how I found W1,W2 ?

But ONLY YOU can choose the values of w1 and w2. They would represent the relative importance of these two functions, as well as the relative scaling of the two functions, to make them both of roughly the same size.

Completely your choice though.

You might want to do some reading here:

https://en.wikipedia.org/wiki/Multi-objective_optimization

https://en.wikipedia.org/wiki/Pareto_efficiency

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ahlem sellami 2018-4-2

my object is to maximize F1 and minimize F2 and formulating a single-objective optimization problem f more explanation please

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Answer 2

Walter Roberson 2018-4-2

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https://ww2.mathworks.cn/matlabcentral/answers/392125-how-i-found-w1-w2#answer_313119

F1(x) and F2(x) are whatever they are. Changing w1 and w2 will not change them.

Is f a fixed value and is the question to fit a bunch of known points? You can use the \ operator for fitting. But that will not have anything to do with minimizing or maximizing.

Are there bounds on the weights or on their sum?

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Walter Roberson 2018-4-3

在 MATLAB Online 中打开

Take the very simple example

F1 = @(x) x;
F2 = @(x) x;

Now try to choose weights so that w1*F1(x) + w2*F2(x) simultaneously maximizes F1 and minimizes F2. Now, for any given pair of weights, as you increase x, w1*F1(x) and w2*F2(x) both increase, so you can maximize F1 by increasing x; and for any given pair of weights, as you decrease x, w1*F1(x) and w2*F2(x) both decrease, so you can minimize F2 by decreasing x. If you increase x then you make F2(x) worse for any given weights; if you decrease x then you make F1(x) worse for any given weights. So what weights and what x do you choose?

For this particular system, w1*F1(x) + w2*F2(x) = w1*x + w2*x = (w1+w2)*x and you defined w1+w2 as 1 so this is 1*x = x. So for weights under the constraints you define, you would be asking to simultaneously maximize and minimize x. How do you choose the x?

What you are asking for simply has no solution in general, even in some simple cases.

Is there a solution in some cases? Possibly. To find the maxima and minima of w1*f1(x) + w2*f2(x) we take its derivative and solve for 0:

solve(diff(w1*f1(x) + w2*f2(x), x)==0, x)

This is

solve(w1 * diff(f1(x),x) + w2 * diff(f2(x),x) == 0, x)

and sometimes that has a solution, x = something. You would then take

subs(diff(f1(x),x,x), x, something)

and check the sign of the result, to find the conditions on w1 and w2 under which the sign is negative, which are the places where x = something would maximize f1. If there are no places the sign can become negative then the task is not possible for that f1. If there are, then

subs(diff(f2(x),x,x), x, something)

and look at the sign under the situations found for diff(f1(x),x,x) and look for a positive. If you can find situations in which it can be positive then you have identified a minima for f2.

These will not necessarily be global minima or maxima, but at least they would be simultaneous.