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More Information about Simplex
John E. Mitchell
Department of Mathematical SciencesRPI, Troy, NY 12180 USA
January 2018
Mitchell More Information about Simplex 1 / 5
Different pivot rules
most negative reduced costbest improvement: for each negative reduced cost, calculate theimprovement from the corresponding simplex step, and take thebeststeepest edge: look at norm of ak
dual simplex: apply simplex to the dual. use same tableau,implement through careful linear algebra. get same choices fordual simplex rules.
Mitchell More Information about Simplex 2 / 5
Different pivot rules
most negative reduced costbest improvement: for each negative reduced cost, calculate theimprovement from the corresponding simplex step, and take thebeststeepest edge: look at norm of ak
dual simplex: apply simplex to the dual. use same tableau,implement through careful linear algebra. get same choices fordual simplex rules.
Mitchell More Information about Simplex 2 / 5
Different pivot rules
most negative reduced costbest improvement: for each negative reduced cost, calculate theimprovement from the corresponding simplex step, and take thebeststeepest edge: look at norm of ak
dual simplex: apply simplex to the dual. use same tableau,implement through careful linear algebra. get same choices fordual simplex rules.
Mitchell More Information about Simplex 2 / 5
Different pivot rules
most negative reduced costbest improvement: for each negative reduced cost, calculate theimprovement from the corresponding simplex step, and take thebeststeepest edge: look at norm of ak
dual simplex: apply simplex to the dual. use same tableau,implement through careful linear algebra. get same choices fordual simplex rules.
Mitchell More Information about Simplex 2 / 5
M i n -Ux, -3×3-5×4s- t . l o x i t h x , +10×4 e x p = 2 0
X i t 3 0 k -40×3+5×5= 3 0
× j30, j-l,...,5
Most negative reduced c o s t : × z e a t e r,Dalit.
Best improvement: calculate (red.cost)x(minrelio)
× , : C-i t ) x (m in{¥,'¥}) =-11x,:(-3) x (min { ¥ , -3) = - i s§: C-5 ) x(min{ ¥ , #D ¥
Steepen. edge: m i n - I lx, -3×3-5×45 .t . l ox , + Lex,410×4-1×5=20
x ,+30×-4%+5×4 = 3O
x j 2 0 , jet,...,5Similardirection: I , - £ "he:L.)
- (g)= D "Want : big improvement for small change.→ #Minimii. E u n t u f f Devi't
Hd"H( ((daffw o o 1616 ¥
Cycling
It is possible to make a degenerate pivot: the value of the minimumratio is 0. In this case, the point doesn’t change, although the set ofbasic variables changes.A sequence of degenerate pivots can result in cycling.Can prevent this through choice of pivot rules:
Bland’s rule: choose the smallest index for incoming variable,and for leaving variable in case of ties.
Without cycling, no BFS is repeated, so simplex is a finite-iterationalgorithm.
Mitchell More Information about Simplex 3 / 5
A
Eg: m i n - " x , r a t i o s :
s - t . l o x i e t x t = 2 0 20/10=2
x ,# = G O 60/30=2
/ X i ,X i , x , -30
Bland's r u l e : pivot here,
since index 1 i , smaller than index5.
Phase I
Simplex requires an initial BFS. Can get an initial BFS by solving anartificial problem (assume b � 0):
minx ,w eT wsubject to Ax + w = b
x , w � 0
Here, e is the vector of ones. This problem has an optimal value ofzero if and only if the original LP is feasible. Furthermore, solving theartificial problem using simplex will lead to a BFS for the originalproblem.
Once we have a BFS for the original problem, we solve it usingsimplex: this is Phase II.
Mitchell More Information about Simplex 4 / 5
Handling free variables
A free variable can be split into two nonnegative variables. However, itis better to leave it as a free variable and treat it as always basic.The variable does not appear in minimum ratio tests.
Mitchell More Information about Simplex 5 / 5
④ + T a - x , # 4 = 5
- X , - Z x , +3×34×4=6- X x ,x , ,X4 3 0
× ,freeGaussian pivot:[email protected]
X c , X s , X 4 7 0
Can delete first conitraint:whatever w e get for x z , X s , 44
i
c a n choose × , s o first constraint hall,.
