Linear Programming Including the Simplex Method

Linear programming serves as a solution approach for optimization problems, designed to identify the optimal solution that either maximizes or minimizes a linear function. The simplex method stands as a widely-used algorithm that iteratively adjusts variable values within the linear problem to converge toward the optimum. In implementation, the simplex algorithm evaluates vertices of the feasible region by pivoting operations, systematically improving the objective function value until no further enhancements are possible. Gradient descent and Newton's method can also be adapted for linear programming; these techniques leverage first-order derivatives (gradients) or second-order derivatives (Hessian matrices) to locate function maxima or minima through iterative steps. For gradient-based approaches, step size control is critical to ensure convergence, while Newton's method employs quadratic approximation for faster optimization. It is essential to note that real-world linear programming applications often incorporate constraint limitations, requiring problem-specific adjustments—such as adding slack variables or implementing barrier methods—to derive feasible optimal solutions. Proper handling of constraints through techniques like the two-phase simplex method or interior-point algorithms ensures robustness in practical scenarios.