LP QZ Duality And Rates Of Change
Math 484 QZ Duality and Rates of Change
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c= −3,4( ) x = x1,x2( )
g1 x( )= 3×1 − 4×2 b1 = −12
g2 x( )= −5×1 + 2×2 b2 = −20
g3 x( )= x1+ x2 b3 = 10
1. Solve LP1 & its dual by pivoting in Mathematica and plot also the feasible set.
2. For each constraint i active in the solution, verify that yi
by explicitly
computing Δz for three values of Δbi = {1,−1,2} by three different methods:
a) By putting the new bi value in the tableau and resolving
b) Using the fact the new optimal x* will be at the intersection of the same constraint
boundaries, and therefore a function x*= x*(bi ), express z* as an explicit
function of bi ; i.e. z*= z(x*) = z(x*(bi )) = z*(bi ).
Use the formulation of z*(bi ) to compute
c) Use the dual solution y* = y1
*( ) to express
Δz= ∇z⋅ Δx = yi
* ∇gi ⋅ Δx( )=
∑ yi* Δgi( )
∑ = yi*Δbi
At each step of this derivation, plug in the numerical values of ∇gi ,Δx,Δgi
a) If b2 remains fixed at − 20, in what range of values (min and max) for b3 does y3
b) If b3 remains fixed at 10, in what range of values (min and max) for b2 does y2
minz= c⋅ x
g1 x( )≥ b1
g2 x( )≥ b2
g3 x( )≥ b3
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