# Determine the function is convex function or not?

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.

• What have you tried so far? Do you know what a linear function is and whether linear functions are convex? – Chris Harshaw Dec 5 '19 at 17:35
• Actually, I have no idea about this question. I've read some doc related to convex function. However, I totally don't understand what are they talking about – chan sum Dec 5 '19 at 17:41
• Hmmm...do you know what the definition of convex function and convex set are? Is this for a homework assignment or for your own personal use? – Chris Harshaw Dec 5 '19 at 17:43
• In general, you have to show/prove that both objective and constraints are convex. – user550103 Dec 5 '19 at 17:50
• I think I know what convex set is, but I am not sure the definition of convex function. This is a question in my lecture note, something like a small quiz after the lecture. – chan sum Dec 5 '19 at 17:52

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.
• give it a try first and I will help to edit your working? You get to learn more if you give a try. For example, try to write out what is $f(\lambda x_1+(1-\lambda)x_2)$ first? – Siong Thye Goh Dec 7 '19 at 13:09