Simplex Method
• LP problem in standard form
Canonical (slack) form
• B  x1 , x2 ,...,xm : basic variables
• N  xm1 ,...,xn  : 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
n
– at most  m 
 
– 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
– m  n  10,000
• 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:
x
0

| Ax  b, x  0
0
0
• 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
T
T
– c x  b w   where b T w is the objective function of
the dual problem