminho.pt/aivaz (under software ). Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 39 / 57

  55. Numerical results with a set of test problems Test problems 122 problems were collected from the global optimization literature. 12 problems of large dimension (between 100 and 300 variables). The others are small ( < 10 ) and medium size ( < 30 ). Majority of objective functions are differentiable, but non-convex. All problems have simple bounds on the variables (needed for the search step — particle swarm). The test problems were coded in AMPL ( A Modeling Language for Mathematical Programming ). Test problems available on http://www.norg.uminho.pt/aivaz (under software ). Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 39 / 57

  56. Numerical results with a set of test problems Test problems 122 problems were collected from the global optimization literature. 12 problems of large dimension (between 100 and 300 variables). The others are small ( < 10 ) and medium size ( < 30 ). Majority of objective functions are differentiable, but non-convex. All problems have simple bounds on the variables (needed for the search step — particle swarm). The test problems were coded in AMPL ( A Modeling Language for Mathematical Programming ). Test problems available on http://www.norg.uminho.pt/aivaz (under software ). Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 39 / 57

  57. Numerical results with a set of test problems Test problems 122 problems were collected from the global optimization literature. 12 problems of large dimension (between 100 and 300 variables). The others are small ( < 10 ) and medium size ( < 30 ). Majority of objective functions are differentiable, but non-convex. All problems have simple bounds on the variables (needed for the search step — particle swarm). The test problems were coded in AMPL ( A Modeling Language for Mathematical Programming ). Test problems available on http://www.norg.uminho.pt/aivaz (under software ). Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 39 / 57

  58. Numerical results with a set of test problems Average objective value Average objective value of 30 runs with maxf=1000 (7500) 1 1 0.9 0.95 0.8 0.9 0.7 0.85 0.6 ρ 0.5 0.8 ρ 0.4 0.75 0.3 0.7 0.2 ASA PSwarm 0.65 PGAPack 0.1 Direct MCS 0 0.6 1 2 3 4 5 6 7 8 9 10 200 400 600 τ τ Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 40 / 57

  59. Numerical results with a set of test problems Average of objective function evaluations Average objective evaluation of 30 runs with maxf=1000 1 1 0.9 0.95 0.8 0.9 0.7 0.85 0.6 ρ 0.5 0.8 ρ 0.4 0.75 0.3 0.7 0.2 ASA PSwarm 0.65 PGAPack 0.1 Direct MCS 0 0.6 5 10 15 20 25 30 35 200 400 600 τ τ Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 41 / 57

  60. Numerical results with a set of test problems Average number of objective function evaluations maxf ASA PGAPack PSwarm Direct MCS 1009 ∗ 1107 ∗ 1837 ∗ 1000 857 686 10009 ∗ 11517 ∗ 10000 5047 3603 4469 Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 42 / 57

  61. Numerical results with a set of test problems Coordinate search vs Particle swarm vs PSwarm Average objective value of 30 runs with maxf=1000 (7500) 1 1 0.98 0.9 0.96 0.94 0.8 0.92 ρ 0.7 0.9 ρ 0.88 0.6 0.86 0.84 0.5 CS 0.82 PSwarm PSOA 0.4 0.8 1 2 3 4 5 6 7 8 9 10 200 400 600 τ τ Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 43 / 57

  62. Parameter estimation in Astrophysics Outline Introduction 1 Particle swarm 2 Coordinate search 3 The hybrid algorithm 4 Numerical results with a set of test problems 5 Parameter estimation in Astrophysics 6 Conclusions and future work 7 Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 44 / 57

  63. Parameter estimation in Astrophysics The problem Objective To determine a set of stellar parameters (that define the star internal structure and evolution) from observable information. Set of parameters to be determined M — stellar mass (relative to Sun mass M ⊙ ). X — abundance of hydrogen ( % ). Y — abundance of helium ( % ). Z — abundance of other elements ( Z = 100% − X − Y ). t — star age (in Gyr = 1000 million years). two other parameters. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 45 / 57

  64. Parameter estimation in Astrophysics The problem Objective To determine a set of stellar parameters (that define the star internal structure and evolution) from observable information. Set of parameters to be determined M — stellar mass (relative to Sun mass M ⊙ ). X — abundance of hydrogen ( % ). Y — abundance of helium ( % ). Z — abundance of other elements ( Z = 100% − X − Y ). t — star age (in Gyr = 1000 million years). two other parameters. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 45 / 57

  65. Parameter estimation in Astrophysics The problem Objective To determine a set of stellar parameters (that define the star internal structure and evolution) from observable information. Set of parameters to be determined M — stellar mass (relative to Sun mass M ⊙ ). X — abundance of hydrogen ( % ). Y — abundance of helium ( % ). Z — abundance of other elements ( Z = 100% − X − Y ). t — star age (in Gyr = 1000 million years). two other parameters. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 45 / 57

  66. Parameter estimation in Astrophysics The problem Objective To determine a set of stellar parameters (that define the star internal structure and evolution) from observable information. Set of parameters to be determined M — stellar mass (relative to Sun mass M ⊙ ). X — abundance of hydrogen ( % ). Y — abundance of helium ( % ). Z — abundance of other elements ( Z = 100% − X − Y ). t — star age (in Gyr = 1000 million years). two other parameters. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 45 / 57

  67. Parameter estimation in Astrophysics The problem Objective To determine a set of stellar parameters (that define the star internal structure and evolution) from observable information. Set of parameters to be determined M — stellar mass (relative to Sun mass M ⊙ ). X — abundance of hydrogen ( % ). Y — abundance of helium ( % ). Z — abundance of other elements ( Z = 100% − X − Y ). t — star age (in Gyr = 1000 million years). two other parameters. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 45 / 57

  68. Parameter estimation in Astrophysics The problem Objective To determine a set of stellar parameters (that define the star internal structure and evolution) from observable information. Set of parameters to be determined M — stellar mass (relative to Sun mass M ⊙ ). X — abundance of hydrogen ( % ). Y — abundance of helium ( % ). Z — abundance of other elements ( Z = 100% − X − Y ). t — star age (in Gyr = 1000 million years). two other parameters. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 45 / 57

  69. Parameter estimation in Astrophysics The problem Objective To determine a set of stellar parameters (that define the star internal structure and evolution) from observable information. Set of parameters to be determined M — stellar mass (relative to Sun mass M ⊙ ). X — abundance of hydrogen ( % ). Y — abundance of helium ( % ). Z — abundance of other elements ( Z = 100% − X − Y ). t — star age (in Gyr = 1000 million years). two other parameters. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 45 / 57

  70. Parameter estimation in Astrophysics The problem Objective To determine a set of stellar parameters (that define the star internal structure and evolution) from observable information. Set of parameters to be determined M — stellar mass (relative to Sun mass M ⊙ ). X — abundance of hydrogen ( % ). Y — abundance of helium ( % ). Z — abundance of other elements ( Z = 100% − X − Y ). t — star age (in Gyr = 1000 million years). two other parameters. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 45 / 57

  71. Parameter estimation in Astrophysics The problem Observable data from spectrum analysis t eff — stellar surface temperature. lum — total stellar luminosity. � Z � — relation between the abundance of other elements and X hydrogen. g — surface gravity (less accurate). Parameters and observable data for Sun M = 1 and t = 4 . 6 Gyr, with t eff = 5777 , lum = 1 and Z/X = 0 . 0245 . This information is only available for Sun. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 46 / 57

  72. Parameter estimation in Astrophysics The problem Observable data from spectrum analysis t eff — stellar surface temperature. lum — total stellar luminosity. � Z � — relation between the abundance of other elements and X hydrogen. g — surface gravity (less accurate). Parameters and observable data for Sun M = 1 and t = 4 . 6 Gyr, with t eff = 5777 , lum = 1 and Z/X = 0 . 0245 . This information is only available for Sun. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 46 / 57

  73. Parameter estimation in Astrophysics The problem Observable data from spectrum analysis t eff — stellar surface temperature. lum — total stellar luminosity. � Z � — relation between the abundance of other elements and X hydrogen. g — surface gravity (less accurate). Parameters and observable data for Sun M = 1 and t = 4 . 6 Gyr, with t eff = 5777 , lum = 1 and Z/X = 0 . 0245 . This information is only available for Sun. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 46 / 57

  74. Parameter estimation in Astrophysics The problem Observable data from spectrum analysis t eff — stellar surface temperature. lum — total stellar luminosity. � Z � — relation between the abundance of other elements and X hydrogen. g — surface gravity (less accurate). Parameters and observable data for Sun M = 1 and t = 4 . 6 Gyr, with t eff = 5777 , lum = 1 and Z/X = 0 . 0245 . This information is only available for Sun. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 46 / 57

  75. Parameter estimation in Astrophysics The problem Observable data from spectrum analysis t eff — stellar surface temperature. lum — total stellar luminosity. � Z � — relation between the abundance of other elements and X hydrogen. g — surface gravity (less accurate). Parameters and observable data for Sun M = 1 and t = 4 . 6 Gyr, with t eff = 5777 , lum = 1 and Z/X = 0 . 0245 . This information is only available for Sun. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 46 / 57

  76. Parameter estimation in Astrophysics The problem Observable data from spectrum analysis t eff — stellar surface temperature. lum — total stellar luminosity. � Z � — relation between the abundance of other elements and X hydrogen. g — surface gravity (less accurate). Parameters and observable data for Sun M = 1 and t = 4 . 6 Gyr, with t eff = 5777 , lum = 1 and Z/X = 0 . 0245 . This information is only available for Sun. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 46 / 57

  77. Parameter estimation in Astrophysics The optimization problem The optimization problem � 2 � 2 � t eff − t eff,obs � lum − lum obs min + δt eff,obs δlum obs M,t,X,Y � 1 − X − Y � Z � 2 � � 2 − � g − g obs X X obs + + � Z � δg obs δ X obs Given M , t and fixing X , Y ( α and ov ) the parameters t eff , lum and g are computed by simulating (CESAM code) a system of differentiable equations. The equations of internal structure are five: conservation of mass and energy, hydrostatic equilibrium, energy transport, production and destruction of chemical elements by thermonuclear reactions. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 47 / 57

  78. Parameter estimation in Astrophysics The optimization problem The optimization problem � 2 � 2 � t eff − t eff,obs � lum − lum obs min + δt eff,obs δlum obs M,t,X,Y � 1 − X − Y � Z � 2 � � 2 − � g − g obs X X obs + + � Z � δg obs δ X obs Given M , t and fixing X , Y ( α and ov ) the parameters t eff , lum and g are computed by simulating (CESAM code) a system of differentiable equations. The equations of internal structure are five: conservation of mass and energy, hydrostatic equilibrium, energy transport, production and destruction of chemical elements by thermonuclear reactions. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 47 / 57

  79. Parameter estimation in Astrophysics The optimization problem The optimization problem � 2 � 2 � t eff − t eff,obs � lum − lum obs min + δt eff,obs δlum obs M,t,X,Y � 1 − X − Y � Z � 2 � � 2 − � g − g obs X X obs + + � Z � δg obs δ X obs Given M , t and fixing X , Y ( α and ov ) the parameters t eff , lum and g are computed by simulating (CESAM code) a system of differentiable equations. The equations of internal structure are five: conservation of mass and energy, hydrostatic equilibrium, energy transport, production and destruction of chemical elements by thermonuclear reactions. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 47 / 57

  80. Parameter estimation in Astrophysics Numerical results Getting t eff , lum and g – CESAM t eff , lum and g are computed by CESAM (Fortran 77 code), which is viewed as a black box function for the optimization process. Optimization solver – PSwarm PSwarm (C code). Solver used with default options. Linking PSwarm and CESAM Optimization solver communicates with CESAM by input and output files. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 48 / 57

  81. Parameter estimation in Astrophysics Numerical results Getting t eff , lum and g – CESAM t eff , lum and g are computed by CESAM (Fortran 77 code), which is viewed as a black box function for the optimization process. Optimization solver – PSwarm PSwarm (C code). Solver used with default options. Linking PSwarm and CESAM Optimization solver communicates with CESAM by input and output files. Vaz, Vicente, Fernandes (Opt 2007) Global optimization in Astrophysics July 22-25, 2007 48 / 57

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