Multi Objective Optimization

CE 588
Structural Optimization

Multi-Objective Optimization
Instructor: Y.Doç.Dr. NİLDEM TAYŞİ
3/12/2013
Samadar Salim Majeed

Introduction
 Is

an area of multiple criteria decision
making, that is concerned with mathmatical
optimization problem involving more than one
objective function to be optimized
simultaneously.
 Most real-world engineering optimization
problems are multi-Objective in nature
 Objectives are often conflicting
 Performance vs. Silicon area
 Quality vs. Cost
 Efficiency vs. Portability
 The notion of ”optimum” has to be redefined

 Multiobjective

optimization (multicriteria,
multiperformance, vector optimization or Pareto
optimization )
 Find a vector of decision variables which
satisfies constraints and optimizes a vector
function whose elements represent the objective
functions
 Objectives are usually in conflict with each other
 Optimize: finding solutions which would give the
values of all the objective functions acceptable to
the designer

Classic Methods :
1- Weighted Sum Method
2- Constraint method
3- Weighted Metric Methods
4- Rotated Weighted Metric Method
5- Benson’s Method
5- Value Function Method
Currently

an Evolutionary Algorithm
Methods are Used For MOOP

All

the problems that we have considered in this class
have been comprised of a single objective function
with perhaps multiple constraints and design variables.

Minimize

Subject To:

F (x)

g ( x) ≤ 0 ....

In such a case, the problem has a 1 dimensional
performance space and the optimum point is the one
that is the furthest toward the desired extreme.
Optimum

0

- +

F

What

happens when it is necessary (or at least
desirable) to optimize with respect to more than one
criteria?

Now

we have additional dimensions in our
performance space and we are seeking the best we
can get for all dimensions simultaneously.

What

does that mean “best in all dimensions”?

Consider the following 2D performance space:
F2

Minimize Both F’s

Optimum

F1

But what happens in a case like this:
F2

Minimize Both F’s

Optimum?

Optimum?

F1

The one on the left is better with respect to F1 but
worse with respect to F2.
And the one on the right is better with respect to F2 and
worse with respect to F1.
How does one wind up in such peril?

That depends on the relationships that exist between
the various objectives.
There are 3 possible interactions that may exist
between objectives in a multi-objective optimization
problem:
1. Cooperation
2. Competition
3. No Relationship

What defines a relationship between objectives? How
can I recognize that two objectives have any
relationship at all?
The relationship between two objectives is defined by
the variables that they have in common.
Two objectives will fight for control of common design
variables throughout a multi-objective design
optimization process.

Just how vicious the fight is depends on what type of
interaction exists (of the 3 we mentioned).
Let’s consider the 1st case of cooperation.
Two objectives are said to “cooperate” if they both wish
to drive all their common variables in the same
direction (pretty much all the time).
In such a case, betterment of one objective typically
accompanies betterment of the other.

In such a case, the optimum is a single point (or
collection of equally desirable points) like in our first
performance plot.

F2

Minimize
Both F’s

Optimum

F1

Now let’s consider the 2nd case of competition.
Two objectives are said to “compete” if they wish to
drive at least some of their common variables in
different directions.
In such a case, betterment of one objective typically
comes at the expense of the other.
This is the most interesting case.

In

such a case, the optimum is no longer a single
point but a collection of points called the Pareto Set.

Optimality

criterion for optimization problems with
multiple objectives. A state (set of parameters) is said
to be Pareto optimal if there is no other state
dominating the state with respect to a set of objective
functions.
State

A dominates state B if A is better than B in at least
one objective function and not worse with respect to all other
objective functions.

So let’s take a look at this:
F2

Minimize Both F’s

F1

For completeness, we will now consider the case in
which there is no relationship between two objectives.
When do you think such a thing might occur?
Clearly this only occurs when the two objectives have
no design variables in common (each is a function of a
different subset of the design variables and the 2
subsets have a null intersection).

In such a case, we are free to optimize each function
individually to determine our optimal design
configuration.
That is why this case is desirable but uninteresting.

So back to competing objectives.

Now that we know what we are looking for, that is, the
set of non-dominated designs, how are we going to go
about generating it?
The most common way to generate points along a
Pareto frontier is to use a weighted sum approach.
Consider the following example:

Suppose I wish to minimize both of the following
functions simultaneously:
F1 = 750x1+60(25-x1) x2+45(25- x1)(25- x2)
F2 = (25- x1) x2
For the typical weighted sum approach, I would assign
a weight to each function such that:

w1 + w2 = 1

and

w1 , w2 ≥ 0

I would then combine the two functions into a single
function as follows and solve:

FT = ∑ wi Fi
i

= w1 F1 + w2 F2

The net effect of our weighted sum approach is to
convert a multiple objective problem into a single
objective problem.
But this will only provide us with a single Pareto point.
How will be go about finding other Pareto points?
By altering the weights and solving again.

As mentioned, such schemes are very common in
multi-objective optimization.
In fact, in an ASME paper published in 1997, Dennis
and Das made the claim that all common methods of
generating Pareto points involved repeated conversion
of a multi-objective problem into a single objective
problem and solving.

Ok, so I march up and down my weights generating
Pareto points and then I’ve got a good representation
of my set.
Unfortunately not. As it turns out it is seldom this easy.
There are a number of pitfalls associated with using
weighted sums to generate Pareto points.

Some of those pitfalls are:
Inability

to generate points in non-convex portions of
the frontier
Inability to generate a uniform sampling of the frontier
A non-intuitive relationship between combinatorial
parameters (weights, etc.) and performances
Poor efficiency (can require an excessive number of
function evaluations).

Let’s consider the 1st pitfall:
What is a non-convex portion of the frontier?
I assume you are all familiar with the concept of
convexity so let’s move on to a pictorial.

F2

Minimize Both F’s

This is a non-convex
region of the frontier

F1

Ok so why do weighted sum approaches have difficulty
finding these points?
As discussed in reference 1, choosing the weights in
the manner that we have can be shown to be
equivalent to rotating the performance axes by an
angle that can be determined from the weights and
then translating those rotated axes until they hit the
frontier.
The effect of this on a convex frontier can be visualized
as follows.

F2

Minimize Both F’s

F1

So I think that you can see already what is going to
happen when the frontier is not convex.
Consider the following animation.

So we missed all the points in the non-convex region.
This also demonstrates one reason why we may not get
a uniform sampling of the Pareto frontier.
As it turns out, a uniform sampling is only possible in
this way for a Pareto set having a very specific shape.
So not even all convex Pareto sets can be sampled
uniformly in this fashion. You can read more about
this in reference 1.

Clearly, if we cannot generate a uniform sampling and
we cannot find non-convex regions, then the
relationship between changes in weights and motion
along the frontier is non-intuitive.
Finally, since with each combination of weights, we are
completing an entire optimization of our system, You
can see how this may result in a great deal of system
evaluations.

Multi Objective Optimization

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