gmalivuk wrote:Yeah, with x(t) and y(t) plotted on an x-y plane, maximizing x and y at the same time means having a cusp that "points" toward the top-right.

cusp.png

Such a cusp is not a general feature of parametric plots, and it's irrelevant (to the general case) that if your graph is just a line segment through the origin then you can identify the top-right endpoint of that segment.

In your picture you can maximise x further, at the cost of taking a hit on y, by choosing the right-most point of that loop at the bottom right.

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Generally, if you are in a situation where you would like to maximise two variables, you have to choose an objective function. In its simplest form you simply assign weights to the variables x and y, as a measure of how much value you attach to increasing one over the other. For example, if increasing x by 1 unit is always worth twice as much to you as increasing y by 1 unit, then you would have the objective function F(x,y)=2x+y. This offers you a way of deciding whether for example (6,20) is better than (16,13).

If you have a picture of your search space (whether it is a line like gmalivuk's picture, or a region of the x-y plane) then you can draw lines F(x,y)=c for various values of c in the picture. My example F gives you parallel lines, with higher values of c the further up and to the right you go. Your optimal point is the furthest point (or points) of your search space in that direction.