The Simplex Method
The Simplex method is one of the most influential algorithms in the history of optimization. Introduced by George Dantzig in the late 1940s, it is a fundamental technique for solving Linear Programming (LP) problems. Despite its age, the Simplex method remains widely used in practice and continues to play a central role in modern optimization solvers. Furthermore, it provides not only optimal solutions but also valuable information such as dual variables and sensitivity analysis.
Problem class and scope
The Simplex method is designed to solve linear optimization problems with continuous decision variables and linear constraints. It applies directly to Linear Programs (LP) and is commonly used as a subroutine within more advanced algorithms for Mixed-Integer Linear Programming (MILP) and related problem classes.
The method relies on the geometric structure of LP feasible regions, which are convex polytopes defined by linear inequalities. Its effectiveness stems from exploiting this structure efficiently.
Basic principle
At a high level, the Simplex method searches for an optimal solution by moving between basic feasible solutions, which correspond to vertices of the feasible polytope. Starting from an initial feasible vertex, the algorithm iteratively improves the objective value by moving along edges of the polytope.
At each iteration, the method selects:
- A non-basic variable to enter the basis, improving the objective,
- A basic variable to leave the basis, maintaining feasibility.
This pivot operation defines a new vertex with a better or equal objective value. The process continues until no improving move exists, at which point an optimal solution has been reached.
Optimality, degeneracy, and pivoting rules
The Simplex method terminates when all reduced costs satisfy the optimality conditions. In practice, issues such as degeneracy and cycling may arise, leading to stalled progress. To address these, various pivot selection rules have been developed, including Bland’s rule and steepest-edge pricing.
While the worst-case complexity of the Simplex method is exponential, it performs extremely well on most practical instances.
Relationship to other solution techniques and algorithms
The Simplex method contrasts with Interior-Point methods, which approach the optimal solution from within the feasible region rather than along its boundary. Both methods are widely used in modern LP solvers.
Branch-and-Bound and Branch-and-Cut frameworks for Mixed-Integer Linear Programming (MILP) also typically use the Simplex Method to solve LP relaxations at each node of the search tree. It is also closely related to dual Simplex, which is particularly effective when constraints are added or modified dynamically.
Further reading and references
Dantzig, G. B. (1947). Maximization of a Linear Function of Variables Subject to Linear Inequalities. In Activity Analysis of Production and Allocation (T. Koopmans, ed.), Wiley.
The original paper introducing the Simplex method. This work laid the foundations of linear programming as a formal optimization discipline.
Chvátal, V. (1983). Linear Programming. W. H. Freeman.
A clear and rigorous introduction to linear programming, with an intuitive geometric explanation of the Simplex method and its behavior on polytopes.
Vanderbei, R. J. (2014). Linear Programming: Foundations and Extensions (4th ed.). Springer.
A comprehensive and practical reference connecting the Simplex method to modern solver implementations, numerical issues, and alternative LP algorithms.