## [Simplex Method](/simplex-method-0) Definition An iterative algorithm for solving linear programming problems by moving along edges of the feasible polytope between basic feasible solutions until an optimal extreme point is reached. ## Principle Principle Exploit the polyhedral structure of a linear program: vertices correspond to basic feasible solutions and optimality can be reached by local pivot operations that improve the objective. ## Demonstration Demonstration Solve a diet problem by initializing a basic feasible solution, compute reduced costs, pivot to adjacent vertices with negative reduced cost, and stop when no improving pivot exists. ## Misapplication Misapplication Applying the simplex method without checking degeneracy or cycling safeguards can lead to infinite loops or repeated tableaux when multiple bases share the same vertex. ## Consequence Consequence When applied correctly the method yields a vertex optimal solution and provides dual information (shadow prices) and certificates of infeasibility or unboundedness when appropriate. ## Reversal Reversal Instead of walking the boundary, interior algorithms traverse the feasible interior using barrier terms; these methods trade explicit vertex pivots for central-path progress. ## Boundary Boundary Applies only to linear programs (affine constraints and linear objective); nonlinear, integer, or nonconvex programs require modification or different algorithms. ## Semantic Tension Semantic Tension Tension exists between 'combinatorial pivoting' (discrete basis changes) and 'continuous path following' (interior methods) as alternative routes to optimum in convex optimization. ## Synthesis Synthesis The simplex method frames linear programming as a combinatorial navigation of a convex polytope: perform algebraic pivots that correspond to geometric moves between vertices until optimality conditions hold.