An explanatory note on a ¡­

An explanatory note on a detail on the example given in Tuesday's lecture:

The issue was whether the set defined by the constraints in the example is closed. It is not -- and the issue is that at certain points, the functional equation defining g has no solution (in the real numbers); consider x=0 and y so that the right hand side is 0 -- that is, y=-2/3. Then g must solve exp g = 0, which is impossible.

This is the similar phenomenon as the following example: max f(x,y) subject to ln x less than or equal to 0. Then x would not only have to be less than or equal to 1, but also >0 in order for the ln to be defined.

Hence, if f were -|x| - |y| (maximum at (0,0)), then the constrained problem would not have any solution; the best we could hope for, would be to choose y=0 and let x approach 0 from above.

For the purposes of this course: If an exam problem states that a function g exists, then you can safely assume it does.