Simplex Method LP problem in standard form

Canonical (slack) form : basic variables : nonbasic variables

Some definitions basic solution – solution obtained from canonical system by setting nonbasic variables to zero basic feasible solution – a basic solution that is feasible – at most – One of such solutions yields optimum if it exists Adjacent basic feasible solution – differs from the present basic feasible solution in exactly one basic variable Pivot operation – a sequence of elementary row operations that generates an adjacent basic feasible solution Optimality criterion – When every adjacent basic feasible solution has objective function value lower than the present solution

Illustrative Example

General steps of Simplex 1. Start with an initial basic feasible solution 2. Improve the initial solution if possible by finding an adjacent basic feasible solution with a better objective function value – It implicitly eliminates those basic feasible solutions whose objective functions values are worse and thereby a more efficient search 3. When a basic feasible solution cannot be improved further, simplex terminates and return this optimal solution

Simplex-cont. Unbounded Optimum Degeneracy and Cycling – A pivot operation leaves the objective value unchanged – Simplex cycles if the slack forms at two different iterations are identical Initial basic feasible solution

Interior Point Methods (Karmarkar’s algorithm)

Interior Point Method vs. Simplex Interior point method becomes competitive for very “large” problems – Certain special classes of problems have always been particularly difficult for the simplex method – e.g., highly degenerate problems (many different algebraic basic feasible solutions correspond to the same geometric extreme point)

Computation Steps 1. Find an interior point solution to begin the method – Interior points: 2. Generate the next interior point with a lower objective function value – Centering: it is advantageous to select an interior point at the “center” of the feasible region – Steepest Descent Direction 3. Test the new point for optimality – where is the objective function of the dual problem