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