Finding dual optimum of a linear problem

A
Thread starter Trollfaz
Start date Oct 26, 2023
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Linear programming

Oct 26, 2023

Trollfaz

A simple linear problem goes
min c'x such that f_i(x)<= 0 and Ax=b
x
Suppose we make all constraints affine. Then

Dx-e<=0 and Ax-b =0
We get the Langrangian function as
c'x + λ'(Dx-e) +ν'(Ax-b) and since Ax-b is 0,
we reduce L to
c'x + λ'(Dx-e)
The dual function g is
inf L(x,λ)
x
Then I differentiate L against x to get c=-D'λ
With that we get g(λ) as -λ.e
So I conclude that d*=sup g(λ) against λ.
Does differentiation of the Lagrangian function give me the dual optimum and does it work for all convex inequality constraints and convex objective functions?

1. How do you find the dual optimum of a linear problem?

To find the dual optimum of a linear problem, you first need to formulate the dual problem by defining the dual variables and setting up the dual objective function. Then, you can solve the dual problem using optimization techniques such as the simplex method or the dual simplex method.

2. What is the significance of finding the dual optimum of a linear problem?

Finding the dual optimum of a linear problem allows you to obtain valuable information about the original primal problem, such as the optimal objective value and the optimal dual variables. This information can help in understanding the duality relationship between the primal and dual problems and in making better decisions in optimization problems.

3. Can the dual optimum be higher than the primal optimum in a linear problem?

Yes, it is possible for the dual optimum to be higher than the primal optimum in a linear problem. This situation occurs when the primal problem is unbounded or infeasible, leading to an unbounded or infeasible dual problem with a higher optimal value.

4. How do you interpret the dual optimum of a linear problem?

The dual optimum of a linear problem represents the maximum value of the dual objective function, which is achieved by choosing the optimal dual variables that satisfy the dual constraints. This value provides insights into the sensitivity of the primal problem to changes in the objective function coefficients or constraints.

5. What are some common applications of finding the dual optimum in linear programming?

Some common applications of finding the dual optimum in linear programming include resource allocation, production planning, portfolio optimization, and network flow problems. By solving the dual problem and obtaining the dual optimum, you can analyze the trade-offs between different objectives and constraints in these real-world optimization problems.