interiorpoint

Interior-point methods are a family of algorithms for constrained optimization that solve problems by moving through the interior of the feasible region rather than along its boundary. They typically augment the objective with a barrier term that becomes large near the boundary, discouraging iterates from approaching infeasible or boundary points.

The modern interior-point paradigm emerged with Karmarkar’s 1984 linear programming algorithm, which spurred widespread interest in barrier and path-following approaches. Since then, primal-dual interior-point methods have become dominant for large-scale linear programs and convex optimization problems. Core ideas include log-barrier functions and following a central path toward optimality, usually via Newton or quasi-Newton steps.

How the method works: A barrier-augmented problem is solved by iterating along a central path corresponding to a decreasing barrier parameter. Each iteration solves a Newton system to obtain a search direction and then applies a step (often with a line search) to maintain feasibility. The overall process reduces the original constrained problem to a sequence of easier subproblems, typically linear or convex in nature.

Applications and variants: Interior-point methods are widely used for linear programming, convex quadratic programming, semidefinite programming, and other large-scale convex problems. Variants include primal, dual, and primal-dual implementations, as well as affine-scaling and barrier-method approaches. They are implemented in many commercial and open-source solvers and are favored for large, sparse instances.

Advantages and limitations: Interior-point methods offer robust, polynomial-time performance and good scaling for large problems. They require efficient linear-system solvers and can be sensitive to conditioning or degeneracy. For small or highly structured problems, specialized methods like the simplex method may be competitive or preferable.