I have a convex optimization problem:
Now to solve this problem, I used lambda to make it one problem:
Maximize lambda * obj1 - (1-lambda) * obj2
For my problem, I considered lambda to be 0.8, which is an excellent answer when I simulate my problem using the "cvxpy" in python. However, if I submit a paper including this problem, I will be questioned about the reason for setting the lambda as 0.8. I know that "It just worked" is not a good answer, and I want a scientific approach to finding the optimal value for lambda. The first idea that came to my mind was making lambda an optimization variable. But, very soon, I realized that by taking lambda as a variable, the problem would no longer count as a linear programming problem. So, I wonder if there is a good way to find lambda in a convex optimization problem that can be justified in a scientific paper.