Find an optimum point in a 2D graph

I hope to get some opinions on this:

I have a 2D graph, with 10 lines, which form constraints. I have the analytical relations of these graphs. Now I want to find the optimum point in the graph, which lies in the lower right corner of the feasible area. Of course I need some kind of weighing factor, to trade between the x an y values, I can specify something myself.

What is the best approach to this? I currently use the fmincon function. Even though the objective value is x(2)*constant-x(1) and there are only two non-linear constraints, it uses 2.4 seconds. Not that much, but I find it still rather slow. Are faster options?