# minimizing a norm and a linear function

Let $y,\lambda\in\mathbb{R}^n$. I want to minimize the following with respect to $y$. $$f(y)=||y|| + \lambda^Ty$$ where $||y||$ is the Euclidean norm. I first take the derivative of the function and get $$\nabla f(y) = \frac{y}{||y||} + \lambda$$ Then I tried to get the Hessian matrix but I could not conclude anything. Is $f(y)$ is a convex function? If not how can I find the minimum of this function?

• It is convex, because it is the sum of two convex functions. – Tomás Feb 2 '13 at 16:16
• Baby case: $n=1$. The minimum is $0$ if $-1\leq \lambda\leq 1$, and there is no minimum otherwise as the infimum is $-\infty$. – Julien Feb 2 '13 at 16:19

## 2 Answers

Note that $$\|y\|+\langle \lambda,y\rangle\geq \|y\|(1-\|\lambda\|)$$

From the last inequality we conclude that if $\|\lambda\|\in [0,1]$, then the minimum of $f$ is zero.

On the other hand, if $\|\lambda\|>1$, you can take $y=-t\lambda$, and make $t\rightarrow\infty$ to conclude that $f(-t\lambda)\rightarrow-\infty$.

• From the last inequality if $||\lambda|| \in [0,1]$, then the minimum of the function in the right is zero. How can you conclude that $f$ is also zero? – neticin Feb 2 '13 at 16:32
• Note that $f(0)=0$. – Julien Feb 2 '13 at 16:35

If $y'$ is a minimum and is not 0, then let $c>0$,

Then $f(cy') = ||cy'|| + c\lambda^Ty'=c( ||y'|| + \lambda^Ty')$

This function is zero at zero, so you're not gonna get a minimum that is less than zero by this calculation.