Set boundary for least-square calculation

Hello,

When calculating least-squares we use the form Xw=y (where X is a matrix and y is a vector of appropriate dimensions).

This gives us an approximation, and it can happen that some resulting w will be such that its Xw<y.

Is there a way how to make every w big enough so that its Xw≥y?

And is it possible, at the same time, to make sure this Xw is smaller than some other given number?

Can MATLAB resolve this?

Thank you Best regards

hugo

Answer 1

[Teja Muppirala]

LSQLIN can do this:

W = rand(10,5);
y = rand(10,1);
someOtherGivenNumber = 1.3;

opts = optimset(optimset('lsqlin'),'LargeScale','off');

x = lsqlin(W,y,[-W; W] , [-y; 0*y+someOtherGivenNumber],[],[],[],[],[],opts)

If it successfully converges to a solution, then you can confirm that X*w >= y and X*w < someOtherGivenNumber (within some tolerances).

[Teja Muppirala]

LSQLIN is a function from the Optimization Toolbox

[hugolove]

Hello Teja, thank you very much for your help. I am very new to MATLIB and do maths occasionally. I tried what you suggest.

When calculating least-squares we use the form Xw=y.

I have created this (10x5)Matrix X:

2 4 1 3 3

3 5 2 2 3

6 6 1 1 4

0 7 3 7 5

2 8 4 10 2

1 0 5 3 3

1 6 0 0 7

2 7 7 5 0

3 8 8 4 7

4 9 0 3 8

and this (10x1) vector y1.....y10:

105

45

36

78

20

60

30

20

79

94

then, when I run your code (for start without someOtherGivenNumber):

opts = optimset(optimset('lsqlin'),'LargeScale','off');

w = lsqlin(X,y,[-X; X] , [-y; 0*y],[],[],[],[],[],opts)

.....I get these w1 ...w5 numbers:

2.6923

5.3846

1.3462

4.0385

4.0385

then for a check, when I create a table by multiplication of these w numbers with transposed X, like:

w1*Xn1,m1..........w1*Xn1,m10

.............................

.............................

.............................

w5*Xn5,m1..........w5*Xn5,m10

and do sum of its columns I should get values same or bigger than y1.....y10:

105 45 36 78 20 60 30 20 79 94 (transposed vector of y)

But I did not get it. I got these numbers:

52.5 57.9 70 90.2 102 33.7 63.3 72.7 106 104

Do you know what have I done wrong please

hugo

[Teja Muppirala]

Since you are not including the second constraint, the function call becomes:

w = lsqlin(X,y,-X , -y,[],[],[],[],[],opts)

In this case, I get a reasonable answer with your numbers.

See the help for LSQLIN to see how the constraints are formulated.

[hugolove]

Dear Teja, you are right. I have set someOtherGivenNumber = 200 and after the multiplication I got:

105 85.8 151 118 180 99.7 42.6 20 98.8 169

so every of these numbers is bigger in comparison to its "column copy"

105 45 36 78 20 60 30 20 79 94

But now, can this someOtherGivenNumber be different for every Xw? So for example we would get sth. like:

105 70 140 115 180 99.7 42 20 98.8 155

[hugolove]

well it is not very good example. off course if it was possible to set someOtherGivenNumber for every Xw all numbers would be shifted appropriately.

[hugolove]

Hello Teja, although my question was not answered fully I thank you very much for your help and time, and for guiding me right direction. I mark your answer as the right one. From my last question I have made another post which can be found here:

http://www.mathworks.com/matlabcentral/answers/34740-how-to-use-lsqlin-to-set-different-upper-limit-for-each-xw

Again, thank you very much and have a nice weekend.

hugo