Interior Point Method Algorithm

Interior point methods solve linear and certain nonlinear optimization problems by moving through the interior of the feasible region defined by the problem's constraints, rather than moving along the boundary edge by edge as the simplex method does, using a barrier function that penalizes points as they approach the boundary and gradually relaxing that penalty as the method converges toward an optimal solution. Narendra Karmarkar published the first practical polynomial-time interior point method for linear programming in 1984, a result that showed such methods could outperform the simplex method on certain large-scale problems. Interior point methods are now standard in commercial optimization software alongside simplex-based solvers.