So in the previous post I was talking about Lagrangian Multipliers and I created an exe file to show the objective function with one constraint. You can reach the file here which includes an exe file and a MCR file in case you don't have Matlab installed on your machine (the exe file compiled and runs just on windows machines). You can visualize the function and the constraint with this file and I will show you an example of that.
The example comes from one of my classes (CE 590 - Modeling and Analysis of Civil Engineering Systems) and called OMAGOSH problem.
Here is the problem description:
"You are an engineer advising the State Department of Natural Resources as it manages Lake Omagosh. The water quality index, x, can be improved, and better fishing would result. Boat launch facilities could also be added (measure in units, y). The benefits of fishing and pleasure boating are 11x+8y-2xy, in million dollars. The costs are x+2y in million dollars. Only 17 million dollars are available. Use the Kuhn-Tucker approach to evaluate this problem. What is the meaning of the value of the Lagrange multiplier for this problem?"
First of all, lets talk abut Kuhn-Tacker conditions. Because we have inequality in our constraint and we would like to solve this problem with Lagrangian approach and since we can solve this problem with Lagrangian method if we have equality, we can change our inequality in our constraint to equality by adding a slack variable to our constraint.
So, if we have equality constraint we will have something like: g(x,y)=b
but here we have inequality: g(x,y)<= b and in the Omagosh problem: g(x,y)<= 17.
we can change the inequality to equality as mentioned by adding a slack variable: g(x,y)+s=17.
's', here is our slack variable and its worth to mention that 's' is always positive, we can just put 's^2'.
You can reach the solution I did by hand here and please let me know if you have any question.
In this picture you can see the objective function and constraint.
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