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
A stochastic augmented Lagrangian algorithm for global optimization
Outline
1 Motivation
2 The augmented Lagrangian population–based globaloptimization algorithm
3 Electromagnetism-like Algorithm for Bound Constraints
4 Numerical Experiments and Conclusions
A stochastic augmented Lagrangian algorithm for global optimization
Motivation
Outline
1 Motivation
2 The augmented Lagrangian population–based globaloptimization algorithm
3 Electromagnetism-like Algorithm for Bound Constraints
4 Numerical Experiments and Conclusions
A stochastic augmented Lagrangian algorithm for global optimization
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.
A stochastic augmented Lagrangian algorithm for global optimization
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∗
A stochastic augmented Lagrangian algorithm for global optimization
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.
A stochastic augmented Lagrangian algorithm for global optimization
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.
A stochastic augmented Lagrangian algorithm for global optimization
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) .
A stochastic augmented Lagrangian algorithm for global optimization
The augmented Lagrangian population–based global optimization algorithm
Outline
1 Motivation
2 The augmented Lagrangian population–based globaloptimization algorithm
3 Electromagnetism-like Algorithm for Bound Constraints
4 Numerical Experiments and Conclusions
A stochastic augmented Lagrangian algorithm for global optimization
The augmented Lagrangian population–based global optimization algorithm
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.
A stochastic augmented Lagrangian algorithm for global optimization
The augmented Lagrangian population–based global optimization algorithm
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 ‖∞ ≤ ∆∗)
A stochastic augmented Lagrangian algorithm for global optimization
The augmented Lagrangian population–based global optimization algorithm
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
A stochastic augmented Lagrangian algorithm for global optimization
Electromagnetism-like Algorithm for Bound Constraints
Outline
1 Motivation
2 The augmented Lagrangian population–based globaloptimization algorithm
3 Electromagnetism-like Algorithm for Bound Constraints
4 Numerical Experiments and Conclusions
A stochastic augmented Lagrangian algorithm for global optimization
Electromagnetism-like Algorithm for Bound Constraints
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.
A stochastic augmented Lagrangian algorithm for global optimization
Electromagnetism-like Algorithm for Bound Constraints
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
A stochastic augmented Lagrangian algorithm for global optimization
Electromagnetism-like Algorithm for Bound Constraints
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.
A stochastic augmented Lagrangian algorithm for global optimization
Electromagnetism-like Algorithm for Bound Constraints
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)
A stochastic augmented Lagrangian algorithm for global optimization
Electromagnetism-like Algorithm for Bound Constraints
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)
A stochastic augmented Lagrangian algorithm for global optimization
Electromagnetism-like Algorithm for Bound Constraints
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
A stochastic augmented Lagrangian algorithm for global optimization
Electromagnetism-like Algorithm for Bound Constraints
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.
A stochastic augmented Lagrangian algorithm for global optimization
Electromagnetism-like Algorithm for Bound Constraints
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.
A stochastic augmented Lagrangian algorithm for global optimization
Numerical Experiments and Conclusions
Outline
1 Motivation
2 The augmented Lagrangian population–based globaloptimization algorithm
3 Electromagnetism-like Algorithm for Bound Constraints
4 Numerical Experiments and Conclusions
A stochastic augmented Lagrangian algorithm for global optimization
Numerical Experiments and Conclusions
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
A stochastic augmented Lagrangian algorithm for global optimization
Numerical Experiments and Conclusions
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
A stochastic augmented Lagrangian algorithm for global optimization
Numerical Experiments and Conclusions
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.
A stochastic augmented Lagrangian algorithm for global optimization
Numerical Experiments and Conclusions
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
A stochastic augmented Lagrangian algorithm for global optimization
Numerical Experiments and Conclusions
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
A stochastic augmented Lagrangian algorithm for global optimization
Numerical Experiments and Conclusions
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.
A stochastic augmented Lagrangian algorithm for global optimization
Numerical Experiments and Conclusions
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
A stochastic augmented Lagrangian algorithm for global optimization
Numerical Experiments and Conclusions
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)
A stochastic augmented Lagrangian algorithm for global optimization
Numerical Experiments and Conclusions
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
A stochastic augmented Lagrangian algorithm for global optimization
Numerical Experiments and Conclusions
Thanks for your attention
Ana Maria A.C. Rocha
Edite M.G.P. Fernandes
www.norg.uminho.pt/NSOS/
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