Interior Point Methods 1

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Lecture claims simple Newton-style primal‑dual schemes solve LP by following the central path and driving the duality gap mu to zero, using only linear solves and diagonal scaling. It gives explicit Newton systems, shows step length selection by solving up to N quadratics, and proves long‑step path‑following yields O(n log(1/epsilon)) iterations while short‑step and predictor‑corrector schemes give O(√n log(1/epsilon)). Initialization in the feasible neighborhood is required; practical methods use adaptive sigma and Mehrotra‑style heuristics.

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" chapters (20)

1. 0:00Start
2. 5:49Reviews of Linear Programming
3. 8:07Lp Duality
4. 8:45Strong Duality
5. 10:03The Simplex Method
6. 10:27Duality and Complementarity
7. 10:59One-Line Proof
8. 13:38Maintaining Strict Feasibility
9. 18:09Newton's Method for Nonlinear Algebraic Equations
10. 18:47Taylor's Theorem Derivation of Newton's Method
11. 19:11First Order Taylor Series Approximation
12. 19:23Taylor's Theorem of Newton's Method
13. 22:02Self Concordance
14. 22:40Newton's Method for Non-Linear Equations
15. 22:51Newton's Method for Minimizing a Function
16. 27:08Ways To Define Neighborhoods
17. 27:25Ways To Measure Neighborhoods
18. 30:21Long Step Path Following
19. 30:33Strategy
20. 57:52Predictor Corrector