A stochastic augmented Lagrangian algorithm for global optimization · 2010-02-13 · A stochastic augmented Lagrangian algorithm for global optimization Motivation The augmented

A stochastic augmented Lagrangian algorithm for global optimization

A stochastic augmented Lagrangian algorithm forglobal optimization

Ana Maria A. C. Rocha and Edite M. G. P. Fernandes

University of Minho, Braga, PORTUGALarocha;[email protected]

7th EUROPT WorkshopAdvances in Continuous Optimization

EUROPT 2009

Remagen, Germany, July 3 - 4, 2009


Outline

1 Motivation

2 The augmented Lagrangian population–based globaloptimization algorithm

3 Electromagnetism-like Algorithm for Bound Constraints

4 Numerical Experiments and Conclusions


Motivation

Outline

1 Motivation





Motivation

Motivation

Many practical engineering problems involve multi-modal andnon-differentiable nonlinear functions of many variables thatare difficult to handle by gradient-based algorithms;

one alternative is to use derivative-free and stochasticmethods.

Most stochastic methods were primary developed for unconstrainedproblems; then extended to constrained optimization problems

by using penalty functions;

by modifying the original procedures.


Motivation

Motivation

When using penalty function, the constraints violation andthe objective function values are combined in the penaltyfunction.

The proposed approach is a practical population-based globaloptimization method based on the augmented Lagrangianframework.

bla bla

Lagrange Existe um large value ...

x∗(ρ) = argminLρ(x, µ) = x∗


Motivation

The augmented Lagrangian methodology

The method solves a sequence of very simple subproblems,with box constraints, whose objective function aims topenalize the equality and inequality constraints violation.

Each objective function is an augmented Lagrangian functionand depends on a positive penalty parameter, as well as onthe multiplier vectors associated with the equality andinequality constraints.

Each of these outer iterations evaluates approximate multipliervectors and an appropriate value for the penalty parameter.

Penalty parameter values and multiplier vectors estimates arecrucial to guarantee global convergence of augmentedLagrangian methods.

Each subproblem is approximately solved by the EM algorithmthat is specially devised for box constraints optimizationproblems.

This is a population-based stochastic method that simulatesthe electromagnetism theory of physics by considering eachpoint in the population associated with an electrical charge.


Motivation

The Constrained Global Optimization Problem

The problems to be addressed are:

minimize f(x)subject to g(x) ≤ 0

h(x) = 0x ∈ Ω,

f : Rn → R,

g : Rn → Rp

h : Rn → Rm

are nonlinear continuous functions

Ω = x ∈ Rn : l ≤ x ≤ u.


Motivation

The Constrained Global Optimization Problem

Equality constraints are converted into inequality constraints by

|hj | − ε ≤ 0, j = 1, . . . ,m.

The problem is rewritten as

minimize f(x)subject to c(x) ≤ 0

x ∈ Ω,

where

c(x) = (g1(x), . . . , gp(x), |h1(x)| − ε, . . . , |hm(x)| − ε) .


The augmented Lagrangian population–based global optimization algorithm

Outline

1 Motivation






The augmented Lagrangian methodology

Based on the Powell–Hestenes–Rockafellar formulae

Lρ(x, µ) = f(x) +ρ

2

p+m∑i=1

max(

0, ci(x) +µi

ρ

)2

x ∈ Ω,

ρ > 0 is a penalty parameter,

µ ∈ Rp+m is the multipliers vector.



The overall algorithm

Initialization of parameters (µ, τ , γ ε, ∆∗)

Generate an initial point x1 ∈ Ω and compute Lρ

(x1, µ

)k ← 1

Do

Run EM algorithm to approximately solve minLρ

(xk, µk

)Update multipliers vector µk

Update penalty parameter ρk

Compute V ki = max

(ci(xk),−µk

i

ρk

)i = 1, . . . ,m + p

k ← k + 1

while (k ≤ max iter AL and ‖V ‖∞ ≤ ∆∗)



Parameters update

Multipliers vector µ update

µ1i = 0 (initial value) for i = 1, . . . ,m + p

µk+1i = min

max(0, µk + ρci(xk)), µmax

Penalty parameter ρ update

ρ1 = max

10−6,min(

10,2|f(x0)|‖c(x0)‖2

)(initial value)

Update penalty parameter ρ

if k = 1 or ‖V k‖∞ ≤ τ‖V k−1‖∞ thenρk+1 ← ρk

elseρk+1 ← γρk


Electromagnetism-like Algorithm for Bound Constraints

Outline

1 Motivation






Electromagnetism-like Algorithm

It starts with a population of psize − 1 points randomlygenerated from Ω;

Each sampled point is considered as a charged particle

the charge of each point is related to its objective functionvaluethe charge determines the magnitude of attraction of eachpoint over the other points in the population;

The charges are used to find the total force exerted on eachpoint, by other points;

The total force is used to move the points.



EM Algorithm

Input: xk

Initialize and select best point based on Lρ

j ← 1

while j ≤ 10k do

Compute charge of each point based on Lρ

Compute force and select direction based on Lρ

Move points and select best point based on Lρ

Local search to improve best point, based on Lρ

j ← j + 1

end while

Output: xk+1 ← xbest



Initialize

Let xi ∈ Rn be the ith point of the population.

A set of psize − 1 points ∈ Ω is randomly generated:

1 Each point xi is componentwise computed by

xi ← l + λ(u− l) where λ← U(0, 1).

2 The augmented Lagrangian function Lρ is computed for eachpoint;

3 Identify the best point - xbest.



Compute charge

According to Coulomb’s law, the force exerted on each point, byother points, is directly proportional to the product of their chargesand is inversely proportional to the square of the distance between

the points.

The charge for each point, based on penalty function value

qi = exp(−n(Lρ(xk

i , µk)− Lρ(xk

best, µk))∑psize

i=1 Lρ(xki , µ

k)− Lρ(xkbest, µ

k)

)i = 1, 2, . . . , psize

Birbil & Fang (2003)



Compute force

The force vector Fi,j between two points xii and xij

F ij =

(xj − xi)qiqj

‖xj − x− i‖3if Lρ(xk

j , µk) < Lρ(xk

i , µk)

(xj attracts xi)

(xi − xj)qiqj

‖xj − xi‖3if Lρ(xk

j , µk) ≥ Lρ(xk

i , µk)

(xj repels xi)



Compute force

The total force exerted on point xi (by other points in population)

F i =psize∑j 6=i

Fi,j , i = 1, 2, . . . , psize,

Example: the total force F 1 exerted on x1, by x2 and x3, is givenby

F 1 = F1,2 + F1,3

F1

F21

F31

x3

x1x2

Lρ(x3, µk) is worse than that of Lρ(x1, µ

k)

F1,3 is an attractive force exerted on x1 by x3

Lρ(x2, µk) is better than that of Lρ(x1, µ

k)

F1,2 is a repulsive force exerted on x1 by x2



Move points

1 Each point xi (except the best) is moved in the direction oftotal force (normalized), componentwise defined by

xi =

xi + λ F i

‖F i‖(u− xi) if F i > 0

xi + λ F i

‖F i‖(xi − l) otherwise, λ ∼ U(0, 1)

2 Penalty function values are computed for each point;

3 The point with least penalty function value is identified as thebest point - xbest.



Local search

Local refinement about xbest

1 Computes the maximum feasible step length,

smax = δ(max(u− l))

2 Computes a new point, componentwise defined by

y = xbest + λ smax,

λ ∼ U(0, 1)3 If y improves over xbest then xbest ← y and the search along

that coordinate ends.


Numerical Experiments and Conclusions

Outline

1 Motivation






Details

µmax = 1012, τ = 0.5, γ = 10, max iter AL = 50

ε = 10−5, ∆∗ = 10−6

population of min(200, 10n) points

30 independent runs for each problem

maximum of EM iterations = 10 ∗ iterAL



Benchmark problems

Problem Type of f n p m nactive

g01 quadratic 13 9 0 6g02 general 20 2 0 1g03 polynomial 10 0 1 1g04 quadratic 5 6 0 2g05 cubic 4 2 3 3g06 cubic 2 2 0 2g07 quadratic 10 8 0 6g08 general 2 2 0 0g09 general 7 4 0 2g10 linear 8 6 0 6g11 quadratic 2 0 1 1g12 quadratic 3 93 0 0g13 general 5 0 3 3g14 general 10 0 3 3g15 quadratic 3 0 2 2g17 general 6 0 4 4g18 quadratic 9 13 0 6g21 linear 7 1 5 6g23 linear 9 2 4 6g24 linear 2 2 0 2



Study of the augmented Lagrangian approach byperformance profiles

Performance profiles as outline in Dolan & More (2002) use:

1 performance ratio (for problem p ∈ P and solver s ∈ S)

r(p,s) =fbest (p,s)

minfbest (p,s) : s ∈ S

2 ρs(τ) - the probability distribution for the ratio rp,s

ρs(τ) =nPτ

nP

nP - number of problems in the set P;nPτ - number of problems in the set such that r(p,s) ≤ τ (forsolver s).

Solver with high values for ρc(τ) is preferable.



Study of γ parameter (in the ρ update)

1 1.05 1.1 1.15 1.2 1.25 1.3 1.350.7

0.75

0.8

0.85

0.9

0.95

1

τ

ρ( τ

)Performance profile on fbest

γ = 10

γ = 2



Modified penalty parameter ρ update

Set ∆k = max(∆∗, 10−k) with ∆∗ = 10−6

if k = 1 or ‖V k‖∞ ≤ τ‖V (k−1‖∞ then

ρk+1 ← ρk

else

if ‖V k‖∞ ≤ ∆k then

ρk+1 ← max10−12,1γ

ρk

else

ρk+1 ← min1012, γ ρk



Modified ε dynamically

Equality constraints are converted into inequality by

|hj | − ε ≤ 0, j = 1, . . . ,m.

where ε = 10−5 (fixed).

⇓

Modified ε along AL iterations (dynamic)

εk ← max(

10−12,1γ

εk

)where ε1 ← 10−3.



Modified ρ and ε updates

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.80.7

0.75

0.8

0.85

0.9

0.95

1

τ

ρ(τ)

Performance profile on fbest

original ρ + ε fixed

ρ modified + ε fixed

ρ modified + ε dynamic



Modified termination condition in EM

Termination conditionin EM

j ≤ 10k (dinamic)

j ≤ 30 (fixed)

1 1.05 1.1 1.15 1.2 1.25 1.30.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

\

ρ(τ)

Performance profile on fbest

j ≤ 30 (fixed)

j ≤ 10k (dynamic)



Conclusions and Future Work

We presented a new version of electromagnetism-likealgorithm for constrained global optimization problem

based on an augmented Lagrangian approach.

the subproblems were approximately solved by the EMalgorithm

Future developments:

extend to use the augmented Lagrangian with equalityconstraint handlingimplementation of other augmented Lagrangian functionscomparison with other techniques



Thanks for your attention

Ana Maria A.C. Rocha

[email protected]

Edite M.G.P. Fernandes

[email protected]

www.norg.uminho.pt/NSOS/

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A stochastic augmented Lagrangian algorithm for global optimization · 2010-02-13 · A stochastic augmented Lagrangian algorithm for global optimization Motivation The augmented