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Examples of Various Formulations of Optimization
Problems
Example 1 (bad formulation)
A chemical factory produces a chemical from two materials, x and y. x can be purchased for $5 per ton and y can be purchased for $1 per ton. The manufacturer wants to determine the amount of each raw material required to reduce the cost per ton of product to a minimum. Formulate the problem as an optimization problem…
Solution (?): Linear problem
0,0 subject to
5 Minimize
yx
yx f(x,y)
Solution (?)
00
0,0 subject to
5 Minimize
, y x
yx
yx f(x,y)
Example 2
Given the perimeter of a rectangle must be at most 16cm, construct the rectangle with maximum area. Formulate this as an optimization problem.
Solution: Nonlinear problem
0,0
1622
Subject to
Maximize
yx
y x
xyf(x,y)
Example 3Suppose we want to maximize the area of
an object, but we have a choice between a square and a circle, where the length of the square is equal to the radius of the circle, and the radius can be at most 4 cm. Formulate this as an optimization problem.
Object 1Object 2
Solution: Mixed integer nonlinear problem
1
large)y arbitraril be chosen to was(1000 1000
40
,
Subject to
, Minimize
..otherwise. 0or square a is
chosen object theif 1 is that riable,binary va a be
,Let radius. theoflength thebe Let circle. and
square theof area the tocorrespond ,Let
21
22
21
21
1
21
21
yy
yA
x
xAxA
A A x,y)f(A
y
yyx
AA
ii
Parameter Identification
Identify the damping, c, and the spring constant, k, of a linear spring by minimizing the difference of a numerical prediction and measured data. Assume that the spring-mass system is set into motion by an initial displacement from equilibrium and measurements of displacement are taken at equally spaced time increments.
Parameter Identification continued
The motion of an unforced harmonic oscillator satisfies the initial value problem,
paramters.unknown of out vector beLet
T].[0,on 0)0(,)0(,0 0
k
cx
uuukuucu
Formulate this as an optimization problem
Nonlinear Least Squares Problem
M
jjj
jj
jj
|u:x)|u(t f(x)
k
cx
t:x)u(t
:Mj
tu
1
2
2
1 Minimize
.given afor
at time prediction numericalour denote Let
.1for
, at times pointsn observatioour denote Let
Example: ‘Black-Box’ Formulation
Suppose there is a contaminated region of groundwater (a plume) that we wish to keep from moving. We can do this by installing wells in the region and changing the direction of groundwater flow. We would like to do this as cheaply as possible…
Hydraulic Capture ModelsGoal: To alter the direction of groundwater
flow to control plume migration
Possible Decision Variables:
• Number of wells• Well rates• Well locations
NniyxQnu iii :1)),,(,,(
Governing Equations
hqhBKt
hS
)( :Equation Flow
hK
v
: Law sDarcy'
CqCvCvDt
CR
))(( Transport
v
vvvD ji
tlijtij )( where
Objective Function: Cost to install and operate wells
simulation flow requires of Evaluation
)(
)()(
cost loperationa
01
31
2
coston installati
1
2min1
10
10
i
t n
nii
n
igsii
n
i
bgs
bmi
n
i
bi
h J(u)
tdQczhQc
hzQcdcuJ
f
e
e
e
Implementation: Simulators
MODFLOW for flow equation• USGS code• Cell centered finite differences• FORTRAN 77, serial
MT3DMS for transport equation• EPA code• Links to flow data from MODFLOW• Cell centered finite differences• FORTRAN 90, serial
Constraints
smQi
30064.00064.0capacity Well
n
1i
3
i 0032.0Q rate pumpingNet sm
)( landout flood/dry t Don' maxmin mhhh i
design from wellRemove10Q
: welluseless a installt Don'6
i
Capture constraint to keep the plume from spreading
“We leave it to the modeler to choose the
physical and mathematical representation of
the constraint.”
+ = ?
A closer look at FBHC
dhh kk 21
hK
v
We can consider head differences in adjacent nodes (aligned in the x,y,or z direction) as constraints on the approximate velocity since Dacry’s Law is
dhh kk 21
For example: A finite difference head gradient in theX direction is
x
xhxxh
)()(
A closer look at FBHC
dhh kk 21
d?k?Locations?
FBHC
Advantages:
• Easy to implement
• Constraint requires flow info onlyDisadvantages
• Not constraining the concentration
• d? k? locations?
Alternate FBHC approach
• Is there another way we can use only flow information to capture the plume?
• A method for choosing d,k,locations?
• Directional derivatives?
• Fix well locations easier?
Optimization
Implementation Issues• Evaluation of J(u) requires a simulation• Parallelism is preferred• Gradient information is unavailable• Removing a well means J(u) discontinuous
Sampling methods look appealing:
Optimization is governed by function values
)(min uJDu