interieurpunt

Interieurpunt, in Dutch often rendered as interior point or interior-point, can refer to two related ideas in mathematics and optimization. As a mathematical concept, an interior point of a set S in Euclidean space is a point for which there exists a small ball around it that is entirely contained in S. For a feasible region defined by linear inequalities Ax ≤ b, a point x with Ax < b is interior to the region, while points with Ax = b lie on the boundary.

In optimization, interior-point methods are a class of algorithms that solve problems by traversing the interior of the feasible region rather than moving along its boundary. These methods use barrier functions to prevent iterates from reaching the boundary, typically adding a penalty that becomes infinite as feasibility is violated. The algorithms follow a central path toward optimality, using Newton-type or related updates to steer progress.

Historically, interior-point ideas gained prominence in the 1980s. Karmarkar’s 1984 algorithm popularized the interior-point approach for linear programming, and subsequent developments by Nesterov, Nemirovski, Khachiyan, and Todd led to practical primal-dual barrier and path-following methods that address large-scale convex problems efficiently.

Applications of interior-point methods span linear programming, convex optimization, and semidefinite programming, where they are valued for handling large problem sizes and strong numerical properties. In these settings, the solution process maintains strict feasibility (interior points) until convergence.

See also: barrier method, primal-dual methods, convex optimization, feasibility.