I've solved a bi-objective optimization problem by means of NOMAD solver from OPTI Toolbox and as a result I've obtained a Pareto frontier: enter image description here

How to interpret the visible "gap" in the Pareto frontier?


I will try to answer myself.

Consider the Schaffer function no. 2:

$\begin{cases} f_{1}\left(x\right) & = \begin{cases} -x, & \text{if } x \le 1 \\ x-2, & \text{if } 1 < x \le 3 \\ 4-x, & \text{if } 3 < x \le 4 \\ x-4, & \text{if } x > 4 \\ \end{cases} \\ f_{2}\left(x\right) & = \left(x-5\right)^{2} \\ \end{cases}$

It is shown in the following figure:

Schaffer function no. 2

For such objective functions the Pareto frontier is discontinuous:

Pareto frontier

If then one denotes on the function plot the corresponding point from the Pareto frontier we obtain:

Function with Pareto points

One can observe that each "part" of Pareto frontier correspond to vicinity of minima of the objective functions. If, now, one considers point $x=2$ it can be observed that for greater $x$ value of $f_1$ e.g. $f_1(2.1)$ is equal to $f_1(4.1)$, but value of $f_2$ decreases significantly. So from optimality point of view this "switch" gives better solution, but results in a discontinuity in the Pareto front.


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