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Say I have two functions of the variable x, f and g. I would like to find a single value of x that gives me the largest value of f and smallest value of g. Is there a procedure to do this? Note: I'm not talking about the global maximum of f and global minimum of g, just some large value of f and some small value of g.

Can this be generalised to an arbitrary number of functions? E.g. I want to find the value of x such that I get the largest values possible of functions f and g, and smallest values possible of functions p, q, and r.

Also, I am not sure what to tag this post as aside from optimization.

Edit: So I might have just asked an XY problem. So here is the initial problem. There are two functions of "w", risk(w) and reward(w). Essentially I want to minimise risk while maximising the reward.

ranky123
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  • In general, this can't be done. What are you trying to do that makes you ask this question? This sounds like an XY problem. – Toby Mak Aug 28 '21 at 10:10
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    In general, you have to decide what you are trying to maximize. Plausible choices might be $f(x) - g(x)$ or $\frac{f(x)}{g(x)}$. For the latter choice, you would have to decide how to interpret (for example) $g(x) = 0$ and (perhaps separately) $g(x) < 0.$ – user2661923 Aug 28 '21 at 10:23
  • How would you differentiate two max-min pars? Say you have by some procedure these (1,4), (2,5), (0,3)... which one would you pick? In any case when you decide how to differentiate these, say |f(x)-g(x)|, then it is the matter of finding the max for that expression and job done. – Alex Peter Aug 28 '21 at 12:47
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    @TobyMak I'm reading about modern portfolio theory. We have two functions of the variable "w", risk(w) and return(w). The goal is to pick a w such that risk is minimised and return is maximised. – ranky123 Aug 28 '21 at 13:06
  • @user2661923 So I guess choosing a ratio or difference is up to the specific real-life application. – ranky123 Aug 28 '21 at 13:15
  • In that case, it would make sense to calculate expected return. – Toby Mak Aug 28 '21 at 13:28
  • You might be interested in reading about Goal Programming. – awkward Aug 28 '21 at 13:30
  • Maximize the function $f-g$. – K.defaoite Aug 28 '21 at 13:36
  • @awkward Would it be that simple though? Like thinking from a real-life perspective, return has units say dollars, but risk doesn't have units of dollars, so a simple subtraction doesn't make physical sense. – ranky123 Aug 28 '21 at 15:10
  • I would not say that goal programming is simple, but it can handle multipole objectives with different units of measurement. It's up to you to define the trade-offs between the objectives. – awkward Aug 28 '21 at 15:22

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