[PPT] - Constraint Approach to Two Approaches Let Us Estimate the . . . PowerPoint Presentation

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Constraint Approach to Multi-Objective Optimization

Martine Ceberio, Olga Kosheleva, and Vladik Kreinovich

University of Texas at El Paso El Paso, TX 79968, USA mceberio@utep.edu, olgak@utep.edu, vladik@utep.edu

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1. Multi-Objective Optimization: Examples

In meteorology and environmental research, it is im-

portant to measure fluxes of heat, H2O, CO2, etc.

To perform these measurements, researchers build tow-

ers with sensors at different heights.

These towers are called Eddy flux towers.
When selecting a location for the Eddy flux tower, we

have several criteria to satisfy: – The station should be located as far away from roads as possible. – Else, gas flux from cars influences our measure- ments. – On the other hand, the station should be located as close to the road as possible. – Otherwise, it is difficult to carrying the heavy parts when building such a station.

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2. Multi-Objective Optimization (cont-d)

In geophysics, different type of data provide comple-

mentary information about the Earth structure: – information from the body waves (P-wave receiver functions) mostly covers deep areas, while – the information about the Earth surface is mostly contained in surface waves.

When we know the relative accuracy of different data

types i, we could apply the Least Squares:

i

1 σ2

i

·

k

( xik − xik(p))2

→ min

p

.

In practice, however, we do not have a good informa-

tion about the relative accuracy of different data types.

In this situation, all we can say that we want to mini-

mize the errors fi(x) corr. to all the observations i.

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3. Multi-Objective Optimization is Difficult

If we want to minimize a single objective f(x) → min,

this has a very precise meaning: – we want to find an alternative x0 for which – f(x0) ≤ f(x) for all other alternatives x.

The difficulty is that multi-objective optimization is

not precisely defined.

For example, when selecting a flight, we want to mini-

mize both travel time and cost.

Convenient direct flights which save on travel time are

more expensive.

On the other hand, a cheaper trip may involve a long

stay-over in between flights

It is therefore necessary to come up with a way to find

an appropriate compromise between several objectives.

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4. Analysis of the Problem

Without losing generality, let us consider a multi-
bjective maximization problem.
Ideally, we should select x0 that satisfies the constraints

fi(x0) ≥ fi(x) for all i and for all x.

So, ideally, if we select an alternative x at random, then

with probability 1, we satisfy the above constraint.

The problem is that we cannot satisfy all these con-

straints with probability 1.

A natural idea is thus to find x0 for which the probabil-

ity of satisfying these constraints is as high as possible.

Let us describe two approaches to formulating this idea

(i.e., the corresponding probability) is precise terms.

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5. Two Approaches

1st approach: maximize the probability that for a ran-

dom x, we have fi(x0) ≥ fi(x) for all i.

2nd approach: maximize the probability that for a ran-

dom x and for a random i, we have fi(x0) ≥ fi(x).

We thus need to estimate:

– the probability pI(x0) that for a randomly selected x, we have fi(x0) ≥ fi(x) for all i, and – the probability pII(x0) that for a randomly selected x and a randomly selected i, we have fi(x0) ≥ fi(x).

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6. Let Us Estimate the First Probability

We do not have any prior information about the de-

pendence between different objective functions fi(x).

So, it is reasonable to assume that the events fi(x0) ≥

fi(x) and fj(x0) ≥ fj(x) are independent.

Thus, pI(x0) =

n

i=1

pi(x0), where pi is the probability that fi(x0) ≥ fi(x) for a random x.

How can we estimate this probability pi(x0)?

– before starting to solve this problem as a multi-

bjective optimization problem,

– we probably tried to simply optimize each of the

bjective functions,

– hoping that the corresponding solution would also

ptimize all other objective functions.

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7. Estimating the First Probability (cont-d)

So, we know the largest possible value Mi of each of

the objective functions: Mi = max

x

fi(x).

The maximum can often be found by equating the

derivatives of the objective function to 0.

This way, we find not only maxima but also minima of

the objective function.

Thus, it is reasonable to assume that we also know

mi = min

x fi(x).

The value fi(x) corresponding to a randomly selected

x lie inside the interval [mi, Mi].

We do not have any reason to believe that some values

from this interval are more probable.

It is thus reasonable to assume that all the values from

this interval are equally probable.

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8. Estimating the First Probability (cont-d)

We do not have any reason to believe that some values

from this interval are more probable.

It is thus reasonable to assume that all the values from

this interval are equally probable.

So, we have a uniform distribution on the interval

[mi, Mi].

This argument can be formalized as selecting the dis-

tribution with the probability density ρ(x) for which S = −

ρ(x) · ln(ρ(x)) dx → max .
For the uniform distribution, we have

pi(x0) = fi(x0) − mi Mi − mi , so pI(x0) =

n

i=1

fi(x0) − mi Mi − mi → max

x0 .

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9. Let Us Estimate the Second Probability

In the second approach, we select the objective func-

tion fi at random.

We have no reason to prefer one of the n objective

functions.

So, it makes sense to select each of these n functions

with equal probability 1 n.

For each i, we know the probability

pi(x0) = Prob(fi(x0) ≥ fi(x)) = fi(x0) − mi Mi − mi .

The probability of selecting each objective function

fi(x) is equal to 1 n.

Thus, we can use the complete probability formula to

compute pII(x0) =

n

i=1

1 n · fi(x0) − mi Mi − mi .

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10. Both Ideas Lead to Widely Used Methods of Multi-Objective Optimization

pII(x0) is a linear combination of n objective functions.
Maximizing linear combinations is indeed the most

widely used approach to multi-objective optimization.

The first idea leads to maximizing a product of nor-

malized objective functions: pI(x0) =

n

i=1

fi(x0) − mi Mi − mi .

Maximizing pI(x0) is exactly what Bellman-Zadeh

fuzzy approach recommends for f&(a, b) = a · b.

This is also exactly what the Nobelist John Nash rec-
mmended:

– for a similar situation of making a group decision – when each participant would like to optimize his/her own utility fi(x) → max.

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11. Acknowledgments This work was supported in part by the National Science Foundation grants:

HRD-0734825 and HRD-1242122

(Cyber-ShARE Center of Excellence), and

DUE-0926721.