Supplied objective function must return a scalar value

When I analyze the stock's return style through share model，I want to determine the sensitive coefficients through the fmincon code of Matlab. However, I encounter some problem about "supplied objective function must return a scalar value", anyone who can help me? Thank you very much.

E=xlsread("data.xlsx")
F=E(:,1)                  % the target stock return time series (with 755*6 matrix)
G=E(:,2)                  % the large cap return time series    (with 755*1 matrix)
H=E(:,3)                  % the mid cap return time series      (with 755*1 matrix)
I=E(:,4)                  % the small cap return time series    (with 755*1 matrix)
J=E(:,5)                  % the aggregate bond return time series (with 755*1 matrix)
K=E(:,6)                  % the treasury time series              (with 755*1 matrix)
Benchmark=[G,H,I,J,K]     
fun=@(x)var(F-Benchmark.*x)  % establishing the objective function (with 1*1 value)
x0=[0,0,0,0,1]            % the initial point
A=[]
b=[]                      % no inequity constraints
Aeq=ones(1,5)
beq=1                       % equity constraints
lb=zeros(1,5)             % lower bound
ub=ones(1,5)              % upper bound
x=fmincon(fun,x0,A,b,Aeq,beq,lb,ub)

In principle, the output of x should be a 5*1 matrix. However, I always get the error that "supplied objective function must return a scalar value". In fact, fun can be handled with value. So I can not understand the "supplied objective function mus t trturn a scalar value". Anyone who can help me? Thank you very much. The data is included in the attachment.

Answer 1

[Matt J]

The objective function must return a scalar value, because fmincon is trying to minimize it. It needs to be able to make comparisons like fun(x1)<fun(x2) and that will only be possible if fun(x) returns a scalar. If you think fun(x) should already be returning a scalar value, you should check to see if that's true.

[惟慎 李]

Excuse me:

fun（x) can be seen as a scalar as the picture indicated. So I can not understand this error.

[Walter Roberson]

You build Benchmark out of a series of column vectors, so Benchmark is a 2D array with 5 columns. You do element-wise multiplication of the 2D vector with x, and x is going to be a row vector, so Benchmark.*x will be a 2D array with 5 columns. F is a column vector with the same number of rows as Benchmark has, so (starting from R2016b) it is valid to do the subtraction, and you would get out a 2D array with the same number of rows and columns as Benchmark has.

Now you take var() of that 2D array. When you apply var() to a 2D array, it operates along the first non-scalar dimension. Provided that the data file had more than one row of data, the first non-scalar dimension would be rows. So var() is going to take the variance of each row, which would give you are result that has one row and 5 columns.

A result that has one row and 5 columns is not a scalar value.

[Matt J]

You should always test your objective function before giving it to fmincon.

E=rand(10,6); %Fake data

F=E(:,1);                
G=E(:,2);              
H=E(:,3);                
I=E(:,4);                
J=E(:,5);            
K=E(:,6);                
Benchmark=[G,H,I,J,K];     
fun=@(x)var(F-Benchmark.*x); 
x0=[0,0,0,0,1];

A simple preliminary test shows that fun does not return a scalar,

fx0 = fun(x0) 

fx0 = 1×5

0.0658 0.0658 0.0658 0.0658 0.1442

[惟慎 李]

Thank you, Benchmark.*x is indeed a 755*5 matrix rather than a 1*1 scalar. Accordingly, I have changed the form of x dividing it into x(1),x(2),x(3),x(4) and x(5), and eventually solved the error.

Thank you very much.

[Walter Roberson]

Benchmark * x.'

might have been appropriate. That would be 755 x 5 * 5 x 1, which would give you a 755 x 1 result. Subtract that from a 755 x 1 result would give you a 755 x 1 result. var() of a 755 x 1 result would give you a scalar.