The simplex method - Matrix + 6 Dual theory
8 important questions on The simplex method - Matrix + 6 Dual theory
How can we interpret the optimal solution in matrix form?
Xbv B^-1*N I B^-1*b
Z CbB^-1*N - Cn O CbB^-1*b
Xbv being vector of basic variables
Xnbv being vector of non basic variables
Cb being vector of coefficients in max function of BV
Cn being vector of coefficients in max function of NBV
Aj being column of variable i in start solution
B = matrix of a,j related to the BV
N = matrix of a,j related to the NBV
b = rhs of constraints
What is weak duality?
How do we proof weak duality?
wAx <= xc
Since w>=0 and Ax >= b, we have
wAx >= wb
which makes
cx >= wAx >= wb
Corollary 1: if x is feasible for P, w is feasible for D and cx = wb then x is an opimal solution for P and w is an optimal solution for D
from WDT we know that xc>=wb for any feasible for P. Since cx = wb, x is optimal for P, similary from WFT we know that xc >= wb for any w feasible for D. Since cx = wb
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Give proof: if P is unbounded then D is infeasible
Which relations can we make between primal and dual?
Optimal Optimal
B optimal basis <-> cbB^-1 is an optimal
unbounded <-> infeasible
infeasible <-> unbounded or infeasible
What can we say about type of constraint i and optimal value of the ith dual variable
>= gives -row 0 coefficient of surplus,i
= gives row 0 coefficient of artificial,i - M
What are the primal-dual pairs?
Min | Max
ith constraint >= | ith variable >= 0
ith constraint <= | ith variable <= 0
ith constraint = | ith variable = 0
jth variable >= 0 | jtth constraint <=
jth variable <= 0 | jtth constraint >=
jth variable free | jtth constraint =
What is complementary slackness?
If a slack variable in one LP is greater than zero, the corresponding variable in the other LP is zero
If a variable in one LP equals zero, the corresponding constraint in the other LP is non-binding
If a slack variable in one LP equals zero, the corresponding variable in the other LP is greater than zero
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