Concept

Simplex method

An algorithm for linear programming that walks from corner to corner of the feasible polytope, always along an edge that improves the objective, until no edge does. It is fast in practice, though some programs force it through exponentially many corners.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

3 moves along the edges to the corner that maximises 2x₁ + 3x₂. The simplex method on a two-variable program with 5 constraints, started at the origin. It visits (0, 0), (0, 8), (1, 8), (5/2, 15/2), with objective values 0, 24, 26, 55/2, and stops where the prices on both binding constraints are non-negative.

Prices at every corner

The duality theorem says a linear program's best value equals its dual's, and says nothing about how to find either. The simplex method finds both at once — it walks from corner to corner, and at each one asks the constraints that meet there for prices. A negative price names an edge that climbs; when none is negative, the prices are the proof.

applied · Duality

Eight corners of a squashed cube, visited in order by the simplex method. Klee and Minty's program in 3 variables drawn as its own deformed cube and as a plain one. The simplex method with the largest-price rule visits all 8 corners, with objective values 0, 4, 6, 10, 15, 19, 21, 25; the optimum is one edge from the start.

The cube that takes every corner

The simplex method is fast on every program anybody meets in practice. In 1972 Victor Klee and George Minty squashed a cube so that the method, choosing the steepest edge each time, visits all of its corners — 2ⁿ − 1 moves in n variables, with the optimum one edge from the start.

applied · Duality

Named alongside it

The objects these essays reach for when they reach for this one.

Linear program Optimisation Certificate Complementary slackness Degeneracy Duality Exponential growth Gray code Hypercube Polytope Shadow price Worst case

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