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

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:

  1. Maximize the minimum eigenvalue of $( X )$.
  2. 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:

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import cvxpy as cp
import numpy as np

# Step 1: Define the size of the matrix
n = 3  # Size of the matrix X (3x3 matrix in this example)

# Step 2: Define the decision variable (X is a symmetric matrix)
X = cp.Variable((n, n), symmetric=True)

# Step 3: Define the slack variable (t) representing the minimum eigenvalue
t = cp.Variable()

# Step 4: Define the objective function (maximize t)
# This corresponds to maximizing the smallest eigenvalue
objective = cp.Maximize(t)

# Step 5: Define the constraints
# The first constraint is X - tI >= 0 (positive semidefinite)
constraints = [X - t * np.eye(n) >> 0]

# Add some additional constraints to make the problem non-trivial
# For example, trace(X) = 10, and X[0, 1] = 1 (arbitrary example constraints)
constraints += [cp.trace(X) == 10]
constraints += [X[0, 1] == 1]

# Step 6: Formulate the problem
problem = cp.Problem(objective, constraints)

# Step 7: Solve the problem
problem.solve()

# Step 8: Output results
print("Status:", problem.status)
print("Optimal value of t (minimum eigenvalue):", t.value)
print("Optimal solution for X:\n", X.value)

Explanation:

Step-by-Step Breakdown:

  1. 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.

  2. 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 $(X)$.

  3. Objective Function:

    • The goal is to maximize $(t)$, i.e., maximize the minimum eigenvalue of $(X)$.

  4. 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.

  5. Problem Definition:

    • The problem is defined using cp.Problem(), with the objective being the maximization of$ (t)$ and the constraints defined above.

  6. Solve:

    • The solve() method is called to solve the semidefinite program.

  7. 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.

Output:

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Status: optimal
Optimal value of t (minimum eigenvalue): 2.6666580666942266
Optimal solution for X:
 [[3.666671   1.00000001 0.        ]
 [1.00000001 3.666671   0.        ]
 [0.         0.         2.666658  ]]
  • 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.