Interior-Point Methods
Interior-Point methods are a class of algorithms for solving continuous optimization problems, most notably Linear Programming (LP) and Convex Optimization problems. Unlike boundary-based methods such as the Simplex algorithm, Interior-Point methods approach optimal solutions from the interior of the feasible region. Since their development in the 1980s, they have become a standard component of modern optimization solvers.
Problem classes and applicability
Interior-Point methods are primarily used for Linear Programs (LP), Quadratic Programs (QP), and more general Nonlinear Programming (NLP) problems. They are also employed as subroutines within more complex frameworks, such as Mixed-Integer Linear Programming (MILP) and Mixed-Integer Nonlinear Programming (MINLP), where they are used to solve continuous relaxations.
When the feasible region and objective function are representable by cones (like LPs), these methods provide strong theoretical guarantees on convergence and efficiency.
Core Idea and algorithmic principle
Interior-Point methods solve optimization problems by maintaining iterates that remain strictly inside the feasible region defined by the constraints. Rather than moving along the boundary of the feasible set, the algorithm progresses through its interior toward an optimal solution.
The most common approach relies on barrier functions. Inequality constraints are incorporated into the objective by adding a penalty term that grows unbounded as a solution approaches the constraint boundary. For example, logarithmic barrier terms are often used to prevent constraint violation while preserving smoothness. The original constrained optimization problem is replaced by a sequence of barrier subproblems, each parameterized by a positive barrier parameter. As the barrier parameter is gradually reduced, the solutions of these subproblems converge toward the optimal solution of the original problem.
This iterative process follows a trajectory known as the central path, which connects the analytic centers of the feasible region to the optimal solution. Convergence is typically achieved in a relatively small number of iterations, although each iteration may involve solving large linear systems.
Comparison with other LP and NLP methods
Interior-Point methods differ fundamentally from the Simplex method, which moves between extreme points of the feasible region. While Simplex often performs well in practice and provides useful combinatorial information, Interior-Point methods exhibit polynomial-time complexity in the worst case and scale efficiently to large, dense problems.
In contrast to Augmented Lagrangian or active-set methods, Interior-Point approaches avoid frequent changes in active constraints and instead solve large linear systems at each iteration.
Role in modern solvers
Modern optimization solvers often include both Interior-Point and Simplex implementations. The choice between them depends on problem structure, size, and numerical properties. Interior-Point methods are particularly effective for large-scale continuous relaxations, making them essential components of MILP, MINLP, and decomposition-based algorithms.
Further reading and references
Karmarkar, N. (1984). A New Polynomial-Time Algorithm for Linear Programming. Combinatorica.
The paper that introduced modern interior-point methods and demonstrated polynomial-time solvability of linear programming.
Boyd, S., & Vandenberghe, L. (2004). Convex Optimization. Cambridge University Press.
A widely used textbook connecting interior-point methods to convex analysis and real-world applications.