Simplex algorithm
Algorithm for linear programming, derived from simplicial cones.
The simplex algorithm, also known as Dantzig's simplex method, is an algorithm for linear programming in mathematical optimization. Its name, suggested by T. S. Motzkin, derives from the concept of a simplex, though simplices are not actually used in the method; instead, it operates on simplicial cones, which become proper simplices with an additional constraint. The algorithm was developed by George Dantzig during the mid-1940s while he was working on planning methods for the US Army Air Force using a desk calculator. Initially, Dantzig formulated the problem as a system of linear inequalities without an objective function, inspired by the work of Wassily Leontief. Without an objective, a vast number of solutions were feasible, and military-specified "ground rules" were used to describe how goals could be achieved. Dantzig’s key insight was that most such ground rules could be translated into a linear objective function to be maximized. After incorporating an objective function in mid-1947, the problem became mathematically more tractable. Dantzig realized that an unsolved problem he had mistakenly taken as homework in a class taught by Jerzy Neyman—concerning the existence of Lagrange multipliers for general linear programs over a continuum of variables—was applicable to finding an algorithm for linear programs. This work, later published as his doctoral thesis, gave Dantzig the column geometry insight that led him to believe the simplex method would be very efficient. The algorithm operates on linear programs in canonical form, where the feasible region defined by constraints is a convex polytope. It begins at a starting vertex, or basic feasible solution, and moves along the polytope’s edges to vertices with progressively higher objective values until the maximum is reached or an unbounded edge indicates no solution. The process is divided into two phases: Phase I finds a starting extreme point, and Phase II applies the algorithm from that point to find the optimum.
- field
- Mathematical optimization
- known_for
- Simplex algorithm for linear programming
- named_by
- T. S. Motzkin
Lore & Background
George Dantzig worked on planning methods for the US Army Air Force during World War II using a desk calculator. Dantzig formulated the problem as linear inequalities inspired by the work of Wassily Leontief, but initially did not include an objective; without one, a vast number of solutions were feasible, requiring military-specified 'ground rules' to describe how goals could be achieved. Dantzig's core insight was that most such ground rules could be translated into a linear objective function to be maximized. The development of the simplex method was evolutionary and occurred over about a year. He realized that one of the unsolved problems he had mistaken as homework in professor Jerzy Neyman's class—and later solved—was applicable to finding an algorithm for linear programs. This problem involved finding the existence of Lagrange multipliers for general linear programs over a continuum of variables, each bounded between zero and one, and satisfying linear constraints expressed as Lebesgue integrals. Dantzig later published this 'homework' as a thesis to earn his doctorate. The column geometry used in that thesis gave him insight that made him believe the simplex method would be very efficient.
Reader's Guide
The simplex algorithm operates on linear programs in canonical form: maximize cᵀx subject to Ax ≤ b and x ≥ 0. Geometrically, the feasible region is a convex polytope; an extreme point or vertex is known as a basic feasible solution (BFS). If the objective function has a maximum on the feasible region, it attains that value at at least one extreme point. The algorithm walks along edges of the polytope to extreme points with greater objective values until the maximum is reached or an unbounded edge is visited, indicating no solution. The algorithm always terminates because the number of vertices is finite. Solution occurs in two phases: Phase I finds a starting extreme point (or determines infeasibility), and Phase II applies the simplex algorithm from that point to find an optimum or an unbounded edge. The transformation to standard form involves introducing new variables for lower bounds other than zero.
Did You Know?
- The name 'simplex algorithm' was suggested by T. S. Motzkin, though simplices are not actually used in the method.
- George Dantzig's core insight was that military-specified 'ground rules' could be translated into a linear objective function to be maximized.
- Dantzig realized that an unsolved problem he had mistaken as homework in Jerzy Neyman's class was applicable to finding an algorithm for linear programs.
- The simplex algorithm always terminates because the number of vertices in the polytope is finite.
Frequently Asked Questions
Who is Simplex algorithm?
The Simplex algorithm, also called Dantzig's simplex method, is a foundational procedure in mathematical optimization built to solve linear programming problems. Its name was proposed by T. S. Motzkin, even though the method itself was developed by George Dantzig.
What are Simplex algorithm's powers and role?
It navigates the edges of a feasible region defined by simplicial cones to locate the optimal vertex for a linear program. Rather than manipulating actual simplices, the method works on those simplicial cones, which only become true simplices once an extra bounding constraint is imposed.
How does Simplex algorithm's story end?
In a typical run, the method halts at a vertex where no neighboring vertex improves the objective, certifying optimality. In the worst case, however, it can require an exponential number of pivots, so its resolution is not guaranteed to be quick.
Why is Simplex algorithm important?
Even though polynomial-time alternatives like the ellipsoid method exist, the simplex approach remains the dominant tool in industrial linear programming because it performs exceptionally well on most practical instances. It set the practical standard for optimization and shaped decades of operations research and applied mathematics.
Why is it called 'simplex' if it never actually uses simplices?
Motzkin chose the name as a geometric nod to the familiar simplex shape, but the algorithm genuinely operates on simplicial cones instead. Only after adding one further constraint do those cones collapse into proper simplices, making the name somewhat aspirational rather than literal.
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