Optimization: Convex Programming & Interior-Point Methods
Optimization: Convex Programming & Interior-Point Methods
💡 Key Takeaways
- Understand the basic ideas of convex optimization: convex sets, convex functions, and global optimality.
- Learn how interior-point methods solve convex problems using barrier functions, the central path, and Newton steps.
- Explore primal–dual formulations and the KKT conditions that certify optimality in convex problems.
- Recognize common problem classes (LPs, QPs) and practical aspects like convergence guarantees and numerical stability.
❓ Frequently Asked Questions
What is convex optimization?
Minimizing a convex objective over a convex feasible set. Any local minimum is global; common forms include f0(x) subject to fi(x) ≤ 0 and hi(x) = 0 with convex/affine functions.
What is an interior-point method?
An algorithm that stays inside the feasible region by adding a barrier term (often a log barrier) to the objective and solving a barrier-augmented problem with Newton steps, gradually removing the barrier.
What is the central path / barrier method idea?
Introduce a barrier parameter and minimize a barrier-augmented objective. The minimizers form the central path; as the barrier is reduced, the solutions converge to the primal optimum of the original problem.
What are the KKT conditions and Slater's condition?
For convex problems with a strictly feasible point (Slater's condition), optimality is characterized by primal feasibility, dual feasibility, and complementary slackness via the Lagrangian; these conditions are necessary (and under convexity, sufficient) for optimality.