Dr. Narendra Karmarkar, an Indian-born mathematician at Bell Laboratories in New Jersey, introduced a groundbreaking algorithm for solving linear programming problems. Known as Karmarkar’s algorithm, it was the first to solve these problems in polynomial time efficiently, offering a practical alternative to the simplex method, which, despite its widespread use, does not guarantee polynomial-time performance in all cases.
Linear programming involves optimizing a linear objective function subject to linear equality and inequality constraints. Prior to Karmarkar’s contribution, the ellipsoid method was known to solve linear programs in polynomial time; however, it proved inefficient in practice. Karmarkar’s algorithm, on the other hand, demonstrated significant practical efficiency, solving certain large-scale problems 50 to 100 times faster than existing methods at the time.
The algorithm operates as an interior-point method, starting from an interior feasible point and iteratively moving towards the optimal solution through a series of projective transformations and steepest-descent steps. This approach contrasts with the simplex method, which traverses the vertices of the feasible region.
Karmarkar’s innovation not only advanced computational optimization but also had substantial practical applications, including in telecommunications and operations research, where efficient solutions to large-scale linear programming problems are essential.
