Particle swarm algorithms for multi-local optimization A. Ismael F - PowerPoint PPT Presentation

Particle swarm algorithms for multi-local optimization A. Ismael F . Vaz Edite M.G.P . Fernandes Production and System Department Engineering School Minho University {aivaz,emgpf}@dps.uminho.pt Work partially supported by FCT grant POCI/MAT/58957/2004 and Algoritmi research center A. Ismael F. Vaz and Edite M.G.P . Fernandes CEIO, Guimarães, 26-28 October, 2005 - p. 1/17

Outline ● Motivation ● The multi-local optimization problem Outline ● The particle swarm paradigm for global optimization ❖ Outline ● Particle swarm variants for multi-local optimization Motivation ● Implementation Multi-local ● Numerical results The PSP ● Conclusions MLPSO Implementation Numerical results Conclusions The end A. Ismael F. Vaz and Edite M.G.P . Fernandes CEIO, Guimarães, 26-28 October, 2005 - p. 2/17

Outline ● Motivation ● The multi-local optimization problem Outline ● The particle swarm paradigm for global optimization ❖ Outline ● Particle swarm variants for multi-local optimization Motivation ● Implementation Multi-local ● Numerical results The PSP ● Conclusions MLPSO Implementation Numerical results Conclusions The end A. Ismael F. Vaz and Edite M.G.P . Fernandes CEIO, Guimarães, 26-28 October, 2005 - p. 2/17

Outline ● Motivation ● The multi-local optimization problem Outline ● The particle swarm paradigm for global optimization ❖ Outline ● Particle swarm variants for multi-local optimization Motivation ● Implementation Multi-local ● Numerical results The PSP ● Conclusions MLPSO Implementation Numerical results Conclusions The end A. Ismael F. Vaz and Edite M.G.P . Fernandes CEIO, Guimarães, 26-28 October, 2005 - p. 2/17

Outline ● Motivation ● The multi-local optimization problem Outline ● The particle swarm paradigm for global optimization ❖ Outline ● Particle swarm variants for multi-local optimization Motivation ● Implementation Multi-local ● Numerical results The PSP ● Conclusions MLPSO Implementation Numerical results Conclusions The end A. Ismael F. Vaz and Edite M.G.P . Fernandes CEIO, Guimarães, 26-28 October, 2005 - p. 2/17

Outline ● Motivation ● The multi-local optimization problem Outline ● The particle swarm paradigm for global optimization ❖ Outline ● Particle swarm variants for multi-local optimization Motivation ● Implementation Multi-local ● Numerical results The PSP ● Conclusions MLPSO Implementation Numerical results Conclusions The end A. Ismael F. Vaz and Edite M.G.P . Fernandes CEIO, Guimarães, 26-28 October, 2005 - p. 2/17

Outline ● Motivation ● The multi-local optimization problem Outline ● The particle swarm paradigm for global optimization ❖ Outline ● Particle swarm variants for multi-local optimization Motivation ● Implementation Multi-local ● Numerical results The PSP ● Conclusions MLPSO Implementation Numerical results Conclusions The end A. Ismael F. Vaz and Edite M.G.P . Fernandes CEIO, Guimarães, 26-28 October, 2005 - p. 2/17

Outline ● Motivation ● The multi-local optimization problem Outline ● The particle swarm paradigm for global optimization ❖ Outline ● Particle swarm variants for multi-local optimization Motivation ● Implementation Multi-local ● Numerical results The PSP ● Conclusions MLPSO Implementation Numerical results Conclusions The end A. Ismael F. Vaz and Edite M.G.P . Fernandes CEIO, Guimarães, 26-28 October, 2005 - p. 2/17

Motivation One of the (many) applications of multi-local optimization is in reduction type methods for semi-infinite programming (SIP) Outline problems. Motivation ❖ Motivation Multi-local The PSP MLPSO Implementation Numerical results Conclusions The end A. Ismael F. Vaz and Edite M.G.P . Fernandes CEIO, Guimarães, 26-28 October, 2005 - p. 3/17

Motivation One of the (many) applications of multi-local optimization is in reduction type methods for semi-infinite programming (SIP) Outline problems. Motivation ❖ Motivation A SIP problems can be posed as: Multi-local y ∈ R q o ( y ) min The PSP s.t. f i ( y, x ) ≥ 0 , i = 1 , . . . , m MLPSO ∀ x ∈ T ⊂ R n , Implementation where o ( y ) is the objective function and f i ( y, x ) , i = 1 , . . . , m , Numerical results are the infinite constraint functions. Conclusions The end A. Ismael F. Vaz and Edite M.G.P . Fernandes CEIO, Guimarães, 26-28 October, 2005 - p. 3/17

Motivation One of the (many) applications of multi-local optimization is in reduction type methods for semi-infinite programming (SIP) Outline problems. Motivation ❖ Motivation A SIP problems can be posed as: Multi-local y ∈ R q o ( y ) min The PSP s.t. f i ( y, x ) ≥ 0 , i = 1 , . . . , m MLPSO ∀ x ∈ T ⊂ R n , Implementation where o ( y ) is the objective function and f i ( y, x ) , i = 1 , . . . , m , Numerical results are the infinite constraint functions. Conclusions A feasible point must satisfy: The end f i ( y, x ) ≥ 0 , i = 1 , . . . , m, ∀ x ∈ T meaning that the global minima of f i must be upper than or equal to zero. A. Ismael F. Vaz and Edite M.G.P . Fernandes CEIO, Guimarães, 26-28 October, 2005 - p. 3/17

Multi-local optimization Assume, for the sake of simplicity, that m = 1 . Then we want to address the following optimization problem Outline x ∈ R n f ( x ) min Motivation s.t. a ≤ x ≤ b Multi-local ❖ Multi-local where f : R n → R is the objective function and a , b are the optimization simple bounds on the variables x (defining the set T ). The PSP MLPSO Implementation Numerical results Conclusions The end A. Ismael F. Vaz and Edite M.G.P . Fernandes CEIO, Guimarães, 26-28 October, 2005 - p. 4/17

Multi-local optimization Assume, for the sake of simplicity, that m = 1 . Then we want to address the following optimization problem Outline x ∈ R n f ( x ) min Motivation s.t. a ≤ x ≤ b Multi-local ❖ Multi-local where f : R n → R is the objective function and a , b are the optimization simple bounds on the variables x (defining the set T ). The PSP MLPSO In each iteration of a reduction type method for SIP we need to obtain all the feasible global and local optima for function f ( x ) . Implementation Numerical results Conclusions The end A. Ismael F. Vaz and Edite M.G.P . Fernandes CEIO, Guimarães, 26-28 October, 2005 - p. 4/17

The Particle Swarm Paradigm (PSP) The PSP is a population (swarm) based algorithm that mimics the social behavior of a set of individuals (particles). Outline An individual behavior is a combination of its past experience Motivation (cognition influence) and the society experience (social Multi-local influence). The PSP ❖ The Particle In the optimization context a particle p , at time instant t , is Swarm Paradigm represented by its current position ( x p ( t ) ), its best ever position (PSP) ❖ The new travel ( y p ( t ) ) and its travelling velocity ( v p ( t ) ). position and velocity ❖ The best ever particle ❖ Features MLPSO Implementation Numerical results Conclusions The end A. Ismael F. Vaz and Edite M.G.P . Fernandes CEIO, Guimarães, 26-28 October, 2005 - p. 5/17

The new travel position and velocity The new particle position is updated by x p ( t + 1) = x p ( t ) + v p ( t + 1) , Outline where v p ( t + 1) is the new velocity given by Motivation Multi-local 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 ) , The PSP ❖ The Particle for j = 1 , . . . , n . Swarm Paradigm (PSP) ❖ The new travel ● ι ( t ) is a weighting factor (inertial) position and ● µ is the cognition parameter and ν is the social parameter velocity ❖ The best ever ● ω 1 j ( t ) and ω 2 j ( t ) are random numbers drawn from the particle ❖ Features uniform (0 , 1) distribution. MLPSO Implementation Numerical results Conclusions The end A. Ismael F. Vaz and Edite M.G.P . Fernandes CEIO, Guimarães, 26-28 October, 2005 - p. 6/17

The new travel position and velocity The new particle position is updated by x p ( t + 1) = x p ( t ) + v p ( t + 1) , Outline where v p ( t + 1) is the new velocity given by Motivation Multi-local 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 ) , The PSP ❖ The Particle for j = 1 , . . . , n . Swarm Paradigm (PSP) ❖ The new travel ● ι ( t ) is a weighting factor (inertial) position and ● µ is the cognition parameter and ν is the social parameter velocity ❖ The best ever ● ω 1 j ( t ) and ω 2 j ( t ) are random numbers drawn from the particle ❖ Features uniform (0 , 1) distribution. MLPSO Implementation Numerical results Conclusions The end A. Ismael F. Vaz and Edite M.G.P . Fernandes CEIO, Guimarães, 26-28 October, 2005 - p. 6/17