5. The central path and path-following IPMs

• The outline above is specifically a path-following interior point method (39:40).
• Central path: the curve λ ↦ x_λ traced as λ runs over [0, ∞) (40:22, 40:52).
  • At λ = 0 the cost is irrelevant and the curve begins at the analytic center — as deep inside the region as possible (40:22, 40:52).
  • As λ → ∞ the curve ends at an optimal vertex (40:52).
• The algorithm starts at the beginning of the curve and follows it (41:18).
• Blackboard cartoon: a feasible polygon with the cost pointing "upward" (minimizing gravity), the optimal vertex at the bottom, the analytic center somewhere inside, and the central path joining them (41:40, 41:58).
• Crucial point: the iterates x̃_{λ0}, x̃_{λ1}, … are never exactly on the central path — they stay near it, refine toward it, then get reused as the warm start for the next λ (41:58, 42:34, 43:00).
• Not every interior point method is described in path-following terms; this course presents the path-following version (43:00, 43:35).

flowchart LR
    AC["Analytic center (lambda = 0)"] -->|central path| V["Optimal vertex (lambda -> infinity)"]
    X0["x_tilde(lambda_0)"] -.near.-> AC
    X0 --> X1["x_tilde(lambda_1)"]
    X1 --> X2["x_tilde(lambda_2)"]
    X2 -.near.-> V