# Analytical test case # 1¶

In this example, we consider a simple optimization problem to illustrate algorithms interfaces and MDOFunction.

## Imports¶

from __future__ import annotations

from gemseo.api import configure_logger
from gemseo.core.mdofunctions.mdo_function import MDOFunction
from numpy import cos
from numpy import exp
from numpy import ones
from numpy import sin
from scipy import optimize

configure_logger()
<RootLogger root (INFO)>

## Define the objective function¶

We define the objective function $$f(x)=\sin(x)-\exp(x)$$ using a MDOFunction defined by the sum of MDOFunction s.

f_1 = MDOFunction(sin, name="f_1", jac=cos, expr="sin(x)")
f_2 = MDOFunction(exp, name="f_2", jac=exp, expr="exp(x)")
objective = f_1 - f_2

The following operators are implemented: $$+$$, $$-$$ and $$*$$. The minus operator is also defined.

print("Objective function = ", objective)
Objective function =  f_1-f_2 = sin(x)-exp(x)

## Minimize the objective function¶

We want to minimize this objective function over $$[-2,2]$$, starting from 1. We use scipy.optimize for illustration.

Note

MDOFunction objects are callable like a Python function.

x_0 = -ones(1)
opt = optimize.fmin_l_bfgs_b(objective, x_0, fprime=objective.jac, bounds=[(-0.2, 2.0)])

print("Optimum = ", opt)