I am virtually new to optimization (saw it years ago in a very shallow course) and now I came across a problem that I believe would require from it. The problem is I don't know exactly how to proceed.
The problem is as follows: Subject to some constrains I want to minimize a function $f(t) = c(t) * g(t)$, where $c(t)$ and $g(t)$ are functions that vary over time. At the same time I want to maximise another function $h(t)$. Moreover, there are some constraints but I don't think that is much of a problem.
In summary, I have two problems (1) to simultaneously minimize one function and maximise another one, and (2) one of the functions depend on the multiplication of other functions.
How should I proceed with this?
Thanks and sorry if this is too basic.
_________________ Edit _________________
I want to minimize the cost of using a service (e.g., CPU time in a Cloud computing service provider). The cost $c$ of the service depends on the demand of this service so I don't know in advance (but we can assume if follows a probability distribution based on previous observations). If at a particular time slot $t$ I use $x$ units of that service (say CPU cycles) that would cost me $x*c$, whatever is the value of $c$ at that particular time.
Nonetheless, I can decide not to use that service at some points in time to reduce the cost because I can get CPU cycles from my own computer or another computer. So, if I am not mistaken, the function I want to minimise would be:
$$ f(x, m, a) = x*c - (m + a) *c$$
Being $m$ and $a$ the number of CPU cycles I get from my own computer and from another computers. So if $f(x,m,a)$ is positive I would end up paying while if negative I am saving money.
Moreover, there are some restrictions. There is a given amount of CPU cycles that I need at every time period to fulfil my task. Again, as with the cost, this is dependent on time but I cannot know in advance (but I could somehow predict or consider it follows some probability distribution).
For now I have simplified the problem not to consider the other function that I need to simultaneously maximise. This is is just to make things easier and clearer.
Do you think the formulation of the problem is correct? Would you formulated it differently? If this is not the case, how do I deal with these variables that vary over time?
Sorry for my poor exposition before!