Transcript Engineering Optimization

Engineering Optimization
Concepts and Applications
Fred van Keulen
Matthijs Langelaar
CLA H21.1
[email protected]
Optimization problem
● Design variables: variables with which the design
problem is parameterized:
x   x1, x2 ,
, xn 
● Objective: quantity that is to be minimized (maximized)
Usually denoted by:
( “cost function”)
f ( x)
● Constraint: condition that has to be satisfied
– Inequality constraint:
g ( x)  0
– Equality constraint:
h ( x)  0
Optimization problem (cont.)
● General form of optimization problem:
min f (x)
x
subject to :
g ( x)  0
h ( x)  0
x X  
x  x  x
n
Classification
● Problems:
– Constrained vs. unconstrained
– Single level vs. multilevel
– Single objective vs. multi-objective
– Deterministic vs. stochastic
● Responses:
– Linear vs. nonlinear
– Convex vs. nonconvex
– Smooth vs. nonsmooth
● Variables:
– Continuous vs. discrete (integer, ordered, non-ordered)
Solving optimization problems
● Optimization problems are typically solved using an
iterative algorithm:
Constants
Model
Design
variables
x
Optimizer
Responses
f , g, h
Derivatives of
responses
(design sensitivities)
f g h
,
,
xi xi xi
Optimization pitfalls!
● Proper problem formulation critical!
● Choosing the right algorithm
for a given problem
● Many algorithms contain lots
of control parameters
● Optimization tends to exploit
weaknesses in models
● Optimization can result in very sensitive designs
● Some problems are simply too hard / large / expensive
Exercises
● Exercise 1: Introduction to the
valve spring design problem
– Study analysis model
– Formulation of spring optimization model
● Exercise 2: Model behavior / optimization formulation
– Study model properties (monotonicity, convexity,
nonlinearity)
– Optimization problem formulation
Course overview
● General introduction, problem formulation, design
space / optimization terminology
● Modeling, model simplification
● Optimization of unconstrained / constrained problems
● Single-variable, zeroth-order and gradient-based
optimization algorithms
● Design sensitivity analysis (FEM)
● Topology optimization
Defining a design model and
optimization problem
1. What can be changed and how can the design be
described?
–
Dimensions
–
Stacking sequence of laminates
–
Ply orientation of laminates
–
Thicknesses
Bridgestone aircraft tire
For structures: distinguish sizing, material and shape
variables
Defining the optimization problem
2. What is “best”? Define an objective function:
– Weight
– Production cost
– Life-time cost
– Profits
3. What are the restrictions? Define the constraints:
– Stresses
– Buckling load
– Eigenfrequency
Defining the optimization problem (cont.)
4. Optimization: find a suitable algorithm to solve the
optimization problem. Choice depends on problem
characteristics:
– Number of design variables, constraints
– Computational cost of function evaluation
– Sensitivities available?
– Continuous / discrete design variables?
– Smooth responses?
– Numerical noise?
– Many local optima? (nonconvex)
Summary
Defining an optimization problem:
1. Choose design variables and their bounds
2. Formulate objective (best?)
3. Formulate constraints (restrictions?)
4. Choose suitable optimization algorithm
Standard forms
● Several standard forms exist:
Negative null form:
Positive null form: g (x)  0
h ( x)  0
min f (x)
x
subject to :
g ( x)  0
Neg. unity form:
g ( x)  1
h ( x)  1
Pos. unity form:
g ( x)  1
h ( x)  1
h ( x)  0
x  X  n
x  x  x
Structural optimization examples
● Typical objective function: weight
W ( x)
f 
W (x 0 )
Note the scaling!
● Typical constraint: maximum stress, maximum
displacement
 max (x)
g
1  0
 allowed
Scaled
g   max (x)   allowed  0
vs.
Unscaled
Example: minimum weight
tubular column design
● Length l given
● Load P given
● Design variables:
P
– Radius R  [Rmin, Rmax]
– Wall thickness t  [tmin, tmax]
● Objective: minimum mass
● Constraints: buckling, stress
R
l
t
Tubular column design
A  2Rt
P

A
I  R 3t
Pcrit 
 2 EI
4l 2
Design problem:
min
R ,t
s.t.
lA
 l  2Rt
R
10
  l  2Rt
s.t.
P
1  0
2Rt   max
R ,t
PP
 max
 max
2ARt
EI 3t
 32 ER
P
44l 2l 2
Rmin  R  Rmax
t min  t  t max
t
min
P  4l 2
1  0
3
3
 ER t
Rmin  R  Rmax
t min  t  t max

10t
1  0
R
Tubular column design (2)
● Alternative formulation:

A   Ro  Ri
2
2

I
4
o

P
Ro
l

R
4

Ri
 Ri
4

min
  l   R
s.t.
P
  max
2
2
 Ro  Ri
Ro , Ri
2
o

P
 Ri
2


 3E
2
R
16l
Ri  Ro  
o
4
 Ri
4
Ro min  Ro  Ro max
Ri min  Ri  Ri max

Multi-objective problems
● Minimize c(x)
Vector!
s.t. g(x)  0, h(x) = 0
● Input from designer required! Popular approach:
replace by weighted sum:
f (x)   wi ci (x)
i
● Optimum, clearly, depends on choice of weights
● Pareto optimal point: “no other feasible point exists that
has a smaller ci without having a larger cj”
Multi-objective problems (cont.)
● Examples of multi-objective problems:
– Design of a structure for
 Minimal weight and
 Minimal stresses
– Design of reduction gear unit for
 Minimal volume
 Maximal fatigue life
– Design of a truck for
 Minimal fuel consumption @ 80 km/h
 Minimal acceleration time for 0 – 40 km/h
 Minimal acceleration time for 40 – 90 km/h
Pareto set
● Pareto point: “Cannot improve an objective without
worsening another”
c2
Attainable set
Pareto set
Pareto point
c1
Pareto set (cont.)
● Alternative view:
c1
c2
Pareto set
x
Pareto set (cont.)
● Pareto set can be disjoint:
Attainable set
c2
Pareto set
c1
Hierarchical systems
● Large system can be decomposed into subsystems /
components:
● Optimization requires specialized techniques,
multilevel optimization
Structural hierarchical systems
● Example: wing box
● Too many design
variables to treat at once
● Global level: global loads,
global dimensions
● Local (rib / stiffner)
level: plate thickness,
fiber orientation
Contents
● Defining an optimization problem
● The design space & problem characteristics
● Model simplification
The design space
● Design space = set of all possible designs
● Example:
kmax
Feasible domain
F
f  k1  k2
k2
F  Fcr
k1  kmax , k2  kmax
k2
k1
Optimum
k1
kmax
Isolines
● Isolines (level sets) connect points with equal function
values:
The design space (cont.)
Problem overconstrained:
no solution exists.
No feasible domain
Dominated constraint
(redundant)
Design space (cont.)
A
A and B inactive
A and B active
B active, A inactive
Objective
function
isolines
Interior
optimum
Optimum
Optimum
B
Active constraint optimization
● Idea of constraint activity at boundary optimum
sometimes used in intuitive design optimization:
– Fully stressed design (sizing / topology optimization)
– Simultaneous failure mode theory
F
F
● Careful: does not always give the optimal solution!
Problem characteristics
● Study of objective and constraint functions:
– simplify problem
– discover incorrect problem formulation
– choose suitable optimization algorithms
● Properties:
– Boundedness
– Linearity
– Convexity
– Monotonicity
Boundedness
● Proper bounds are necessary to avoid unrealistic
solutions:
– Example: aspirin pill design
Objective: minimize dissolving time
= maximize surface area
(fixed volume)
h
r

max 2r 2  2rh
r ,h
s.t. r 2 h  1
Boundedness (cont.)
● Volume equality constraint can be substituted, yielding:
1
h 2
r
2
max 2r 
r
r

2
500
450
f
400
350
300
250
200
150
100
50
0
0
1
2
3
4
r
5
6
7
8
9
10
Linearity
“A function f is linear if it satisfies
f(x1+ x2) = f(x1)+ f(x2)
and
f(a x1) = a f(x1)
for every two points x1, x2 in the domain, and all a”
Linearity (2)
● Nonlinear objective functions can have multiple
local optima:
f
x2
f
x2
x
x1
● Challenge: finding the global optimum.
x1
Problem characteristics
● Study of objective and constraint functions:
– simplify problem
– discover incorrect problem formulation
– choose suitable optimization algorithms
● Properties:
– Boundedness
– Linearity
– Convexity
– Monotonicity
Boundedness
● Surface maximization of aspirin pill not well bounded:
1
h 2
r
2
max 2r 
r
r

2
500
450
f
400
350
300
250
200
150
100
50
0
0
1
2
3
4
r
5
6
7
8
9
10
Linearity
● Nonlinear objective functions can have multiple
local optima:
f
x2
f
x2
x
x1
● Challenge: finding the global optimum.
x1
Convexity
● Convex function: any line connecting any 2 points on
the graph lies above it (or on it):
● Linearity implies convexity (but not strict convexity)
Convexity (cont.)
● Convex set [Papalambros 4.27]:
“A set S is convex if for every two points x1, x2 in S, the
connecting line also lies completely inside S”
Convexity (cont.)
● Nonlinear constraint functions can result in nonconvex
feasible domains:
x2
x1
● Nonconvex feasible domains can have multiple local
boundary optima, even with linear objective functions!
Monotonicity
● Papalambros p. 99:
– Function f is strictly monotonically increasing if:
x2 > x1
f2
– weakly monotonically increasing if:
f1
f(x2) > f(x1) for
f(x2)  f(x1) for
x2 > x1
– Similar for mon. decreasing
● Similar:
df
0
dx
● Note: monotonicity  convexity!
● Linearity implies monotonicity
x1
x2
Optimization problem
characteristics
● Responses:
– Boundedness
– Linearity
– Convexity
– Monotonicity
● Feasible domain:
– Convexity
Example: tubular column design
P
R
t
l
min
f  2lRt
s.t.
P
g1 
1  0
2Rt max
R ,t
R
g1
g3
f
4l 2 P
g2  3 3 1  0
 ER t
10t
g3 
1  0
R
Rmin  R  Rmax
t min  t  t max
g2
t
Optimization problem analysis
● Motivation:
– Simplification
– Identify formulation errors early
– Identify under- / overconstrained problems
– Insight
● Necessary conditions for existence of optimal solution
● Basis: boundedness and constraint activity
Well-bounded functions –
some definitions
● Lower bound:
l  f ( x)
x  
● Greatest lower bound (glb):
g l
l  f (x)
f
g
● Minimum:
f ( x*)  g
● Minimizer:
x*
x*
x
Boundedness checking
● Assumption: in engineering optimization problems,
design variables are positive and finite
● Define
N  x : 0  x  
P  x : 0  x  
● Boundedness check:
– Determine
g+ for
x N
– Determine minimizers
X   x : f ( x)  g  
– Well bounded if
 X P
Examples:
f ( x)  x
g  0
X  0  P
1
f ( x) 
x
g  0
X    P
f ( x)  ( x 1)

g 0
2

g
 3
f ( x)  ( x 1) ( x  2)  3
2
2
Bounded
at zero
Asymptotically
bounded
X  1  P
X  1,2  P
f ( x)  ( x 1)  2
g  2
X  1,1  P
f ( x1, x2 )  ( x2 1)2 1
g 1
X  ( N ,1)  P2
2
2
Air tank design
● Objective: minimize mass


  2rt  t l  2(r  t ) h
f (x)   (r  t )2  r 2 l  2 (r  t )2 h
2
 g  0
l
2
X  0  P
● Not well bounded: constraints needed
r
h
t
Air tank constraints
● Minimum volume:
● Min. head/radius ratio
(ASME code):
● Min. thickness/radius ratio
(ASME code):
● Room for nozzles:
min. length
● Space limitations:
max. outside radius
7 27
r 21l.48
 2.12
10
g1  1
10
r l 0
h
h
g 2  1 7.07.13 0
r
r
t
t
g 3  1 104
0.00959
0
r
r
g 4  1l  010
.1l  0
r t
gg5 5r  t 150
1 0
150
Partial minimization &
bounding constraints
● Partial minimization: keep all variables constant but
one. Example: air tank wall thickness t:


min f   2rt  t 2 l  2(r  t ) 2 h
h ,l , r ,t
s.t.
7

g1  1  1.48 10 r l  0
h
g 2  1  7.7  0
r
t
g 3  1  104  0
r
g 4  1  0.1l  0
r t
g5 
1  0
150
2
min f   2 Rt  t 2 L  2( R  t ) 2 H 
t
s.t.
g 3  1  104
t
0
R
Rt
g5 
1  0
150
Conclusion:
• f not well bounded from below
• g3 bounds t from below
Constraint activity
● Removing constraint = relaxing problem
● Solution set of relaxed problem without
1. X i  X  gi inactive
gi is Xi
A
A and B active
2. X i  X    gi active
3. X i  X  gi semiactive
●
Activity information can
simplify problem: ● Active: eliminate variable
●
Inactive: remove constraint
B
Constraint activity checking
● Example:
min f  x12  ( x2  1) 2 ( x2  3) 2 ( x2  4) 2  ( x3  5) 2
x1 , x2 , x3
s.t.
g1  1  x1  0
g 2  2  x2  0
g3  x2  5  0
g 4  1  x3  0
Conclusion:
• g1 active
• g2 semiactive
• g3 and g4 inactive
f(1,x2,5)
g2
g3
x2
Activity and Monotonicity Theorem
● “Constraint
gi is active if and
only if the minimum of the
relaxed problem is lower than
that of the original problem”
f(x)
f
g(x)
x
g2
● “If
f(x) and gi(x) all increase or
decrease (weakly) w.r.t. x, the
g1
f(x)
f
g(x)
domain is not well constrained”
x
g
First Monotonicity Principle
● “In a well-constrained minimization problem every
variable that increases f is bounded below by at least
one non-increasing active constraint”
● This principle can be
used to find active
constraints.
f(x)
f
g(x)
x
● Exactly one bounding
constraint: critical constraint
g
Air tank design
● Monotonicity analysis:


min f   2rt  t 2 l  2(r  t ) 2 h
h ,l , r ,t
s.t.
7

g1  1  1.48 10 r l  0
h
0
r
t
g 3  1  104  0
r
g 4  1  0.1l  0
g 2  1  7.7
g5 
r t
1  0
150
2

g l , r 
g h , r 
g r , t 
g l 
g r , t 
f h , l  , r  , t 


1


2


3

Critical w.r.t. r
Critical w.r.t. h
Critical w.r.t. t

4


5
What about l? Unclear.
Optimizing variables out
● Critical constraints must be active:
min f  
h ,l , r ,t
s.t.
 2rt  t  l  2(r  t ) h 
2
2
g1  1  1.48  107 r 2l 
0
h
g 2  1  7.7 
0
r
t
g3  1  104 
0
r
g 4  1  0.1l  0
r t
g5 
1  0
150

r  1 (l )

h  2 (r )

t  3 (r )
Optimizing variables out
● Critical constraints must be active:
min f  
h ,l , r ,t
s.t.
 2rt  t  l  2(r  t ) h 
2
2
g1  1  1.48  107 r 2l  0
h
g 2  1  7.7  0
r
t
g3  1  104  0
r
g 4  1  0.1l  0
g5 
r t
1  0
150
g 2  1  7.7
h
0
r

h  0.13r
Optimizing variables out
● Critical constraints must be active:


min f   2rt  t 2 l  0.26(r  t ) 2 r
l , r ,t
s.t.

g1  1  1.48107 r 2l  0
t
g 3  1  104  0
r
g 4  1  0.1l  0
g5 
r t
1  0
150
t
g 3  1  104  0
r

r
t
104
Optimizing variables out
● Critical constraints must be active:
2
 209 2
 105 3 

min f  
r l  0.26
 r 
 1042
l ,r
 104 

s.t.
g1  1  1.48107 r 2l  0
g1  1  1.48107 r 2l  0

g 4  1  0.1l  0
105
g5 
r 1  0
104*150
r
2600
l
Optimizing variables out
● Critical constraints must be active:
2
3
 209* 26002

2600
105



min f   
 0.26*


2
 104
l
l l  104 

s.t.
g 4  1  0.1l  0
105* 2600 1
g5 
1  0
104*150 l

4.65109 
4

min f   1310 
l
l l 

s.t.
g 4  1  0.1l  0
g5  1 
l
0
306
Problem!
● Length not well bounded:

4.65109 
4

min f   1310 
l
l l 

s.t.
g 4  1  0.1l  0
g5  1 

4.65109 
4

min f l   1310 
l
l l


s.t.
g 4  l  10
 

g 5  l  306
l
0
306
● Additional constraint from above is needed:
● Maximum plate width:
l  610
Air tank solution
● Length constraint is critical: must be active!
● Solution:
l  610
r  105
t 1
h  13.6
l
r
h
t
● Result of Monotonicity Analysis:
● Problem found, and fixed
● Solution found without numerical optimization
Recognizing monotonicity
● Some useful properties:
– Sums:

1
f3  f  f 2


f3

Sums of similarly monotonic functions have the same
monotonicity
– Products:
f3  f1 * f 2

f3 '  f1 ' f 2  f1 f 2 '
Products of similarly monotonic functions have:
– same monotonicity if
f1  0, f 2  0
– opposite monotonicity if
f1  0, f 2  0
Recognizing monotonicity
● More properties:
– Powers:
 
f 3  f1
 a



a  0 : f 3



a

0
:
f
3

Positive powers of monotonic functions have the same
monotonicity, negative powers have opposite
monotonicity
f3  f1  f 2 
– Composites:



f
,
f
 1 2
  

 f1 , f 2
f3


f3 '  f1 ' f 2 '



f
,
f
 1 2
  

 f1 , f 2
f3

Recognizing monotonicity
f1
● Integrals:
b
– w.r.t. limits:
f 3 (a, b)   f1 ( x)dx
a
f1 (a  x  b)  0


f1 , f1 (a)  0, f1 (b)  0
0
a
b x
f 3 (a  , b  )

f3 (a , b )
b
– w.r.t. integrand:
f 3 (a, b, y )   f1 ( x, y )dx
y
f1
a
f1 ( y  ) 
f3 ( y  )
a
b
x
Criticality
Refined definitions:
# of variables
critically bounded
by constraint i
0
1
>1
Uncritical
constraint
Uniquely critical
constraint
Multiple critical
constraint
Conditionally
critical constraint
# of constraints
possibly critically
bounding variable j
1
>1
Air tank example


min f   2rt  t 2 l  2(r  t ) 2 h
h ,l , r ,t
s.t.
7

g1  1  1.48 10 r l  0
h
g 2  1  7.7  0
r
t
g 3  1  104  0
r
g 4  1  0.1l  0
r t
g5 
1  0
150
2

g l , r 
g h , r 
g r , t 
g l 
g r , t 
f h , l  , r  , t 


1


2


3

4



Critical w.r.t. r
Critical w.r.t. h
Critical w.r.t. t
Conditionally
critical w.r.t. l
5
Multiple critical!
Multiple critical constraint can obscure boundedness!
Eliminate if possible
Air tank example
● Starting with eliminating r:


min f   2rt  t 2 l  2(r  t ) 2 h
h ,l , r ,t
s.t.

g1  1  1.48 107 r 2l  0
h
0
r
t
g 3  1  104  0
r
g 4  1  0.1l  0
g 2  1  7.7
g5 
r t
1  0
150
g1  1  1.45107 r 2l  0

r
2600
l
Air tank example
● New problem:




 26002
t
2

min f    t 5200 l  tl  2h
 5200  t  
h , l ,t
l
 l


s.t.
h l
g 2  1  7.7
0
2600
t l
g 3  1  104
0
2600
g 4  1  0.1l  0
g5 
52
t

1  0
3 l 150
h , l ? , t 
Critical for h
h , l 
Critical for t
l,t
l
l,t
?
Air tank example
● Finally, after also eliminating h and t:

4.65109 

min f   130625
3/ 2
l
l


s.t.
g 4  1  0.1l  0
g5 
35
1  0
2 l
l
l
Not well bounded!
l
● Conclusion: multiple critical constraint
obscured ill-boundedness in l
Summary
● Optimization problem checking:
– Boundedness check of objective
 Identify underconstrained problems
– Monotonicity analysis
 Identify not properly bounded problems
 Identify critical constraints
 Eliminate variables
 Remove inactive constraints
But what about …
● Equality constraints:
– Active if all constraint variables in objective
– Otherwise semi-active
● Example:
min f  3x1
x1 , x2
s.t.
g1  1  x1  0
h1  x2  3  0
x2
f
3
Relaxed problem: min f  3x1
x1 , x2
s.t.
g1  1  x1  0
1
x1
More on nonobjective variables
● Monotonicity Principle for nonobjective variables:
“In a well-constrained minimization problem every
nonobjective variable is bounded below by at least one
non-increasing semiactive constraint and above by at
least one non-decreasing semiactive constraint”
g(x)
0
gi
gj
x
Nonobjective variables (2)
● Other options:
– Equality constraint
– Single nonmonotonic constraint
h(x)
0
g(x)
hi
0
gi
x
● See example in book (Papalambros p. 114)
x
Nonmonotonic functions
● Monotonicity analysis difficult!
– Sometimes regional monotonicity can be used
– Concave constraints can split feasible domain:
g(x)
0
gj
gi
x
Model preparation procedure (3.9)
● Remove dominated constraints
● Check boundedness for each design variable:
– Objective monotonic? Constraints monotonic?
– Critical constraints?
Uniquely / conditionally / multiply?
● If possible, eliminate active constraints,
and repeat steps
Spending time on model checking usually pays off!