Maximize The Euclidean Norm Of A Matrix Times A Vector On Unit Sub-spheres

Introduction

In this article, we will explore the problem of maximizing the Euclidean norm of a matrix times a vector on unit sub-spheres. This problem is a classic example of a non-convex optimization problem, which is a type of problem that is difficult to solve using traditional convex optimization techniques. We will discuss the problem formulation, the constraints, and the objective function, and then provide a solution using a combination of mathematical techniques and numerical methods.

Problem Formulation

Given a matrix $A \in \mathbb{R}^{r \times Nm}$, a vector $X = (x_1^T,\ldots,x_N^T)^T \in \mathbb{R}^{Nm \times 1}$, where $x_i \in \mathbb{R}^{m \times 1}$ for $i \in \{1,\ldots,N\}$, and a scalar $r$, we want to maximize the Euclidean norm of the product $AX$ subject to the constraint that each $x_i$ lies on a unit sub-sphere.

Mathematically, this problem can be formulated as:

$$\max_{X} \quad \|AX\|_2^2$$

subject to:

$$\|x_i\|_2 = 1, \quad i \in \{1,\ldots,N\}$$

where $\| \cdot \|_2$ denotes the Euclidean norm.

Constraints

The constraints in this problem are the unit sub-sphere constraints, which require each $x_i$ to lie on a unit sub-sphere. This means that each $x_i$ must satisfy the equation:

$$\|x_i\|_2 = 1$$

This constraint can be rewritten as:

$$x_i^T x_i = 1, \quad i \in \{1,\ldots,N\}$$

where $x_i^T$ denotes the transpose of $x_i$.

Objective Function

The objective function in this problem is the Euclidean norm of the product $AX$, which is given by:

$$\|AX\|_2^2 = (AX)^T (AX)$$

This can be rewritten as:

$$\|AX\|_2^2 = X^T A^T A X$$

where $A^T$ denotes the transpose of $A$.

Solution

To solve this problem, we can use a combination of mathematical techniques and numerical methods. One approach is to use the method of Lagrange multipliers to enforce the unit sub-sphere constraints.

Let $\lambda_i$ be the Lagrange multiplier associated with the constraint $x_i^T x_i = 1$. Then, the Lagrangian function can be written as:

$$L(X, \lambda) = X^T A^T A X - \sum_{i=1}^N \lambda_i (x_i^T x_i - 1)$$

To find the maximum of the Lagrangian function, we can take the partial derivatives of $L$ with respect to $X$ and $\lambda_i$ and set them equal to zero.

Taking the partial derivative of $L$ with respect to $X$, we get:

$$\frac{\partial L}{\partial X} = 2 A^T A X - 2 \sum_{i=1}^N \lambda_i x_i = 0$$

Taking the partial derivative of $L$ with respect to $\lambda_i$, we get:

$$\frac{\partial L}{\partial \lambda_i} = -x_i^T x_i + 1 = 0$$

Solving these equations, we get:

$$X = (A^T A)^{-1} \sum_{i=1}^N \lambda_i x_i$$

$$x_i^T x_i = 1, \quad i \in \{1,\ldots,N\}$$

Substituting the expression for $X$ into the objective function, we get:

$$\|AX\|_2^2 = X^T A^T A X = \sum_{i=1}^N \lambda_i x_i^T A^T A x_i$$

To find the maximum of this expression, we can use the method of Lagrange multipliers again.

Let $\mu_i$ be the Lagrange multiplier associated with the constraint $x_i^T x_i = 1$. Then, the Lagrangian function can be written as:

$$L(\lambda, \mu) = \sum_{i=1}^N \lambda_i x_i^T A^T A x_i - \sum_{i=1}^N \mu_i (x_i^T x_i - 1)$$

To find the maximum of the Lagrangian function, we can take the partial derivatives of $L$ with respect to $\lambda_i$ and $\mu_i$ and set them equal to zero.

Taking the partial derivative of $L$ with respect to $\lambda_i$, we get:

$$\frac{\partial L}{\partial \lambda_i} = x_i^T A^T A x_i = 0$$

Taking the partial derivative of $L$ with respect to $\mu_i$, we get:

$$\frac{\partial L}{\partial \mu_i} = -x_i^T x_i + 1 = 0$$

Solving these equations, we get:

$$x_i^T A^T A x_i = 0, \quad i \in \{1,\ldots,N\}$$

$$x_i^T x_i = 1, \quad i \in \{1,\ldots,N\}$$

Substituting the expression for $x_i$ into the objective function, we get:

$$\|AX\|_2^2 = \sum_{i=1}^N \lambda_i x_i^T A^T A x_i = 0$$

This means that the maximum value of the objective function is zero.

Conclusion

In this article, we have discussed the problem of maximizing the Euclidean norm of a matrix times a vector on unit sub-spheres. We have formulated the problem, discussed the constraints and the objective function, and provided a solution using a combination of mathematical techniques and numerical methods. The solution involves using the method of Lagrange multipliers to enforce the unit sub-sphere constraints and to find the maximum of the objective function.

References

• [1] Boyd, S., & Vandenberghe, L. (2004). Convex optimization. Cambridge University Press.
• [2] Bertsekas, D. P. (1999). Nonlinear programming. Athena Scientific.
• [3] Nocedal, J., & Wright, S. J. (2006). Numerical optimization. Springer.

Code

The code for this problem can be written in MATLAB as follows:

function [X, fval] = maximize_norm(A, r)
    % Define the number of variables
    N = size(A, 2);
    m = size(A, 1);
% Define the number of constraints
n = N * m;

% Define the Lagrange multipliers
lambda = zeros(n, 1);

% Define the constraints
constraints = cell(n, 1);
for i = 1:n
    constraints{i} = @(x) x(i)^2 - 1;
end

% Define the objective function
f = @(x) sum(x.^2);

% Define the bounds for the variables
bounds = cell(n, 1);
for i = 1:n
    bounds{i} = [-1, 1];
end

% Define the options for the solver
options = optimoptions(@fmincon, 'Display', 'iter', 'TolX', 1e-6);

% Solve the problem
[x, fval] = fmincon(f, zeros(n, 1), [], [], [], [], [], [], constraints, bounds, options);

% Compute the solution
X = reshape(x, m, N);

end

Q: What is the problem of maximizing the Euclidean norm of a matrix times a vector on unit sub-spheres?

A: The problem of maximizing the Euclidean norm of a matrix times a vector on unit sub-spheres is a non-convex optimization problem that involves maximizing the Euclidean norm of the product of a matrix and a vector, subject to the constraint that each component of the vector lies on a unit sub-sphere.

Q: What is the mathematical formulation of the problem?

A: The mathematical formulation of the problem is:

$$\max_{X} \quad \|AX\|_2^2$$

subject to:

$$\|x_i\|_2 = 1, \quad i \in \{1,\ldots,N\}$$

where $\| \cdot \|_2$ denotes the Euclidean norm.

Q: What are the constraints in the problem?

A: The constraints in the problem are the unit sub-sphere constraints, which require each component of the vector $X$ to lie on a unit sub-sphere. This means that each component of $X$ must satisfy the equation:

$$x_i^T x_i = 1, \quad i \in \{1,\ldots,N\}$$

Q: What is the objective function in the problem?

A: The objective function in the problem is the Euclidean norm of the product $AX$, which is given by:

$$\|AX\|_2^2 = (AX)^T (AX)$$

This can be rewritten as:

$$\|AX\|_2^2 = X^T A^T A X$$

Q: How can the problem be solved?

A: The problem can be solved using a combination of mathematical techniques and numerical methods. One approach is to use the method of Lagrange multipliers to enforce the unit sub-sphere constraints.

Q: What is the solution to the problem?

A: The solution to the problem is:

$$X = (A^T A)^{-1} \sum_{i=1}^N \lambda_i x_i$$

$$x_i^T x_i = 1, \quad i \in \{1,\ldots,N\}$$

Substituting the expression for $X$ into the objective function, we get:

$$\|AX\|_2^2 = X^T A^T A X = \sum_{i=1}^N \lambda_i x_i^T A^T A x_i$$

Q: How can the solution be computed?

A: The solution can be computed using the following steps:

1. Define the number of variables $N$ and the number of constraints $n$.
2. Define the Lagrange multipliers $\lambda$.
3. Define the constraints $x_i^T x_i = 1$ for each $i \in \{1,\ldots,N\}$.
4. Define the objective function $\|AX\|_2^2 = X^T A^T A X$.
5. Use the method of Lagrange multipliers to enforce the constraints and find the maximum of the objective function.

Q: What is the code for the solution?

A: The code for the solution is:

function [X, fval] = maximize_norm(A, r)
    % Define the number of variables
    N = size(A, 2);
    m = size(A, 1);
% Define the number of constraints
n = N * m;

% Define the Lagrange multipliers
lambda = zeros(n, 1);

% Define the constraints
constraints = cell(n, 1);
for i = 1:n
    constraints{i} = @(x) x(i)^2 - 1;
end

% Define the objective function
f = @(x) sum(x.^2);

% Define the bounds for the variables
bounds = cell(n, 1);
for i = 1:n
    bounds{i} = [-1, 1];
end

% Define the options for the solver
options = optimoptions(@fmincon, 'Display', 'iter', 'TolX', 1e-6);

% Solve the problem
[x, fval] = fmincon(f, zeros(n, 1), [], [], [], [], [], [], constraints, bounds, options);

% Compute the solution
X = reshape(x, m, N);

end

This code defines a function maximize_norm that takes as input the matrix A and the scalar r, and returns the solution X and the value of the objective function fval. The function uses the fmincon solver to find the maximum of the objective function subject to the constraints.