Is optimizing a convex function $f(x,y)$ that has a equality constraint $x+y=1$ a convex optimization problem?

Question

I have a convex function $f(x,y)$, with the equality constraint $x+y=1$. Is this still a convex optimization problem, despite the equality constraint? or is it a nonlinear optimization problem?

Answer 1

The standard definition of a convex optimization problem is:

Minimize $f(x)$ subject to $x\in S$
where (1) $S$ is a convex set and (2) $f$ is a convex function on set $S$

• The set $(x,y)\in R^2$ with $x+y=1$ is clearly convex; hence (1) holds.

• And $f$ is convex by your statement; hence (2) holds.

Answer 2

Linear equality constraints are OK in a convex optimization problem because the set of solutions to a linear system of equality constraints is convex. When you have equality constraints involving a nonlinear function of the variables, then you're out of luck.