How can I model my objective function for optimization?

I have an objective function in microsoft excel, which is expressed as: SQRT(MMULT(MMULT(TRANSPOSE((AI25:AI54)*E5:E34),G5:AJ34),(AI25:AI54)*E5:E34)),

• AI25:AI54 - is a 29x1 matrix of unknowns
• E5:E34 - is a 29x1 matrix of constants
• G5:AJ34 - is 29x29 covariance matrix of constants

I am not sure how to write this function in MATLAB terminology, but I can explain mathematically of what is happening:

• Step 1: ((AI25:AI54)*E5:E34) , here suppose the first matrix is [1,2,3] and second matrix is [3,2,1], then the resultant matrix will be [1x3,2x2,3x1], which is [3,4,3] . I think * operation is similar to .* operation of MATLAB with matrices. Resultant is a 29x1 matrix.
• Step 2: A transpose of resultant of step 1 is taken, which results in 1x29 matrix.
• Step 3: Resultant of step 2 is matrix multiplied(normal matrix multiplication) with G5:AJ34, which results into a 1x29 matrix. You can observe all the elements are linear.
• Step 4: Resultant of step 3 is matrix multiplied with the resultant of step 1, which results into a 1x1 matrix. You can observe the element in the matrix is in quadratic form
• Step 5: Square root of the resultant of step 4, which results into my objective function

I want to model the resultant of step 5 in MATLAB with something like quadprog as my objective function, which is to be minimized. I choose quadprog, as my objective function is quadratic and all my constraints are linear in nature.

Can someone please help me with putting this objective function in the form of quadprog objective function?

Answer 1

[Matt J]

Let

 x=AI25:AI54
 e=E5:E34
 G=G5:AJ34

Then you would call QUADPROG as

 quadprog(diag(e)*G*diag(e), zeros(29,1) , ... constraints ...)

[Siddharth]

Thanks a lot for your response, it really did help, as I had no idea how to go around it!

Regarding quadprog(diag(e)*G*diag(e), zeros(29,1) , ... constraints ..., I have few questions:

• Do you think the first parameter of quadprog should also be multiplied by 2? I have a very limited knowledge about MATLAB, but after going through the documentation of quadprog and few forum questions, I just tried multiplying expression passed as first parameter by 2. After putting the values of unknowns received from Initial optimization(without multiplying the first parameter by 2) in the excel formula without square root(i.e. (MMULT(MMULT(TRANSPOSE((AI25:AI54)*E5:E34),G5:AJ34),(AI25:AI54)*E5:E34))) the resultants of function from matlab and excel didn't match. But after multiplying the first parameter by 2 the value of excel formula without square root started to match with matlab's.
• Do you think the step 5 of the objective function, i.e. is the square root of the whole objective function is not included in the model you have suggested? If so, can you please suggest a way we can include it in there? or do we have to you use some other optimization function for that form?

[Matt J]

Do you think the first parameter of quadprog should also be multiplied by 2?

When QUADPROG is called as I proposed, it is the function

   f1(x) = x.'*diag(e)*G*diag(e)*x/2

that is being minimized (see also here). So, if you instead want to receive as output the minimum of

   f2(x) = x.'*diag(e)*G*diag(e)*x

then multiplying the input by 2 will give that to you. You could also wait until the optimization of f1(x) is finished and multiply that final output by 2, which involves less computation.

However, both f1(x) and f2(x) will be minimized by the same x, which is often what people are more interested in, so it's questionable if the factor of 2 really matters.

Do you think the step 5 of the objective function, i.e. is the square root of the whole objective function is not included in the model you have suggested

Again, f1(x) and sqrt(f1(x)) are monotonically related and are minimized at the same x. You can take the sqrt of the final value for f1(x) given to you by QUADPROG if it's important to you for some reason that the objective is measured that way.

[Siddharth]

Thank you for your invaluable inputs, I was able to model the desired optimization model.