Solving structure optimization with Lagrange duality

[ENgez]

Homework Statement

I need to optimize the following structure with respect to compliance $P\cdot u_y$. the constraint is that the volume of the truss must not exceed $V_0$. The design variables are the bar cross sections $A_1,A_2,A_3$

Homework Equations

The mathematical programming problem I got is:

The Attempt at a Solution

The book wants me to solve this with both KKT conditions and lagrange duality.
Applying the KKT conditions leads to a set of 4 non linear equations that are complicated but solvable for the optimal design variables.

What i don't understand is how to use the Lagrange duality to solve this problem. As i understand, it is supposed to be easier to apply to more complex structural optimization problems, such as this indeterminate truss.

The book only shows how to use the Lagrange duality for separable programming problems, but my objective function is non separable, so how do I go about it?

 

[KataKoniK]

First, let's define the Lagrangian function for this problem:

L = P*u_y - lambda*(V_0 - V) - mu_1*(A_1 - A_1_max) - mu_2*(A_2 - A_2_max) - mu_3*(A_3 - A_3_max)

where lambda, mu_1, mu_2, mu_3 are the Lagrange multipliers corresponding to the constraint (volume), and the design variables (A_1, A_2, A_3) respectively.

The Lagrange dual function can then be defined as:

g(lambda, mu_1, mu_2, mu_3) = min L

To find the optimal design variables, we need to maximize the dual function g(lambda, mu_1, mu_2, mu_3) with respect to the Lagrange multipliers. This can be done by setting the partial derivatives of g with respect to each multiplier equal to 0:

∂g/∂lambda = 0
∂g/∂mu_1 = 0
∂g/∂mu_2 = 0
∂g/∂mu_3 = 0

Solving these equations will give us the optimal values for the Lagrange multipliers, which can then be substituted back into the Lagrangian function to find the optimal design variables.

Note that the dual function g(lambda, mu_1, mu_2, mu_3) is a concave function, and therefore can be maximized using standard convex optimization techniques.

In summary, to solve this problem using Lagrange duality, we need to:

1. Define the Lagrangian function.
2. Define the Lagrange dual function.
3. Maximize the dual function with respect to the Lagrange multipliers.
4. Substitute the optimal values of the multipliers back into the Lagrangian function to find the optimal design variables.

I hope this helps!