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Simplex Method

Simplex Method

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Simplex Method

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  1. Simplex Method • LP problem in standard form

  2. Canonical (slack) form • : basic variables • : nonbasic variables

  3. 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

  4. Illustrative Example

  5. 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

  6. 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

  7. Interior Point Methods(Karmarkar’s algorithm)

  8. 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)

  9. 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