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🔗 Mathematics

In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain "shadowy" techniques used to "prove" them. These techniques were introduced by John Blissard (1861) and are sometimes called Blissard's symbolic method. They are often attributed to Édouard Lucas (or James Joseph Sylvester), who used the technique extensively.

🔗 Mathematics

Karmarkar's algorithm is an algorithm introduced by Narendra Karmarkar in 1984 for solving linear programming problems. It was the first reasonably efficient algorithm that solves these problems in polynomial time. The ellipsoid method is also polynomial time but proved to be inefficient in practice.

Denoting ${\displaystyle n}$ as the number of variables and ${\displaystyle L}$ as the number of bits of input to the algorithm, Karmarkar's algorithm requires ${\displaystyle O(n^{3.5}L)}$ operations on ${\displaystyle O(L)}$-digit numbers, as compared to ${\displaystyle O(n^{6}L)}$ such operations for the ellipsoid algorithm. The runtime of Karmarkar's algorithm is thus

${\displaystyle O(n^{3.5}L^{2}\cdot \log L\cdot \log \log L),}$

using FFT-based multiplication (see Big O notation).

Karmarkar's algorithm falls within the class of interior-point methods: the current guess for the solution does not follow the boundary of the feasible set as in the simplex method, but moves through the interior of the feasible region, improving the approximation of the optimal solution by a definite fraction with every iteration and converging to an optimal solution with rational data.