Multidisciplinary System
System
Design Optimization (MSDO)
(MSDO)
How to Break “Robust” Optimizers
Recitation 6
Andrew March
1
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Today’s Topics
Topics
• fmincon
– What the options are
– What it’s doing
– How to break it
• GA toolbox on Stellar
• MATLAB® GA Toolbox
– Some options
– How to break it
• Questions
2
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
Sequential Quadratic Programming
• Create Quadratic Approximation: B(x k ) ≈ ∇ xx L(x k );
B (x k ) f 0
sk = xk +1 − xk
mineq
meq
mineq
meq
⎛
⎞ ⎛
⎞
qk = ⎜⎜ ∇f (xk +1 ) + ∑ λi ∇gi (xk +1 ) + ∑ λ j ∇h j (xk +1 ) ⎟⎟ − ⎜⎜ ∇f (xk ) + ∑ λi ∇gi (xk ) + ∑ λ j ∇h j (xk ) ⎟⎟
i=1
j=1
i=1
j=1
⎝
⎠ ⎝
⎠
qk qkT BkT skT sk Bk
Bk +1 = Bk + T − T
qk sk
sk Bk sk
• Solve Quadratic Program:
1
minn d T Bk d + ∇f (xk )T d
d∈ℜ 2
s.t. Aeq d = beq
Ad ≤ b
• Evaluate merit function:
meq
mineq
i=1
i=1
Ψ(x) = f (x) + ∑ ri hi (x) + ∑ ri ⋅ max[gi (x),0]
Ψ ( xk + αd k )
• Compute step length: α k = arg α∈A(min
αd )≤b
• Take step: xk +1 = xk + α k d k
• Repeat until αkdk≤ε
Interior Point Algorithm
•
Two equivalent optimization problems:
mineq
minn f (x)
min f (x) − μ ∑ log si
x,s∈ℜ
x,s∈ℜn
s.t. h(x) = 0
g(x) + s = 0
s ≥ 0
•
KKT Conditions:
m
ineq
s.t.
meq
∇f (x) + ∑ zi ∇gi (x) + ∑ y j ∇h j (x)
i=1
j=1
sz − μe = 0
i=1
h(x) = 0
g(x) + s = 0
mineq
meq
i=1
j=1
∇f (x) + ∑ zi ∇gi (x) + ∑ y j ∇h j (x)
− μS −1e + z = 0
h(x) = 0
g(x) + s = 0
h(x) = 0
g(x) + s = 0
• Solve KKT system with Newton’s method, for directions.
• Compute step length: α s = max{0 < α ≤ 1: s + αd s ≥ (1 − τ )s}
(τ~0.995)
α = max{0 < α ≤ 1: z + αd ≥ (1 − τ )z}
z
xk +1 = xk + α s d x sk +1 = sk + α s d s
yk +1 = yk + α z d y zk +1 = zk + α z d z
Repeat until KKT Conditions satisfied to within μk
Repeat entire process with μk+1=σμk ,0<σ<1
• Take step:
•
•
z
Genetic Algorithm
Initialize Population (initialization)
next generation
Select individual for mating (selection)
Mate individuals and produce children (crossover)
Mutate children (mutation)
Insert children into population (insertion)
n
Are stopping criteria satisfied ?
y
Finish
5
Ref: Goldberg (1989)
© Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox
Engineering Systems Division and Dept. of Aeronautics and Astronautics
GA Toolbox
• BUG!!!
• Add into genetic.m
– Line 118: stats=[];
Fitness Function
• Only has Roulette Wheel selection
– You can add others…
• Roulette Wheel
f (x k )
Ρ=
∑ f (x k )
k
• Are there any restrictions on f(x)?
Matlab GA Toolbox
• Constraint Handling
• Augmented Lagrangian/Penalty method:
mineq
minn f (x) − ∑ λi si log(gi (x) − si )
x∈ℜ
i=1
gi (x) − sk < 0
sk =
1
μk
λk
– μ0≥1
– σ>1; μk+1=σμk
• Disclaimer: this is one of three papers
referenced, it may not be exactly what’s in
MATLAB
Summary
• There are many optimization toolboxes
available to you.
• Everyone of them has limitations.
• Know what they are doing.
• Assure your problem fits within the
assumptions that algorithm makes!
MIT OpenCourseWare
http://ocw.mit.edu
ESD.77 / 16.888 Multidisciplinary System Design Optimization
Spring 2010
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.