Minimise $\|h-h_i\|+\lambda\|h-u\|$ for $\lambda \in [0,100]$

Question

I would be thankful if anyone can answer my question. This is a very basic question. Let's say we wish to minimise the quantity

$$\hat{h}= \|h-h_i\|+\lambda\|h-u\|,$$

where:

$$h=[13,17,20, 17, 20, 14, 17, 18, 16, 15, 15, 12, 19, 13, 17, 13]^\top,\\ h_i=[18, 17, 14, 13, 17, 15, 17, 19, 12, 20, 15, 13, 16, 17, 20, 13]^\top, \\u = [16, 16,16,16,16,16,16,16,16,16,16,16,16,16,16,16]^\top,\\ \text{with }\lambda \in [0,100].$$

I know this is a very basic question, but please help me to understand. Also, please suggest me any book where I can start from zero to learn to solve these kinds of problems.

There is no equation in your question. An equation is `something = something else' and you never used the '=' sign. — CiaPan, Aug 23 '17 at 12:05
I have edited the question, is it still not complete? I found the same equation in an article. — Muhammad Imran, Aug 23 '17 at 12:06
@S.vanNigtevecht $\lambda$ can take any value between 0 and 100. — Muhammad Imran, Aug 23 '17 at 12:13
@S.vanNigtevecht Thank you so much. Basically, this is my first time posting any question. WIll definitely improve with time. — Muhammad Imran, Aug 23 '17 at 12:21

score 0 · Answer 1 · answered Aug 23 '17 at 12:04

0

$\lambda = -\frac {\|h-h_i\|}{\|h-u\|}$ seems to make $\|h-h_i\|+\lambda\|h-u\|$ equal zero.

But I'm unsure if that is what you are looking for...

answered Aug 23 '17 at 12:04

CiaPan

13,049

Can you please solve it numerically for the values given above, if $\lambda =0.5$. Thanks in anticipation. – Muhammad Imran Aug 23 '17 at 12:13

score 0 · Answer 2 · answered Aug 23 '17 at 12:31

0

Because $\lambda$ needs to be positive, immediately you see that $\| h-h_i\|$ will be the smallest value you can obtain (because $\| h-u\|$ is positive). Now simply compute that.

answered Aug 23 '17 at 12:31

SvanN

2,307

Minimise $\|h-h_i\|+\lambda\|h-u\|$ for $\lambda \in [0,100]$

2 Answers2