Simplex Method for Linear Programming Optimization

The Simplex Method is a classic algorithm for solving linear programming problems, specifically designed for canonical form linear programs. The canonical form requires the objective function to be in maximization form, all constraints to be equations, and all decision variables to be non-negative.

The algorithm starts from a vertex of the feasible solution and iteratively moves to adjacent vertices, progressively improving the objective function value until reaching the optimal solution. Each iteration involves selecting entering and leaving basis variables, followed by updating the simplex tableau using Gaussian elimination to maintain solution feasibility throughout the process.

When all non-basic variables in the objective function have non-positive coefficients, the current solution is optimal. At this point, the optimal value and values of all decision variables can be directly read from the simplex tableau. The Simplex Method proves to be efficient and reliable, serving as the core approach for linear programming optimization. From an implementation perspective, key components include pivot element selection, tableau transformation operations, and optimality condition checking through coefficient analysis.