Optimization of 2 equations with 2 independent variables

Dears, I have a sort of problem where I have 2 different equations which are functions of 2 independent variables. Let say F1(x,y) and F2(x,y). "x" represents the temperature which is constrained between 200 deg.C and 300 deg.C and "y" represents the time which is constrained between 0 and 60 minutes. The problem is that the two functions increases (non-linearly) whenever any of the two variables increases but what I really want is the optimum variables values that minimize F1 while maximizing F2.

[Matt J]

what I really want is the optimum variables values that minimize F1 while maximizing F2.

Even if you do this on a bounded region of (x,y), what if F1 doesn't have a minimum at the same x,y where F2 has a maximum?

Perhaps you really mean you want to minimize something like F1(x,y)-F2(x,y) or F1(x,y)-10*F(x,y) ? If not, it's not clear what you mean. If yes, then how would you choose the relative weights of F1 and F2?

[John D'Errico]

Since you cannot assume the minimum of F1 will happen in the same place as the maximum of F2, this problem is poorly-posed, in the sense that a simultaneous solution will generally never exist.

The common solution is to optimize some sum of the two objectives. This is the idea behind multi-criteria optimization. You decide the relative importance of each competing objective. That weight will also factor out any scale differences between the two (or in general, N) components. Then you minimize (or maximize as you wish) the weighted sum of the components. Since you want to minimize one and maximize the other, one of those weights will be negative.

A tool which does the work for you is fgoalattain, from the optimization toolbox. But you can do it yourself using any optimizer.

[Mohamed Talaat]

Dear @Matt J, for my case F1 represents mass loss in kg while F2 is the heating value of the remaining mass in J/kg, so I am not sure if I add or subtracting the two functions may add something as they are not of the same domain.

[Mohamed Talaat]

@John D'Errico, it seems as you said "you cannot assume the minimum of F1 will happen in the same place as the maximum of F2" as it depends on the chemical behaviour of the sample I deal with based on its constituents. And the way I though about to solve this was to multiply two functions (F1 here will represent the remaining mass of the sample in kg while F2 is the heating value of the remaining mass in J/kg) to get one function represents the energy yield from the sample to be maximized but the problem is that it will give me the lower bounds of my constraints which means the process shall not be done at all.

[Matt J]

@Mohamed Talaat

The problem definition is your responsibility. No one besides you know can know what optimization problem you need to solve. If you are having difficulty solving the optimization problem once an objective and constraints have been selected, then the community may be able to give you advice.