A family of optimization algorithms that traverse the interior of the feasible region, using barrier functions, primal-dual systems, or path-following strategies to approach optimal solutions without explicitly visiting vertices.

Principle

Replace hard constraints by smooth barrier terms or solve coupled primal-dual KKT systems so iterates remain strictly feasible and follow a central path toward optimality with controlled step updates.

Demonstration

Solve a quadratic program by adding logarithmic barrier terms to inequality constraints, compute Newton steps for the barrier-augmented system, reduce the barrier parameter, and repeat until optimality conditions are met.

Misapplication

Using interior-point steps without maintaining numerical conditioning (poor linear solves or insufficient barrier reduction control) can produce inaccurate iterates or stall far from an optimal solution.

Consequence

Produces polynomial-time performance guarantees for many convex problems, yields high-quality primal and dual approximations, and scales well for large sparse instances when implemented with robust linear algebra.

Reversal

Contrasts with boundary-following combinatorial methods (e.g., simplex): interior-point methods avoid explicit basis changes and instead trade combinatorial moves for continuous nonlinear solves.

Boundary

Primarily applicable to convex programs (linear, quadratic, conic); nonconvex problems may only receive local solutions and require globalization strategies or heuristics.

Semantic Tension

Tension with the simplex view arises over sparsity and warm-starting: simplex offers warm starts and extreme-point sparsity while interior methods emphasize smooth progress and often denser iterates.

Synthesis

Interior-point methods reframe convex optimization as continuous path following inside the feasible set: maintain strict feasibility via barrier or primal-dual conditions and advance with Newton-like updates toward optimality.