In general, numerical optimization, multi-objective is transformed into a single functional called as a preference function. There are several types of preference function such as weighed summation type, weighted distance type and weighted min-max type.
•Weighted Summation Type
•Weighted Distance Type
•Min-Max Type
Conceptually, each local optimum 
is preferred as a 
in the above formulations. In
practical design, no one knows them until solving each single objective
optimization. One guesses them properly or replaces them as 
, where 
is the initial design point.
Now, we compare the optimization results for multi-objective
formulations. Suppose that 
and 
will be minimized simultaneously.
Figure 1 shows them.
Figure 1 Graphical representation of two objectives
If the same weightings are used in these two objectives,
Figure 1 shows that the pareto optimum is 
and 
.
•Weighted Summation Type
As the optimum satisfies 
, it gives
Thus, the optimum is 
, which is different from the Pareto
optimum.
•Weighted Distance Type
Let the value of 
be 2. Then, the distance function
is
By solving ,
This gives that 
, which is different from the Pareto
optimum.
•Weighted Min-Max Type
This functional is a composite non-smooth function as follows:
Figure shows that this formulation gives the Pareto optimum
but it requires some special
techniques to overcome the non-smoothness of functional. This is the reason that
the min-max type is not used in the gradient-based optimization, even though it
can guarantee a local Pareto optimum.