Least Squares problem?

Question

Manifold $W$ is divided into 9 points. The value of the parameter $P$ at each point in manifold is represented by a certain number, and the partition itself is represented as a matrix:

P = {{1, 2, 3}, {-1, 0, 1}, {-3, -2, -1}}

There is also a matrix $J$ that depends on the value from the matrix $P$ at the selected point and on the variables $x,y$

J = {{x^2, 1, P[[i, j]]}, {0, x, -1}, {1, 2, y^4}}

I need help solving an optimization problem:

1 Take each point from matrix $P$.

2 Substitute it into the matrix $J$ and use the matrix norm $||J||$ formula at the given point.

3 After that, I need to sum the squares of all the norms obtained at each point and divide by the number of partitions, i.e. use the formula:

$G=\frac{1}{n} q^Tq $,

where $q=\begin{bmatrix} ||J(x,y,P(1,1))|| & ||J(x,y,P(1,2))|| & ||J(x,y,P(1,3))|| & \cdots & ||J(x,y,P(3,3))|| \end{bmatrix}$

4 Find a combination of parameters $x,y$ that minimizes $G$, i.e.:

$\min_{x,y}G$

On the one hand, here is a ready-made algorithm. On the other hand, I have heard a lot about the possibilities of Mathematica in the field of optimization (and solving least-squares problems). Teach me how to use these possibilities.

I will be happy and grateful!

Answer 1

If I understand you description correctly, then there is no need to consider P as a matrix, a vector will do. (note I am using lower case names):

P = {1, 2, 3, -1, 0, 1, -3, -2, -1};
J = {{x^2, 1, #}, {0, x, -1}, {1, 2, y^4}} & /@ P

Unfortunately you did not specify which matrix norm you use. I will use the Frobenius norm,: sqrt of the sum of the squares of the matrix elements. For this we need the squares of the elements:

j^2

Our target function we want to minimize is then:

target = Total[Total[Flatten[#]^2] & /@ J]
(* 93 + 9 x^2 + 9 x^4 + 9 y^8 *)

It is now obvious, as only even powers of x and y appear, that the minimum is achieved for:

{x->0,y->0}