Optimization of 4 cost functions altogether with specified constraints

How can I optimize 4 simple cost functions altogether with specified constraints on each of them using MATLAB and optimization toolbox?

[Matt J]

When you say "altogether", do you mean the sum of the 4 four functions? Otherwise, what do you mean?

[Ali Rezaei]

I mean simultaneously.

[Matt J]

Are they all functions of different sets of variables? So each function f_i(x_i), i=1,2,3,4 will have a different minimizing x_i?

[Ali Rezaei]

All 4 functions use 4 same variables.

[Matt J]

Maybe I wasn't clear. Do you mean you want to calculate

 [argmin(f1), argmin(f2), argmin(f3), argmin(f4)]

without using a for-loop? If instead, you're looking for a single x that simultaneously minimizes f1(x), f2(x), f3(x), and f4(x), then see Jan's Answer.

[Ali Rezaei]

Yes, that is true. Each f_i (i=1,2,3,4) can be shown as f_i(x1,x2,x3,x4). The x1, x2, x3 and x4 are the same for all of 4 functions.

Answer 1

[Jan]

You cannot optimize for 4 different criteria at the same time. An optimization requires a scalar cost function, and this is related to the nature of the term "optimimum".

If you want to consider 4 parameters, you still need a norm to create a scalar: E.g. a weighted or unweighted sum(), max() or mean(). Using max() is not smooth anymore and the optimization method need to consider this explicitly.

Example:

 Find the price of an article, which maximizes the income of the sellers and minimizes the costs for the buyers simultaneously.

This must fail for obvious reasons.

[Ali Rezaei]

Thanks for the proper answer, Is there another methodology like MCDM to solve the problem ?