Variational calculus - dual problem

In summary: Alternatively, you can also take the partial derivative of the dual problem and set it to 0, then solve for r and substitute into the formula for x*.
  • #1
braindead101
162
0
the primal problem was:
min (x^T)Px
i found g(r) and the partial derivative of g(r) w.r.t. x to be: x=-1/2(P^-1)(A^T)r

i have found the dual problem to be:
max -1/4(r^T)A(P^(-1))(A^T)r - (b^T)r
subject to r>= 0
I am told to find x* and r* (which i think is just x and r):
i have not shown my work going from primal to dual as i know it is correct but i have just shown what i think is the necessary information to do this problem.

I am given the following:
A=P=2x2 identity matrix. and b = (1,0)
How do I go about computing x* and r*?
do i just set the max = 0 and calculate r like that, then substitute this r value into the x= formula.
or do i need to partial derivative the dual problem and set to 0 and calculate r like that, then substituting this r value into the x= formula.

please let me know. and am i correct in thinking that x* and r* is the same as x and r...
 
Physics news on Phys.org
  • #2
just x* and r* is the optimal values.The answer is yes, you are correct in thinking that x* and r* are the same as x and r, but with the optimal values. To compute x* and r*, set the maximum of the dual problem to 0 and solve for r. Then, substitute this value of r into the formula x = -1/2(P^-1)(A^T)r to get x*.
 

1. What is the dual problem in variational calculus?

The dual problem in variational calculus is a mathematical formulation that allows us to find the optimal solution to a given variational problem. It involves finding the maximum or minimum value of a function, known as the Lagrangian, subject to certain constraints.

2. How is the dual problem related to the primal problem?

The dual problem is closely related to the primal problem, as they both involve the same set of variables and constraints. However, the dual problem aims to find the optimal values of these variables that maximize or minimize the Lagrangian, while the primal problem aims to find the optimal values that satisfy the constraints.

3. What is the significance of the dual problem in optimization?

The dual problem is significant in optimization as it provides a way to solve complex optimization problems that may be difficult to solve directly. It also allows us to obtain a lower bound for the optimal solution of the primal problem, which can be useful in certain applications.

4. How is the dual problem solved?

The dual problem is typically solved using a technique called duality theory. This involves transforming the primal problem into its dual form, which can then be solved using various optimization methods such as the simplex algorithm or gradient descent.

5. What are the applications of the dual problem in real-world problems?

The dual problem has a wide range of applications in various fields, including economics, engineering, and physics. It is commonly used in optimization problems involving resource allocation, portfolio optimization, and control theory, among others.

Similar threads

  • Calculus and Beyond Homework Help
Replies
1
Views
518
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
465
  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
437
  • Calculus and Beyond Homework Help
Replies
9
Views
921
  • Calculus and Beyond Homework Help
Replies
13
Views
147
  • Calculus and Beyond Homework Help
Replies
24
Views
672
  • Calculus and Beyond Homework Help
Replies
9
Views
699
  • Calculus and Beyond Homework Help
Replies
8
Views
757
Back
Top