Standard form LPs
3 important questions on Standard form LPs
What is the simplex method in a simple form?
From the fundamental theorem of LP, we know that if an LP has an optimal solution, then one of the extreme points is optimal.
We start at an extreme point. If it has a better neighbour, we go to such a neighbour and repeat. Else, we stop, the extreme point is optimal.
We start at an extreme point. If it has a better neighbour, we go to such a neighbour and repeat. Else, we stop, the extreme point is optimal.
What is the standard form?
LPs can have equality and inequality constraints. They can have nonnegative, nonpositive and free variables. To us the simplex method everything should be converted to standard form.
In standard form, all constraints are equalities and all variables are nonnegative.
This enables the algebraic characterization of extreme points
In standard form, all constraints are equalities and all variables are nonnegative.
This enables the algebraic characterization of extreme points
What is a basic solution?
- A solution to Ax=b is called a basic solution if it is obtained by setting n-m variables equal to - and solving for the remaining m variables whose columns are linearly independent.
- The n-m variables whose values are set to 0 are called nonbasic variables.
- The remaining m variables are called basic variables.
Ax = b = [B N] (Xb, Xn) = b -> BXb + NXn = b
If we set Xn = 0, then Xb = B^-1*b will be a unique solution
If a basic solution x = (B^-1*b, 0) >= 0
Then x is called a basic feasible solution
- The n-m variables whose values are set to 0 are called nonbasic variables.
- The remaining m variables are called basic variables.
Ax = b = [B N] (Xb, Xn) = b -> BXb + NXn = b
If we set Xn = 0, then Xb = B^-1*b will be a unique solution
If a basic solution x = (B^-1*b, 0) >= 0
Then x is called a basic feasible solution
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