# Minimal surface problem#

`surface.py`

```"""
## GAMSSOURCE: https://www.gams.com/latest/noalib_ml/libhtml/noalib_surface.html
## MODELTYPE: NLP

Minimal surface problem

Find a function f that minimizes the array of its graph subject to some
constraints on the boundary of the domain of f.

Boyd, S., Vandenberghe, L., Convex Optimization, Cambridge University Press,
Cambridge, 2004.
"""

from __future__ import annotations

import os

import gamspy.math as gams_math
from gamspy import Card
from gamspy import Container
from gamspy import Domain
from gamspy import Model
from gamspy import Ord
from gamspy import Parameter
from gamspy import Set
from gamspy import Sum
from gamspy import Variable
from gamspy.math import sqr

def main():
m = Container(
system_directory=os.getenv("SYSTEM_DIRECTORY", None),
delayed_execution=int(os.getenv("DELAYED_EXECUTION", False)),
)

# SETS #
X = Set(m, name="X", records=[f"I{i}" for i in range(1, 22)])
Y = Set(m, name="Y", records=[f"J{j}" for j in range(1, 22)])
inside = Set(m, name="inside", domain=[X, Y])

# Exclude i1 and i21 from inside
inside[X, Y].where[~((Ord(X) == 1) & (Ord(X) == Card(X)))] = True

#   display inside

# SCALAR #
K = Parameter(m, name="K", records=10)

# VARIABLES #
f = Variable(m, name="f", domain=[X, Y], type="positive")

# Bounds on variables, initial conditions, fixing conditions:
f.up[X, Y] = 1
f.l[X, Y] = 1.0
f.fx[X, Y].where[(Ord(X) == 1) | (Ord(X) == Card(X))] = 1

# EQUATION #
objfun = (1 / sqr(K)) * Sum(
Domain(X, Y).where[inside[X, Y]],
gams_math.sqrt(
sqr((f[X.lead(1), Y] - f[X, Y]) / K)
+ sqr((f[X, Y.lead(1)] - f[X, Y]) / K)
+ 1
),
)

surface = Model(
m,
name="surface",
equations=m.getEquations(),
problem="nlp",
sense="MIN",
objective=objfun,
)
surface.solve()

print("Objective Function Value:  ", round(surface.objective_value, 4))
print("f(X,Y):  \n", f.pivot())

# End surface

if __name__ == "__main__":
main()
```