simplexalgoritmi

Simplexalgoritmi, in the context of optimization, refers to a family of algorithms used to solve linear programming problems by moving along the edges of the feasible region in search of an optimum. The most well-known member is the simplex method, which navigates from one vertex (basic feasible solution) to another, improving the objective function at each step in maximization problems.

Typically, a linear program is stated in standard form as maximize c^T x subject to Ax ≤ b, x ≥ 0. The simplex method converts inequalities to equalities by introducing slack variables, creating a basic and a nonbasic set of variables. At each iteration, a nonbasic variable with a positive reduced cost enters the basis (entering variable) and a basic variable leaves (leaving variable) according to a minimum ratio test, preserving feasibility. The process continues until no improving direction exists, at which point the current vertex is optimal. If a direction allows unbounded growth, the program is declared unbounded.

Variants and practical considerations include primal simplex, dual simplex, and revised simplex methods. Anti-cycling rules, such as Bland’s rule, are used to prevent cycles in degenerate cases. Although the simplex algorithm has exponential worst-case complexity, it performs very efficiently on a wide range of problems in practice, and many modern solvers implement refined versions to handle numerical stability and degeneracy.

History and impact: the method was developed by George Dantzig in 1947 and has become foundational in operations research and optimization. It underpins numerous applications in resource allocation, production planning, transportation and network design, scheduling, and economic analysis, and remains a core technique in both academic and industry software for linear programming.