Problem when dealing with Multi-Objective Optimization

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Ali 2014-4-10

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评论： Alberto 2014-4-11

Hi programmers,

I've faced a problem that may needs a special formulation before using MATLAB Optimization ToolBox:

If we have a problem with the following expression:

OBJ1 = min f(X)

OBJ2 = max f(X)

Both are subjected to the same inequality constraints:

g_1(X)<=0

g_2(X)<=0

...

g_m(X)<=0

From the above mathematical model .. Is the correct way to go with multi-objective technique since we have two objective functions?

If yes, then how can we do that since we have similar design function f(X) with maximization and minimization (i.e., dealing with Min-Max Problem)?

Thanks

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Alberto 2014-4-11

Hi Ali:

In theory (and I`m talking about Maths), you cant obtain a value x that can maximize one function and minimize another; most of times that's not possible because the dots in a plane has not a proper order to do that.

The usual way is to create another function (the one that you really will maximize or minimize ) with a function that measures the importance of both functions.

For example, if you would like to max f and min g, try to maximize the function f-g if the relevance of both are the same.

You can use weights if some of them are more relevant. If f is more relevant you can try to compute max for the function 3f-2g (the weights depends of experience); also can use functions as weigth, like xf(x)-x^2g(x) or so.

All depends on the meaning of f and g and your knowledge about the problem.

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