MultiStart optimization problem structure

I currently have a 9x9 double matrix called coeff and another 1x9 double matrix called new_data. I am trying to use MultiStart and GlobalSearch in the Global Optimization Toolbox to find the minima of the following optimization problem:

min (new_data-(coeff*test_weights))^2.

In words, I am trying to minimize the squared distance between the new_data and coeff multiplied by test_weights where test_weights is what I am changing. test_weights should be a 9x1 matrix when finished. I would appreciate if anybody could help me come up with the createOptimProblem statement in MATLAB.

[Matt J]

and another 1x9 double matrix called new_data.

Presumably, you really mean new_data is 9x1 like test_weights, and not 1x9. Otherwise, the subtraction

new_data-(coeff*test_weights)

is not defined.

Answer 1

[Matt J]

It seems strange to be using the Global Optimization Toolbox for this when there is a one-line analytical solution

test_weights=coeff\new_data

Furthermore, unless you have some nonlinear constraints that you haven't mentioned, the problem is convex. There can be no non-global minima.

[Aleks]

I'm not sure if I explained it well up above. Below is exactly what I need to do

1) Do PCA for market moves from T-n to T 2) Eliminate factors which explain less than 5% of variance 3) gradient descent to get sets of factor weights most accurately describing the market move from T to T+1 4) Record top 10-20 sets of factor weights and percentage of variance explained by each set of weights.

I already completed steps 1 and 2 and "coeff" is my matrix of principal components while new_data is the return data for day T+1.

How would you go about doing Steps 3 and 4?

[Matt J]

I understood your first explanation better...

If the thing you are trying to minimize as a function of test_weights is

 ||new_data-(coeff*test_weights)||^2

and you have no constraints on test_weights, then the most efficient solution is,

 test_weights=coeff\new_data

[Aleks]

That's not really what I am looking for but thank you anyways. Keep in mind that coeff, new_data and test_weights are all matrices.

[Matt J]

No, you said that test_weights and new_data are 9x1 vectors, not matrices. It doesn't matter, though. It would work for matrices, too, as long as they are of compatible dimensions so that the expression

 new_data-(coeff*test_weights)

can be evaluated.