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).

Exploring the Barrier Function

Use the Log-Determinant Barrier module to observe how the barrier restricts variables.

1. 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.
2. Increase Correlation t step by step towards 1. The determinant drops towards 0, causing the barrier value to rapidly increase.
3. 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).
4. 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.