Optimization of the weighted average sum of matrix norms in the workspace using the Optimization Tool

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Andrew Sol 2022-7-26

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https://ww2.mathworks.cn/matlabcentral/answers/1768010-optimization-of-the-weighted-average-sum-of-matrix-norms-in-the-workspace-using-the-optimization-too

编辑： Matt J 2022-7-26

There are workspace from three 3x3 matrices with random elements:

A=rand(3,3);
B=rand(3,3);
C=rand(3,3);

There are 9 vectors of the following form:

u1=[cos(a);sin(a);1-cos(0)]
u2=[cos(a);sin(a);1-cos(2*pi/3)]
u3=[cos(a);sin(a);1-cos(-2*pi/3)]
v1=[sin(b)*cos(C);sin(a);-cos(0)-1]
v2=[sin(b)*cos(C);sin(a);-cos(2*pi/3)-1]
v3=[sin(b)*cos(C);sin(a);-cos(-2*pi/3)-1]
w1=[cos(A);sin(B);-sin(0)]
w2=[cos(A);sin(B);-sin(2*pi/3)]
w3=[cos(A);sin(B);-sin(-2*pi/3)]

here

- variables;

The matrix of the following form is formed from the vectors:

J=[cross(v1,w1)/dot(cross(u1,v1),w1) cross(v2,w2)/dot(cross(u2,v2),w2) cross(v3,w3)/dot(cross(u3,v3),w3)];

Problem: We need to find parameters

that minimize the weighted average sum of matrix J norms at each of the workspace points, i.e.:

How to solve this problem using Optimization Tool ?

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Matt J 2022-7-26

编辑：Matt J 2022-7-26

在 MATLAB Online 中打开

There are 9 vectors of the following form:

If A,B,C are 3x3 matrices and a,b are scalars, then v and w cannot be vectors. Nor can you run the code you've posted without concatenation error messages.

A=rand(3,3);
B=rand(3,3);
C=rand(3,3);
a=pi; b=pi;
u1=[cos(a);sin(a);1-cos(0)]
u1 = 3×1
   -1.0000
    0.0000
         0
u2=[cos(a);sin(a);1-cos(2*pi/3)]
u2 = 3×1
   -1.0000
    0.0000
    1.5000
u3=[cos(a);sin(a);1-cos(-2*pi/3)]
u3 = 3×1
   -1.0000
    0.0000
    1.5000
v1=[sin(b)*cos(C);sin(a);-cos(0)-1]
Error using vertcat
Dimensions of arrays being concatenated are not consistent.
v2=[sin(b)*cos(C);sin(a);-cos(2*pi/3)-1]
v3=[sin(b)*cos(C);sin(a);-cos(-2*pi/3)-1]
w1=[cos(A);sin(B);-sin(0)]
w2=[cos(A);sin(B);-sin(2*pi/3)]
w3=[cos(A);sin(B);-sin(-2*pi/3)]

Andrew Sol 2022-7-26

No. Please listen carefully. I will explain. Matrices A, B, C - the same dimension 3x3. That is, the workspace is divided into 9 points.

1. We take from each matrix

one parameter at each point of the workspace and substitute it into the vectors.

2. It is necessary to determine the norm of the matrix J at each of the 9 points.

3. Sum up all 9 norms of J.

4. Divide by the number of points in the workspace, i.e. calculate the average sum of norms.

5. It is necessary to choose the optimal values of the parameters

that minimize the average sum of norms.

I do not know how to properly formalize this task, so the incorrect notation caused a misunderstanding.

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Optimization of the weighted average sum of matrix norms in the workspace using the Optimization Tool

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