Interior Point Methods

Optimization · Axiom Academy

Modern polynomial-time algorithms for constrained optimization using barrier functions and the central path 1. Barrier Method Fundamentals The barrier method transforms inequality-constrained problems into a sequence of unconstrained problems by adding a barrier function that prevents solutions from leaving the feasible region. We replace this with a sequence of barrier problems parameterized by μ > 0: As μ → 0, the barrier solutions x*(μ) converge to the constrained optimum. The logarithmic barrier grows to infinity as we approach the constraint boundary, keeping iterates strictly feasible. 2. The Central Path Trajectory The central path is the curve traced by optimal solutions x*(μ) as the barrier parameter μ varies from large to small values. This smooth trajectory starts deep in the interior and converges to the optimal solution. Each point on the central path satisfies the KKT conditions with complementarity slackness parameter μ. The path is characterized by the equation ∇f(x) + Σλ_i∇g_i(x) = 0 with λ_i·g_i(x) = -μ. 3. Primal-Dual Interior Point Methods Modern interior point methods simultaneously update both primal variables (x) and dual variables (λ) using Newton steps on the perturbed KKT conditions. This is the key to achieving polynomial-time complexity. At each iteration, we solve a linear system to compute Newton directions for both primal and dual variables, then take steps that maintain strict feasibility while reducing the duality gap.

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