A convex objective function will be convex regardless of constraints?

Question

Let's say I have a convex objective function. The boilerplate example is $z=x^2+y^2$. Now, I also have some constraint, $f(x,y)=0$. Is it true that the constrained optimization problem must be convex as well? I suspect this is the case, but have only rough intuition to back it up and was wondering if there is a proof.

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My intuitive argument is that if the objective function is convex, it is always "curved downwards" (where "down" is in the direction of the objective function). Now, since the constraint can only add a condition in the x-y plane, it can't introduce any curvature in the z-direction (the direction of the objective function). So, the convexity of the objective function in that direction must be preserved.

Answer 1

The concept that delivers results in convex optimization is that the objective function have a convex epigraph, that is, the set of points $\{(x,f(x)):x\in\text{ constraint set}\}$ be convex. This will fail if the constraint set is non-convex.

Indeed, Rockafellar's 1970 book Convex Analysis defines the term convex function (on p.23) as follows

Let $f$ be a function whose values are real or $\pm\infty$ and whose domain is a subset $S$ of $R^n$. The set $$\{(x,\mu)|x\in S,\mu\in R,\mu\ge f(x)\}$$ is called the epigraph of $f$ and is denoted by $\text{epi } f$. We define $f$ to be a convex function on $S$ if $\text{epi }f$ is convex as a subset of $R^{n+1}$.