How to define an objective function in the fmincon?

I would like to solve a large scale non-convex QCQP problem using fmincon, such that

min_x x.'H*x

s.t. x.'*R*x = 1

Ax<=b

where H and R are positive definite matrices. Because x has dimension of 5000, how can i define the objective function? I tried to pass

obj = @(x)x.'*H*x;

to the fmincon, it won't work.

many thanks

[Matt J]

There's nothing obviously incorrect in what you've described. You should attach a .mat file with your H,R,A,b matrices and demonstrate what you've done using the Run button

Answer 1

[Rik]

Your objective function should return a scalar. To borrow a term from machine learning: it is a cost function.

One of the often used functions to compute a single cost for a set of observations is the ordinary least squares:

OLS=@(x) sum((fun(x)-target).^2);

[Mingyang Sun]

thank you for your replying, if x is a 5000 x 1 vector, then x.'H*x is a scalar. Because of the dimension of the x, it is impossible to define the objective function using anonymous function such that

@(x) x(1) + x(2) + x(3).....

how can I define the objective function efficiently?

[Rik]

If it already returns a scalar, what is your question?

From the documentation:

fun — Function to minimize

Function to minimize, specified as a function handle or function name. fun is a function that accepts a vector or array x and returns a real scalar f, the objective function evaluated at x.

fmincon passes x to your objective function and any nonlinear constraint functions in the shape of the x0 argument. For example, if x0 is a 5-by-3 array, then fmincon passes x to fun as a 5-by-3 array. However, fmincon multiplies linear constraint matrices A or Aeq with x after converting x to the column vector x(:).