2.
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
3.
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
11.
Multi-Objective Optimization
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
12.
Multi-Objective Optimization
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 ....
13.
Multi-Objective Optimization
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
14.
Multi-Objective Optimization
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”?
15.
Multi-Objective Optimization
Consider the following 2D performance space:
F2
Minimize Both F’s
Optimum
F1
16.
Multi-Objective Optimization
But what happens in a case like this:
F2
Minimize Both F’s
Optimum?
Optimum?
F1
17.
Multi-Objective Optimization
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?
18.
Multi-Objective Optimization
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
19.
Multi-Objective Optimization
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.
20.
Multi-Objective Optimization
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.
21.
Multi-Objective Optimization
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
22.
Multi-Objective Optimization
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.
23.
Multi-Objective Optimization
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.
24.
Multi-Objective Optimization
So let’s take a look at this:
F2
Minimize Both F’s
F1
25.
Multi-Objective Optimization
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).
26.
Multi-Objective Optimization
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.
27.
Multi-Objective Optimization
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:
28.
Multi-Objective Optimization
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
29.
Multi-Objective Optimization
I would then combine the two functions into a single
function as follows and solve:
FT = ∑ wi Fi
i
= w1 F1 + w2 F2
30.
Multi-Objective Optimization
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.
31.
Multi-Objective Optimization
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.
32.
Multi-Objective Optimization
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.
33.
Multi-Objective Optimization
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).
34.
Multi-Objective Optimization
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.
35.
Multi-Objective Optimization
F2
Minimize Both F’s
This is a non-convex
region of the frontier
F1
36.
Multi-Objective Optimization
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.
37.
Multi-Objective Optimization
F2
Minimize Both F’s
F1
38.
Multi-Objective Optimization
So I think that you can see already what is going to
happen when the frontier is not convex.
Consider the following animation.
39.
Multi-Objective Optimization
F2
Minimize Both F’s
F1
40.
Multi-Objective Optimization
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.
41.
Multi-Objective Optimization
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.
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