## Maximizing the Minimum Eigenvalue of a Matrix Using Semidefinite Programming in CVXPY

### Semidefinite Programming (SDP) in CVXPY

$Semidefinite programming (SDP)$ is an optimization problem where the decision variable is a matrix, and the constraints include that this matrix must be positive semidefinite.

$SDP$ problems arise in various fields, such as control theory, signal processing, quantum mechanics, and machine learning.

#### Problem Setup:

The general form of a semidefinite program is:

$$

\text{minimize } \quad \text{tr}(CX)

$$

$$

\text{subject to } \quad \text{tr}(A_iX) = b_i \quad \text{for each } i

$$

$$

X \succeq 0

$$

Where:

- $(X)$ is the decision variable (a symmetric matrix).
- $(\text{tr}(C X))$ is the objective function (trace of matrix multiplication).
- The matrix $(X \succeq 0)$ indicates that $(X)$ is positive semidefinite, meaning all its eigenvalues are non-negative.

### Example: Maximizing the Minimum Eigenvalue of a Matrix

Let’s solve an $SDP$ example where we want to maximize the minimum eigenvalue of a symmetric matrix $( X )$, subject to some constraints.

#### Problem:

Given a symmetric matrix $( X )$, we want to:

- Maximize the minimum eigenvalue of $( X )$.
- Ensure that $( X )$ satisfies some linear constraints.

#### Formulation:

We want to maximize $( \lambda_{\text{min}}(X) )$, the smallest eigenvalue of $( X )$, which can be written as an $SDP$ using a slack variable $( t )$ and ensuring $( X - tI \succeq 0 )$ (i.e., the matrix $( X - tI )$ is positive semidefinite).

### Step-by-Step Solution using CVXPY:

1 | import cvxpy as cp |

### Explanation:

#### Step-by-Step Breakdown:

**Decision Variable**:- We define
`X`

as a symmetric matrix variable of size $(n \times n)$.

In our case, we chose $(n = 3)$, so $(X)$ is a $(3 \times 3)$ symmetric matrix.

- We define
**Slack Variable $(t)$**:- The variable
`t`

represents the smallest eigenvalue of the matrix $(X)$.

The objective is to**maximize $(t)$**, which corresponds to maximizing the minimum eigenvalue of $(X)$.

- The variable
**Objective Function**:- The goal is to maximize $(t)$, i.e., maximize the minimum eigenvalue of $(X)$.

**Constraints**:**Semidefinite Constraint**: $(X - tI \succeq 0)$ (i.e., the matrix $(X - tI)$ must be positive semidefinite). This ensures that $(t)$ is a lower bound on the eigenvalues of $(X)$.

All eigenvalues of $(X)$ must be greater than or equal to $(t)$.**Additional Constraints**:- We impose $( \text{tr}(X) = 10 )$ (i.e., the sum of the diagonal elements of $(X) equals 10)$.

This is an arbitrary constraint to ensure the matrix is not trivial. - We also constrain $( X[0, 1] = 1 )$, just to add more structure to the problem.

- We impose $( \text{tr}(X) = 10 )$ (i.e., the sum of the diagonal elements of $(X) equals 10)$.

**Problem Definition**:- The problem is defined using
`cp.Problem()`

, with the objective being the maximization of$ (t)$ and the constraints defined above.

- The problem is defined using
**Solve**:- The
`solve()`

method is called to solve the semidefinite program.

- The
**Results**:- The status of the solution is printed (
`Optimal`

,`Infeasible`

, or`Unbounded`

). - The optimal value of $(t)$, which corresponds to the maximum minimum eigenvalue of (X), is printed.
- The optimal solution matrix $(X)$ is also displayed.

- The status of the solution is printed (

### Output:

1 | Status: optimal |

**Optimal value of $(t)$**: The minimum eigenvalue of the matrix $(X)$ is approximately $2.67$.**Optimal solution for $(X)$**: The matrix $(X)$ that satisfies the given constraints and maximizes the minimum eigenvalue is shown.

### Explanation of Results:

**Semidefinite Constraint**: The constraint $$( X - tI \succeq 0 )$$ ensures that all eigenvalues of $(X)$ are greater than or equal to $(t)$.

Thus, maximizing $(t)$ corresponds to maximizing the smallest eigenvalue of $(X)$.**Trace Constraint**: The sum of the diagonal elements of $(X)$ is constrained to be 10.**Off-Diagonal Constraint**: The constraint $( X[0, 1] = 1 )$ ensures that there’s a relationship between the elements of $(X)$, adding complexity to the optimization.

### Use Cases of Semidefinite Programming:

**Control Systems**: $SDP$ can be used in control theory to design controllers that guarantee system stability by finding matrices that satisfy Lyapunov stability criteria.**Structural Design**: Optimizing stiffness matrices for structures to maximize strength while minimizing material usage.**Quantum Information**: $SDP$ is used to analyze quantum states, where matrices representing states must be positive semidefinite.**Finance**: Portfolio optimization and risk management, where covariance matrices must satisfy certain positive semidefinite constraints.

### Conclusion:

This example demonstrates how $CVXPY$ can be used to solve semidefinite programming problems by formulating the problem with matrix variables, semidefinite constraints, and other linear constraints.

The power of $SDP$ lies in its ability to optimize over matrices while ensuring positive semidefiniteness, making it a crucial tool in many advanced applications across different fields.