# Determining the optimize lambda in Multi-Objective Optimization

I have a convex optimization problem:

Maximize    obj1
Minimize    obj2
Some constraint


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.

There is no mathematical way to derive (or justify) a value for $$\lambda$$. The justification has to be made in the context of a specific problem and a specific (reasonably credible) decision maker. Basically, the choice of $$\lambda$$ says that the decision maker would be indifferent between an increase (decrease) of one unit of obj1 and an increase (decrease) of $$\frac{\lambda}{1-\lambda}$$ units of obj2. So, for example, if obj1 measures number of traffic accidents avoided and obj2 is thousands of dollars spent on safety measures, choosing $$\lambda=0.6$$ says that the decision maker would be indifferent (neither particularly in favor nor particularly against) spending 1,500 per accident avoided.
In addition I'd say you can also try with a common method like defining a variable $$z \le$$ obj2. Then maximize obj1 + $$z$$. Or you can try with priority based optimization and goal programming.