l***o
发帖数: 6 | 1 Hi there,
Sorry I can't input Chinese right now.
Please help me with the following question. Thank you so much!
To solve a problem, I have two objectives:
1. choose xi to max f(xi),
2. choose xi to min g(xi).
where f(x) and g(x) are two functions of x.
Since there are two objectives, I want to combine them to one objective
function to ease the problem.
Two possible choices of the objective functions are:
1. choose xi to max[f(xi) - g(xi)],
2. choose xi to max[ f(xi)/g(xi) ].
My question is: will | N***l
发帖数: 52 | 2 i think the problem is the optimizer for f and g may be different,
and if they ARE different, any composite objective function can only
be a ``guess" of the solution, actually there is no such thing as
``the" solution in your problem.
As in the f-g case, you are maximizing the difference, but the thing
is the maximizer for the difference function may not simultaneously
maximize f and minimize g, all you know is it maximize the difference.
I think there is no best solution in general, but in your
【在 l***o 的大作中提到】
: Hi there,
: Sorry I can't input Chinese right now.
: Please help me with the following question. Thank you so much!
: To solve a problem, I have two objectives:
: 1. choose xi to max f(xi),
: 2. choose xi to min g(xi).
: where f(x) and g(x) are two functions of x.
: Since there are two objectives, I want to combine them to one objective
: function to ease the problem.
: Two possible choices of the objective functions are:
| l***o
发帖数: 6 | 3 Yeah, you are right! There is a compromise bwtween maxf(x) and maxg(x) for my
problem.
So the 1st objective function maximize the absolute difference, and the 2nd
objective function maximize the proportional difference, right?
【在 N***l 的大作中提到】
: i think the problem is the optimizer for f and g may be different,
: and if they ARE different, any composite objective function can only
: be a ``guess" of the solution, actually there is no such thing as
: ``the" solution in your problem.
: As in the f-g case, you are maximizing the difference, but the thing
: is the maximizer for the difference function may not simultaneously
: maximize f and minimize g, all you know is it maximize the difference.
: I think there is no best solution in general, but in your
| N***l
发帖数: 52 | 4 according to your definition, yes.
my
【在 l***o 的大作中提到】
: Yeah, you are right! There is a compromise bwtween maxf(x) and maxg(x) for my
: problem.
: So the 1st objective function maximize the absolute difference, and the 2nd
: objective function maximize the proportional difference, right?
| l***o
发帖数: 6 | 5 Thank you!
You only mentioned the linear combination of the two functions, but not the
nonlinear combination, such as f(x)/g(x). Is this combination correct? | N***l
发帖数: 52 | 6 I think there are counter examples (bad examples) for both of your
composite ``optimal" objectives.
Hi there,
Sorry I can't input Chinese right now.
Please help me with the following question. Thank you so much!
To solve a problem, I have two objectives:
1. choose xi to max f(xi),
2. choose xi to min g(xi).
where f(x) and g(x) are two functions of x.
Since there are two objectives, I want to combine them to one objective
function to ease the problem.
Two possible choices of the objective functi
【在 l***o 的大作中提到】
: Hi there,
: Sorry I can't input Chinese right now.
: Please help me with the following question. Thank you so much!
: To solve a problem, I have two objectives:
: 1. choose xi to max f(xi),
: 2. choose xi to min g(xi).
: where f(x) and g(x) are two functions of x.
: Since there are two objectives, I want to combine them to one objective
: function to ease the problem.
: Two possible choices of the objective functions are:
| D*******a
发帖数: 3688 | 7 you can do whatever you want as long as it is meaningful
for linear, you can view 'a' as a price
【在 l***o 的大作中提到】
: Thank you!
: You only mentioned the linear combination of the two functions, but not the
: nonlinear combination, such as f(x)/g(x). Is this combination correct?
| s***t
发帖数: 113 | 8 You also need to think about how to solve the resulted model.
In general, f(x) - g(x) is much easier than f(x)/g(x). You can also
model the problem in 2-level optimization problem, e.g.,
min g(x)
s.t. max f(x)
s.t. x \in X
The resultant problem is however usually very hard to solve.
【在 l***o 的大作中提到】
: Hi there,
: Sorry I can't input Chinese right now.
: Please help me with the following question. Thank you so much!
: To solve a problem, I have two objectives:
: 1. choose xi to max f(xi),
: 2. choose xi to min g(xi).
: where f(x) and g(x) are two functions of x.
: Since there are two objectives, I want to combine them to one objective
: function to ease the problem.
: Two possible choices of the objective functions are:
| l***o
发帖数: 6 | 9 Yeah, you are right! There is a compromise bwtween maxf(x) and maxg(x) for my
problem.
So the 1st objective function maximize the absolute difference, and the 2nd
objective function maximize the proportional difference, right?
【在 N***l 的大作中提到】
: i think the problem is the optimizer for f and g may be different,
: and if they ARE different, any composite objective function can only
: be a ``guess" of the solution, actually there is no such thing as
: ``the" solution in your problem.
: As in the f-g case, you are maximizing the difference, but the thing
: is the maximizer for the difference function may not simultaneously
: maximize f and minimize g, all you know is it maximize the difference.
: I think there is no best solution in general, but in your
| l***o
发帖数: 6 | 10 So you mean that the two objective functions are not equal and may give
different results for a given problem?
【在 N***l 的大作中提到】
: I think there are counter examples (bad examples) for both of your
: composite ``optimal" objectives.
:
: Hi there,
: Sorry I can't input Chinese right now.
: Please help me with the following question. Thank you so much!
: To solve a problem, I have two objectives:
: 1. choose xi to max f(xi),
: 2. choose xi to min g(xi).
: where f(x) and g(x) are two functions of x.
| D*******a
发帖数: 3688 | 11 you can do whatever you want as long as it is meaningful
for linear, you can view 'a' as a price
【在 l***o 的大作中提到】
: Thank you!
: You only mentioned the linear combination of the two functions, but not the
: nonlinear combination, such as f(x)/g(x). Is this combination correct?
| w******o
发帖数: 442 | 12 f(xi)/g(xi) is good for persentage change.
f(xi)-g(xi) is good for aboslute value change.
It depend on which one do you preffer (persentage or aboslute value), which is
better.
【在 l***o 的大作中提到】
: Hi there,
: Sorry I can't input Chinese right now.
: Please help me with the following question. Thank you so much!
: To solve a problem, I have two objectives:
: 1. choose xi to max f(xi),
: 2. choose xi to min g(xi).
: where f(x) and g(x) are two functions of x.
: Since there are two objectives, I want to combine them to one objective
: function to ease the problem.
: Two possible choices of the objective functions are:
| D*******a
发帖数: 3688 | 13 there is a trade off between two objectives
usually, you can combine them like
max f(x)-a*g(x)
or
max f(x)
s.t. g(x)>b
there is no universal rules
【在 l***o 的大作中提到】
: Hi there,
: Sorry I can't input Chinese right now.
: Please help me with the following question. Thank you so much!
: To solve a problem, I have two objectives:
: 1. choose xi to max f(xi),
: 2. choose xi to min g(xi).
: where f(x) and g(x) are two functions of x.
: Since there are two objectives, I want to combine them to one objective
: function to ease the problem.
: Two possible choices of the objective functions are:
| w****r
发帖数: 1046 | 14 You may take derivative with respect to xi to find your optimum f and g. But
the values of xi for the two optimization functions may differ.
Well, the optimum solution for maximizing the function of f(x)-g(x) probably
does not optimize f and g, simutaneously. Sometimes, it does.
Your problem is very similar to firm theory in microeconomics. |
|
