Log-Determinant Barrier
Approach a positive-definite boundary through a convex matrix barrier.
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Log-Determinant Barrier
In convex optimization, particularly interior-point methods, a barrier function prevents iterates from leaving a feasible region by approaching positive infinity near the boundary. The log-determinant barrier, f(A) = -log det A, is defined strictly on the interior of the positive semidefinite cone (the set of positive definite matrices).
Example: Evaluating the Barrier
Consider a symmetric matrix:
A = [1 0.5] [0.5 1]
- Calculate the determinant: det A = 1 · 1 - 0.5 · 0.5 = 1 - 0.25 = 0.75.
- Evaluate the barrier function: f(A) = -log(0.75).
- Numerically, -log(0.75) ≈ 0.287682.
- Since det A > 0 and the diagonal entries are positive, the matrix is positive definite, so it lies strictly within the domain.
Exploring the Barrier Function
Use the Log-Determinant Barrier module to observe how the barrier restricts variables.
- Observe the starting matrix with Diagonal scale set to 1 and Correlation t set to 0.5. Notice the finite barrier value and its derivatives.
- Increase Correlation t step by step towards 1. The determinant drops towards 0, causing the barrier value to rapidly increase.
- Select Correlation t exactly at 1 or -1. The matrix becomes singular, placing it exactly on the boundary of the positive semidefinite cone where the barrier is undefined (positive infinity).
- Change the Diagonal scale to 2 and set Correlation t to 0. Notice that the barrier function becomes negative. The log-determinant is not a distance or probability, and negative values simply indicate that det A > 1.
Domain and Scaling
The function f(A) = -log det A requires the matrix A to be strictly positive definite. If A has any zero or negative eigenvalues, it falls on or outside the boundary of the positive semidefinite cone, meaning the barrier function and its derivatives are undefined. It is important to note that the barrier function can be negative when the determinant is greater than 1; it is not restricted to non-negative values like a distance metric.
Further Reading
This reference contains the formal definition and properties of the logarithmic barrier function for the semidefinite cone.
- Convex Optimization by Stephen Boyd and Lieven Vandenberghe. See Chapter 11 on Interior-point Methods, particularly the logarithmic barrier function for the semidefinite cone.
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Log-Determinant Barrier module.
Try Log-Determinant Barrier on Riano →