How to use the gradient with respect to a given vector?

I am trying to implement an algorithm that uses the matrices R - ( m x m matrix) and K - ( n x n ) that are the inputs of a function f(R,K). I am trying to use a starting point z = [r,n] where r is the vectorization of R and n is the vectorization of K. Starting values of R and K are given thus z is also known. I need to calculate r(z) = [gradient_r(f(R,K)) gradient_n(f(R,K))].

Once this is calculated I will do more calculations and use my next starting point in the algorithm, so z is trying to optimize to r(z) = 0 which is the Kurush-Khan-Tucker condition. Thus optimizing my matrices R and K in the process.

I do not know whether I should use numerical or symbolic gradient or even if what I am trying to do is possible. The paper I am referencing appears to use this method.

I have been trying something like this:

syms z [48 1]

r = z([1:16]);
n = z([17:48]);

R_hat = Dr*r;
R = reshape(R_hat, [4,4]);

K_hat = Dn*n;
K = reshape(K_hat, [8,8]) + eye(8);


g = solve(gradient(ft_RK));

Any guidance would be greatly appreciated.

Answer 1

[Shishir Reddy]

Hi Zac

To implement the algorithm you're describing, the gradient of the function f(R, K) should be computed with respect to the vectorized forms of R and K. The symbolic computation ismore accurate than numerical approximationsto find these gradients.

Kindly refer to the folloowing steps to understand the workflow of the solution -

1. Define the symbolic variables to represent the vectorized forms of ( R ) and ( K ).

syms z [48 1] real
r = z(1:16);
n = z(17:48); 

2. Convert the vectorized forms back into matrices ( R ) and ( K ).

Dr = eye(16); 
Dn = eye(32); 

R_hat = Dr * r;
R = reshape(R_hat, [4, 4]);

K_hat = Dn * n;
K = reshape(K_hat, [8, 8]) + eye(8); 

3. Symbolically define your function ( f(R, K) ).

f_RK = trace(R * K); % example function

4. Use symbolic differentiation to compute the gradient with respect to ( r ) and ( n ).

grad_r = gradient(f_RK, r);
grad_n = gradient(f_RK, n);

r_z = [grad_r; grad_n];

5. Use the gradient results to solve for the conditions

solutions = solve(r_z == 0, z);
disp(solutions);

I hope this helps.