Linear Programming
Linear programming, also known as operations research, optimization theory, convex optimization theory, or linear optimization, is the problem of maximizing a linear function over a convex polyhedron. Linear programming is extensively used in economics and engineering. Examples from economics include Leontief's input-output model, the determination of shadow prices, etc., while an example of an engineering application would be maximizing profit in a factory that manufactures a number of different products from the same raw material using the same resources.
Linear programming can be solved using the simplex method (Wood and Dantzig 1949, Dantzig 1949) which runs along polytope edges of the visualization solid to find the best answer. Khachian (1979) found a polynomial time algorithm. A much more efficient polynomial time algorithm was found by Karmarkar (1984). This method goes through the middle of the solid (making it a so-called interior point method), and then transforms and warps. Arguably, interior point methods were known as early as the 1960s in the form of the barrier function methods, but the media hype accompanying Karmarkar's announcement led to these methods receiving a great deal of attention.