Indrepunktmetoderne

Indrepunktmetoderne, also known as interior-point methods, are a class of algorithms used for solving mathematical optimization problems, particularly linear programming and convex quadratic programming. They are a significant alternative to the simplex method for linear programming. The core idea behind interior-point methods is to approach the optimal solution by moving through the interior of the feasible region, rather than along the boundary as the simplex method does.

These methods typically involve iterative steps. In each iteration, the algorithm takes a step towards the optimal solution from a point strictly inside the feasible region. This step is usually determined by solving a system of linear equations that approximates the Karush-Kuhn-Tucker (KKT) conditions, which characterize the optimality of a solution. A key feature is the use of a barrier function, which discourages the iterates from approaching the boundary of the feasible region too closely. As the iterations progress, the barrier effect is gradually reduced, guiding the solution towards the optimal point.

Interior-point methods have proven to be very efficient for large-scale problems. They often exhibit polynomial time complexity in the worst case, making them theoretically attractive. While the simplex method can perform better on some smaller or specially structured problems, interior-point methods have become a dominant technique in many practical applications due to their robustness and scalability. They are widely implemented in optimization software.