Optimizing a Paraboloid – The TL;DR Version

import openmdao.api as om

# build the model
prob = om.Problem()

prob.model.add_subsystem('paraboloid', om.ExecComp('f = (x-3)**2 + x*y + (y+4)**2 - 3'))

# setup the optimization
prob.driver = om.ScipyOptimizeDriver()
prob.driver.options['optimizer'] = 'SLSQP'

prob.model.add_design_var('paraboloid.x', lower=-50, upper=50)
prob.model.add_design_var('paraboloid.y', lower=-50, upper=50)
prob.model.add_objective('paraboloid.f')

prob.setup()

# Set initial values.
prob.set_val('paraboloid.x', 3.0)
prob.set_val('paraboloid.y', -4.0)

# run the optimization
prob.run_driver()

# minimum value
print(prob.get_val('paraboloid.f'))

# location of the minimum
print(prob.get_val('paraboloid.x'))
print(prob.get_val('paraboloid.y'))

Optimization terminated successfully    (Exit mode 0)
            Current function value: -27.33333333333333
            Iterations: 5
            Function evaluations: 6
            Gradient evaluations: 5
Optimization Complete
-----------------------------------
[-27.33333333]
[6.66666667]
[-7.33333333]