Simplex Method

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Simplex Method

• LP problem in standard form

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Canonical (slack) form

• basic variables
• nonbasic variables

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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

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Illustrative Example
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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

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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

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Interior Point Methods(Karmarkars algorithm)
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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)

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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