Simpleksimenetelmää

Simpleksimenetelmää is a Finnish term that translates to "simplex method" in English. It is a well-known algorithm used in linear programming to find the optimal solution to a linear objective function subject to linear equality and inequality constraints. The method was developed by George Dantzig in 1947.

The simplex method works by systematically moving from one feasible corner point (vertex) of the feasible region to an adjacent one, improving the objective function value at each step. The feasible region is a convex polytope defined by the intersection of the constraint inequalities. The algorithm terminates when it reaches a vertex where no further improvement in the objective function is possible, indicating that the optimal solution has been found.

Geometrically, the simplex method explores the vertices of the feasible region. It starts at a basic feasible solution and iteratively moves to a better-connected vertex until optimality is reached. This process guarantees finding the global optimum if one exists.

In practice, the simplex method is implemented using algebraic manipulations of a tableau, which represents the constraints and the objective function. The algorithm involves pivoting operations, which correspond to moving from one vertex to another. While efficient for many problems, its worst-case time complexity can be exponential, although this is rarely encountered in practice. Despite the existence of polynomial-time algorithms for linear programming, such as the interior-point methods, the simplex method remains widely used due to its conceptual simplicity and strong performance on typical real-world problems.