Determine the function is convex function or not?

Question

I've got some trouble in determining the function, which contains vectors and matrix, is a convex function or not.

\begin{aligned}\min _{x}k^{T}x\\ s.t. Ax\leq y\end{aligned}

$x$ is $n\times1$ vector, $y$ is $m\times1$ fixed vector and $A$ is $m\times n$ fixed matrix

Is the above problem is convex optimization problem? and why? For this question, should we have to prove both $k^{T}x$ and $Ax\leq y$ are convex functions? If yes, please tell me the detailed steps.

Answer 1

Guide:

A function $f:\mathbb{R}^n \to \mathbb{R}$ is convex if for $\forall \lambda \in (0,1), x_1, x_2 \in \mathbb{R}^n$, $$f(\lambda x_1 + (1-\lambda)x_2) \le \lambda f(x_1) + (1-\lambda )f(x_2).$$

Now let $f(x)=k^Tx$, verify that the condition above holds.

Also prove that $\{x \in \mathbb{R}^n:Ax \le y\}$ is a convex set.