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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

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simplex method
Simplex Method
  • LP problem in standard form
canonical slack form
Canonical (slack) form
  • : basic variables
  • : nonbasic variables
some definitions
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
general steps of simplex
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
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 method vs simplex
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
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