I am new to the optimization theory and want to figure out the following problem about convexity.
Suppose that function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is a convex function. Consider another function $f^{\prime}: \mathbb{R}^m\rightarrow \mathbb{R}$ which satisfy the following conditions:
- $m<n$;
- For a fix $x_2\in\mathbb{R}^{n-m}$, which satisfy $\forall x_1\in dom(f^{\prime}), x=[x_1^T, x_2^T]^T\in dom(f)$, $g(x_2) = f(x)$;
Can we ensure that function $g$ is also convex?
The above mathematical formulation might be abstract. What I want to express here is that, if a function is convex in $\mathbb{R}^n$ space, can we prove that the function of the partial variable is still convex?
To make things clear, let us consider an example function $f$ which is defined in $\mathbb{R}^3$. The function $f$ calculate the squared norm of $x\in \mathbb{R}^3$, which is $x(1)^2+x(2)^2+x(3)^2$, so it is a convex function. Now, suppose that we fix $x(3) = 1$, then this function becomes $g:\mathbb{R}^2\rightarrow \mathbb{R}$, which is $x(1)^2+x(2)^2 + 1$. This function is also convex. My question is that if this property often holds?
BTW: I think the expression above the rule maybe a little bit ambiguous, can someone also help one better illustrate what I want to express in formal mathematical formulation?