## BNH

Binh and Korn defined the following test problem in <cite data-cite="bnh"></cite> with 2 objectives and 2 constraints:

**Definition**

\begin{equation}
\newcommand{\boldx}{\mathbf{x}}
\begin{array}
\mbox{Minimize} & f_1(\boldx) = 4x_1^2 + 4x_2^2, \\
\mbox{Minimize} & f_2(\boldx) = (x_1-5)^2 + (x_2-5)^2,    \\
\mbox{subject to} & C_1(\boldx) \equiv (x_1-5)^2 + x_2^2 \leq 25, \\
& C_2(\boldx) \equiv (x_1-8)^2 + (x_2+3)^2 \geq 7.7, \\
& 0 \leq x_1 \leq 5, \\
& 0 \leq x_2 \leq 3.
\end{array}
\end{equation}

**Optimum**

The Pareto-optimal solutions are constituted by solutions 
$x_1^{\ast}=x_2^{\ast} \in [0,3]$ and $x_1^{\ast} \in [3,5]$,
$x_2^{\ast}=3$. These solutions are marked by using bold 
continuous
curves.  The addition of both constraints in the problem does not make any solution
in the unconstrained Pareto-optimal front infeasible. 
Thus, constraints may not introduce any additional difficulty
in solving this problem.

**Plot**


```python
from pymoo.factory import get_problem
from pymoo.util.plotting import plot

problem = get_problem("bnh")
plot(problem.pareto_front(), no_fill=True)
```

51cb4766ae7a57ad3470c22eb276a0e5538468aa
ipynb
Jupyter Notebook
doc/source/problems/multi/bnh.ipynb
gabicavalcante/pymoo
1711ce3a96e5ef622d0116d6c7ea4d26cbe2c846
Apache-2.0
2020-09-18T19:33:31.000Z
2020-09-18T19:33:33.000Z
doc/source/problems/multi/bnh.ipynb
gabicavalcante/pymoo
1711ce3a96e5ef622d0116d6c7ea4d26cbe2c846
Apache-2.0
doc/source/problems/multi/bnh.ipynb
gabicavalcante/pymoo
1711ce3a96e5ef622d0116d6c7ea4d26cbe2c846
Apache-2.0
2022-03-31T08:19:13.000Z
2022-03-31T08:19:13.000Z
Qwen/Qwen-72B
1. YES
2. YES