Direct Multisearch for Multiobjective Optimization Ana Lusa Custdio 1 - PowerPoint PPT Presentation

Direct Multisearch for Multiobjective Optimization Ana Luísa Custódio 1 José F. Aguilar Madeira 2 A. Ismael F. Vaz 3 Luís Nunes Vicente 4 2 IDMEC-IST, ISEL 1 Universidade Nova de Lisboa 3 Universidade do Minho 4 Universidade de Coimbra Optimization 2011 July 24-27, 2011 A.I.F. Vaz (Optimization 2011) DMS July 24-27, 2011 1 / 53

Outline Outline Introduction and motivation 1 Direct MultiSearch 2 Numerical results 3 Further improvements on DMS 4 Conclusions and references 5 A.I.F. Vaz (Optimization 2011) DMS July 24-27, 2011 2 / 53

Outline Outline Introduction and motivation 1 Direct MultiSearch 2 Numerical results 3 Further improvements on DMS 4 Conclusions and references 5 A.I.F. Vaz (Optimization 2011) DMS July 24-27, 2011 2 / 53

Outline Outline Introduction and motivation 1 Direct MultiSearch 2 Numerical results 3 Further improvements on DMS 4 Conclusions and references 5 A.I.F. Vaz (Optimization 2011) DMS July 24-27, 2011 2 / 53

Outline Outline Introduction and motivation 1 Direct MultiSearch 2 Numerical results 3 Further improvements on DMS 4 Conclusions and references 5 A.I.F. Vaz (Optimization 2011) DMS July 24-27, 2011 2 / 53

Outline Outline Introduction and motivation 1 Direct MultiSearch 2 Numerical results 3 Further improvements on DMS 4 Conclusions and references 5 A.I.F. Vaz (Optimization 2011) DMS July 24-27, 2011 2 / 53

Introduction and motivation Outline Introduction and motivation 1 Direct MultiSearch 2 Numerical results 3 Further improvements on DMS 4 Conclusions and references 5 A.I.F. Vaz (Optimization 2011) DMS July 24-27, 2011 3 / 53

Introduction and motivation 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 (Optimization 2011) DMS July 24-27, 2011 4 / 53

Introduction and motivation 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 (Optimization 2011) DMS July 24-27, 2011 4 / 53

Introduction and motivation 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 (Optimization 2011) DMS July 24-27, 2011 4 / 53

Introduction and motivation 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 (Optimization 2011) DMS July 24-27, 2011 4 / 53

Introduction and motivation 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 (Optimization 2011) DMS July 24-27, 2011 4 / 53

Direct MultiSearch Outline Introduction and motivation 1 Direct MultiSearch 2 Numerical results 3 Further improvements on DMS 4 Conclusions and references 5 A.I.F. Vaz (Optimization 2011) DMS July 24-27, 2011 5 / 53

Direct MultiSearch 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. Keeps a list of feasible nondominated points. Poll centers are chosen from the list. Successful iterations correspond to list changes. A.I.F. Vaz (Optimization 2011) DMS July 24-27, 2011 6 / 53

Direct MultiSearch 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. Keeps a list of feasible nondominated points. Poll centers are chosen from the list. Successful iterations correspond to list changes. A.I.F. Vaz (Optimization 2011) DMS July 24-27, 2011 6 / 53

Direct MultiSearch 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. Keeps a list of feasible nondominated points. Poll centers are chosen from the list. Successful iterations correspond to list changes. A.I.F. Vaz (Optimization 2011) DMS July 24-27, 2011 6 / 53

Direct MultiSearch 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. Keeps a list of feasible nondominated points. Poll centers are chosen from the list. Successful iterations correspond to list changes. A.I.F. Vaz (Optimization 2011) DMS July 24-27, 2011 6 / 53

Direct MultiSearch 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. Keeps a list of feasible nondominated points. Poll centers are chosen from the list. Successful iterations correspond to list changes. A.I.F. Vaz (Optimization 2011) DMS July 24-27, 2011 6 / 53

Direct MultiSearch 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. Keeps a list of feasible nondominated points. Poll centers are chosen from the list. Successful iterations correspond to list changes. A.I.F. Vaz (Optimization 2011) DMS July 24-27, 2011 6 / 53

Direct MultiSearch 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. Keeps a list of feasible nondominated points. Poll centers are chosen from the list. Successful iterations correspond to list changes. A.I.F. Vaz (Optimization 2011) DMS July 24-27, 2011 6 / 53

Direct MultiSearch 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. Keeps a list of feasible nondominated points. Poll centers are chosen from the list. Successful iterations correspond to list changes. A.I.F. Vaz (Optimization 2011) DMS July 24-27, 2011 6 / 53

Direct MultiSearch DMS example x 2 x 1 f 2 x 2 x 1 f 1 A.I.F. Vaz (Optimization 2011) DMS July 24-27, 2011 7 / 53

Direct MultiSearch DMS example x 2 x 1 f 2 x 2 x 1 f 1 A.I.F. Vaz (Optimization 2011) DMS July 24-27, 2011 8 / 53

Direct MultiSearch DMS example x 2 x 1 f 2 x 2 x 1 f 1 A.I.F. Vaz (Optimization 2011) DMS July 24-27, 2011 9 / 53

Direct MultiSearch DMS example x 2 x 1 f 2 x 2 x 1 f 1 A.I.F. Vaz (Optimization 2011) DMS July 24-27, 2011 10 / 53