engineering optimization
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Concepts and Applications. Engineering Optimization. Fred van Keulen Matthijs Langelaar CLA H21.1 [email protected]. Optimization problem. Design variables: variables with which the design problem is parameterized: - PowerPoint PPT PresentationTRANSCRIPT
Engineering Optimization
Concepts and Applications
Fred van Keulen
Matthijs Langelaar
CLA H21.1
Optimization problem
● Design variables: variables with which the design
problem is parameterized:
● Objective: quantity that is to be minimized (maximized)
Usually denoted by:
( “cost function”)
● Constraint: condition that has to be satisfied
– Inequality constraint:
– Equality constraint:
( ) 0g x
( ) 0h x
( )f x
1 2, , , nx x xx
Optimization problem (cont.)
● General form of optimization problem:
xxx
x
xh
xg
xx
nX
f
0)(
0)(
)(
:to subject
min
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:
Model
Optimizer
Designvariables
Constants Responses
Derivatives ofresponses(design sensi-tivities)
hgf ,,
iii x
h
x
g
x
f
,,
x
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
For structures: distinguish sizing, material and shape variables
Bridgestone aircraft tire
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:
xxx
x
xh
xg
xx
nX
f
0)(
0)(
)(
:to subject
min
Negative null form:
0)(
0)(
xh
xgPositive null form:
1)(
1)(
xh
xgNeg. unity form:
1)(
1)(
xh
xgPos. unity form:
Structural optimization examples
● Typical objective function: weight
● Typical constraint: maximum stress, maximum
displacement
)(
)(
0x
x
W
Wf Note the scaling!
01)(max
allowed
g x 0)(max allowedg x
Scaled vs. Unscaled
Example: minimum weight
tubular column design● Length l given
● Load P given
● Design variables:
– Radius R [Rmin, Rmax]
– Wall thickness t [tmin, tmax]
● Objective: minimum mass
● Constraints: buckling, stress
R
t
R
t
P
l
maxmin
maxmin
2
2
max
,
4
..
min
ttt
RRRl
EIP
A
Pts
lAtR
Design problem:
Tubular column design
maxmin
maxmin
2
33
max
,
4
2..
2min
ttt
RRRl
tERP
Rt
Pts
RtltR
2
23
42
l
EIP
A
PtRIRtA crit
maxmin
maxmin
33
2
max
,
014
012
..
2min
ttt
RRRtER
lP
Rt
Pts
RtltR
0110
10
R
tRt
Tubular column design (2)● Alternative formulation:
4422
4 ioio RRIRRA
Ro
Ri
P
l
maxmin
maxmin
44
2
3
max22
22
,
16
..
min
iii
ooo
oi
io
io
ioRR
RRR
RRR
RR
RRl
EP
RR
Pts
RRlio
Multi-objective problems
● Minimize c(x)
s.t. g(x) 0, h(x) = 0
● Input from designer required! Popular approach:
replace by weighted sum:
Vector!
)()( xx i
iicwf
● 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”
c1
c2
Attainable set
Pareto setPareto point
Pareto set (cont.)
● Pareto set can be disjoint:
c1
c2
Attainable set
Pareto set
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:
F
k2
k1
1 2f k k
crFF k2
k1
max2max1 , kkkk
kmax
kmax
Feasible domain
Optimum
The design space (cont.)
Dominated constraint(redundant)
No feasible domain
Problem overconstrained:no solution exists.
A
B
Design space (cont.)
A and B active
Objective functionisolines
Optimum
B active, A inactive
Optimum
Objective functionisolines
A and B inactive
Interior optimum
Objective functionisolines
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
● Careful: does not always give the optimal solution!
FF
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)
1
22
2
2
,
hr
rhrhr
s.t.
maxrh
Boundedness (cont.)
● Volume equality constraint can be substituted, yielding:
rr
rh
r
22
1 22
max
0
50
100
150
200
250
300
350
400
450
500
0 1 2 3 4 5 6 7 8 9 10
r
f
Linearity
“A function f is linear if it satisfies
f(x1+ x2) = f(x1)+ f(x2)
and
f( x1) = f(x1)
for every two points x1, x2 in the domain, and all ”
Linearity (2)
● Nonlinear objective functions can have multiple local optima:
f
x1
x2
x
x1x2
f
● Challenge: finding the global optimum.
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:
rr
rh
r
22
1 22
max
0
50
100
150
200
250
300
350
400
450
500
0 1 2 3 4 5 6 7 8 9 10
r
f
Linearity
● Nonlinear objective functions can have multiple local optima:
f
x1
x2
x
x1x2
f
● Challenge: finding the global optimum.
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”
● Nonlinear constraint functions can result in nonconvex
feasible domains:
Convexity (cont.)
x1
x2
● 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:
f(x2) > f(x1) for x2 > x1
– weakly monotonically increasing if:
f(x2) f(x1) for x2 > x1
– Similar for mon. decreasing
f2
f1
x1 x2
0dx
df● Similar:
● Note: monotonicity convexity!
● Linearity implies monotonicity
Optimization problem
characteristics● Responses:
– Boundedness
– Linearity
– Convexity
– Monotonicity
● Feasible domain:
– Convexity
Example: tubular column designR
t
P
l
maxmin
maxmin
3
33
2
2
max1
,
0110
014
012
..
2min
ttt
RRRR
tg
tER
Plg
Rt
Pgts
lRtftR
f
t
R g3g1
g2
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:
xxfl )(
● Greatest lower bound (glb):
)(xfllg g
f
xx*● Minimum:
● Minimizer:
gxf *)(
*x
Boundedness checking
● Assumption: in engineering optimization problems,
design variables are positive and finite
● Define
● Boundedness check:
– Determine g+ for
– Determine minimizers
– Well bounded if
Nx
: ( )X x f x g PX
xxP 0: xxN 0:
Examples:
xxf )(
xxf
1)(
Bounded at zero
Asymptotically bounded
2)1()( xxf
3)2()1()( 22 xxxf
2)1()( 22 xxf
0g PX 0
0g PX
0g PX 1
3g PX 2,1
2g PX 1,1
21 2 2( , ) ( 1) 1f x x x 1g 2)1,( PNX
Air tank design
● Objective: minimize mass
t
r
h
l htrlrtrf 222 )(2)()( x
htrltrt 22 )(22
PXg 00
● Not well bounded: constraints needed
Air tank constraints
● Minimum volume:72
1 1012.2 lrg
● Min. head/radius ratio
(ASME code):13.02
r
hg
● Min. thickness/radius ratio
(ASME code):
00959.03 r
tg
● Room for nozzles:
min. length104 lg
● Space limitations:
max. outside radius
1505 trg
01048.11 271 lrg
07.712 r
hg
010413 r
tg
01.014 lg
011505
tr
g
Partial minimization &
bounding constraints● Partial minimization: keep all variables constant but
one. Example: air tank wall thickness t:
01150
01.01
01041
07.71
01048.11
)(22
5
4
3
2
271
22
,,,
trg
lgr
tg
r
hg
lrg
htrltrtftrlh
s.t.
min
Conclusion: • f not well bounded from below
• g3 bounds t from below
01150
01041
)(22
5
3
22
tRg
R
tg
HtRLtRtft
s.t.
min
Constraint activity
● Removing constraint = relaxing problem
● Solution set of relaxed problem without gi is Xi
1.
2.
3.
inactive ii gXX
active ii gXX
semiactive ii gXX
A
B
A and B active
● Activity information can
simplify problem: ● Active: eliminate variable
● Inactive: remove constraint
Constraint activity checking
● Example:
1 2 3
2 2 2 2 21 2 2 2 3, ,
1 1
2 2
3 2
4 3
( 1) ( 3) ( 4) ( 5)
1 0
2 0
5 0
1 0
min
s.t.
x x xf x x x x x
g x
g x
g x
g x
x2
f(1,x2,5)
g3
g2
Conclusion:
• g1 active
• g2 semiactive
• g3 and g4 inactive
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”x
f
g1
x
f
g
f(x)
g(x)
● “If f(x) and gi(x) all increase or
decrease (weakly) w.r.t. x, the
domain is not well constrained”
f(x)
g(x)
g2
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”
x
f
g
● This principle can be
used to find active
constraints.
● Exactly one bounding
constraint: critical constraint
f(x)
g(x)
Air tank design
● Monotonicity analysis:
01150
01.01
01041
07.71
01048.11
)(22
5
4
3
2
271
22
,,,
trg
lgr
tg
r
hg
lrg
htrltrtftrlh
s.t.
min
trg
lg
trg
rhg
rlg
trlhf
,
,
,
,
,,,
5
4
3
2
1Critical w.r.t. r
Critical w.r.t. h
Critical w.r.t. t
What about l? Unclear.
Optimizing variables out
● Critical constraints must be active:
2 2
, , ,
7 21
2
3
4
5
2 2( )
1 1.48 10 0
1 7.7 0
1 104 0
1 0.1 0
1 0150
h l r tf rt t l r t h
g r l
hg
rt
gr
g l
r tg
min
s.t.
2 2
, , ,
7 21 1
2 2
3 3
4
5
2 2( )
1 1.48 10 0 ( )
1 7.7 0 ( )
1 104 0 ( )
1 0.1 0
1 0150
min
s.t.
h l r tf rt t l r t h
g r l r l
hg h r
rt
g t rr
g l
r tg
2 2
, , ,
7 21
2
3
4
5
2 2( )
1 1.48 10 0
1 7.7 0
1 104 0
1 0.1 0
1 0150
min
s.t.
h l r tf rt t l r t h
g r l
hg
rt
gr
g l
r tg
rhr
hg 13.007.712
Optimizing variables out
● Critical constraints must be active:
01150
01.01
01041
01048.11
)(26.02
5
4
3
271
22
,,
trg
lgr
tg
lrg
rtrltrtftrl
s.t.
min
104010413
rt
r
tg
Optimizing variables out
● Critical constraints must be active:
01150*104
105
01.01
01048.11
104
10526.0
104
209
5
4
271
32
22,
rg
lg
lrg
rlrfrl
s.t.
min
lr
lrg
2600
01048.11 271
Optimizing variables out
● Critical constraints must be active:
011
150*104
2600*105
01.01
104
1052600*26.0
104
2600*209
5
4
23
2
2
lg
lg
llf
l
s.t.
min
0306
1
01.01
1065.41013
5
4
94
lg
lg
llf
l
s.t.
min
Optimizing variables out
● Critical constraints must be active:
Problem!
● Length not well bounded:
0306
1
01.01
1065.41013
5
4
94
lg
lg
llf
l
s.t.
min
306
10
1065.41013
5
4
94
lg
lg
lllf
l
s.t.
min
● Additional constraint from above is needed:
● Maximum plate width: 610l
Air tank solution
● Length constraint is critical: must be active!
● Solution:
6.13
1
105
610
h
t
r
l
t
r
h
l
● Result of Monotonicity Analysis:
● Problem found, and fixed
● Solution found without numerical optimization
Recognizing monotonicity
● Some useful properties:
– Sums: 3213 ffff
– Products: '''* 21213213 ffffffff
Sums of similarly monotonic functions have the same
monotonicity
Products of similarly monotonic functions have:
– same monotonicity if
– opposite monotonicity if
0,0 21 ff
0,0 21 ff
Recognizing monotonicity
● More properties:
– Powers:
3
313
:0
:0
f
fff
Positive powers of monotonic functions have the same
monotonicity, negative powers have opposite
monotonicity
– Composites: ''' 213213 ffffff
3
21
213
21
21
,
,
,
,f
ff
fff
ff
ff
Recognizing monotonicity
● Integrals:
– w.r.t. limits: dxxfbafb
a
)(),( 13
f1
x0 a b
),(0)( 31 bafbxaf
),(0)(,0)(, 3111 bafbfaff
)()( 31 yfyf
– w.r.t. integrand: b
a
dxyxfybaf ),(),,( 13 f1
x
y
ab
Criticality
Refined definitions:
# of variables critically bounded
by constraint i
# of constraints possibly critically
bounding variable j
0
Uncritical constraint
1 1Uniquely critical
constraint
>1 Multiple critical constraint
>1
Conditionally critical constraint
Air tank example
Multiple critical constraint can obscure boundedness!
Eliminate if possible
01150
01.01
01041
07.71
01048.11
)(22
5
4
3
2
271
22
,,,
trg
lgr
tg
r
hg
lrg
htrltrtftrlh
s.t.
min
trg
lg
trg
rhg
rlg
trlhf
,
,
,
,
,,,
5
4
3
2
1Critical w.r.t. r
Critical w.r.t. h
Critical w.r.t. t
Conditionally critical w.r.t. l
Multiple critical!
Air tank example
● Starting with eliminating r:
01150
01.01
01041
07.71
01048.11
)(22
5
4
3
2
271
22
,,,
trg
lgr
tg
r
hg
lrg
htrltrtftrlh
s.t.
min
lr
lrg
2600
01045.11 271
Air tank example
● New problem:
011503
52
01.01
02600
1041
02600
7.71
52002600
25200
5
4
3
2
22
,,
t
lg
lg
ltg
lhg
tl
t
lhtlltf
tlh
s.t.
min tlh ,, ?
lh ,
tl ,
l
tl ,
Critical for t
Critical for h
?
Air tank example
● Finally, after also eliminating h and t:
012
35
01.01
1065.4130625
5
4
2/3
9
lg
lg
lf
l
s.t.
min l
l
l
● Conclusion: multiple critical constraint
obscured ill-boundedness in l
Not well bounded!
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:
03
01
3
21
11
1, 21
xh
xg
xfxx
s.t.
min
01
3
11
1, 21
xg
xfxx
s.t.
minRelaxed problem:x1
x2 f
3
1
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”
x
0gi gj
g(x)
Nonobjective variables (2)
● Other options:
– Equality constraint
– Single nonmonotonic constraint
x
0hi
x
0gi
● See example in book (Papalambros p. 114)
g(x)h(x)
Nonmonotonic functions
● Monotonicity analysis difficult!
– Sometimes regional monotonicity can be used
– Concave constraints can split feasible domain:
x
0gj gi
g(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!