numerical problem of quadprog

Hi, I am trying to solve a badly scaled quadprog problem

H = [5e15, -1.66e15, 1.02e9, -1.23e10;

-1.66e15, 1.11e15, 1.42e6, 8.3e9;

1.02e9, 1.42e6, 2.5e6, 93;

-1.23e10, 8.32e9 , 93, 5e5 ]

f = [-4.76e7, 3.49e7, -2.19, 2.31e3]'

Aieq = [1/2, -1/3, 0, 0]

bieq = 0

when I try to use quadprog(H,-f,Aieq,bieq), MATLAB indicates problem successfully solved, but if you check inequality constraint manually, Aieq * x = a very very small positive number.

What is the best way to solve this numerical problem?

1. set 'ConstraintTolerance' to a small number?
2. set bieq = a very small negative number?
3. rescale matrix H and f?

[Bruno Luong]

I think this must be the priority:

• rescale matrix H and f?

[Mingyang Sun]

thank you for your replying. Is there a general formula to re-scale my problem? How to determine the rescale factor? Because I need to run thousands of similar problems, all of them can give "optimal solution found" message, but some of them violate the inequality constraint, some of them don't.

[Bruno Luong]

There is MATLAB function normalize to help you to do such thing.

It is on recent version only though.

Roughly the "ideal" method is to change the decision variable to y, where

sqrtm(H)*y = x

so that the Hessian becomes identity with respect to y.

A quick and dirty rescaling is then

sqrt(diag(H)).*y = x

[Mingyang Sun]

Thanks a lot for your replying, can I let

scal = sqrt(vecnorm(H))

and do

H_adj = H./(scal.'*scal); f_adj = f./scal.'

and

quadprog(H_adj,-f_adj,Aieq,bieq)?

[Bruno Luong]

You have also to modify Aieq.