"""Problem D in paper "Outer approximation algorithms for separable nonconvex mixed-integer nonlinear programs"

Ref:
Kesavan P, Allgor R J, Gatzke E P, et al. Outer approximation algorithms for separable nonconvex mixed-integer nonlinear programs[J]. Mathematical Programming, 2004, 100(3): 517-535.

Problem type:   nonconvex MINLP
        size:   3  binary variable
                2  continuous variables
                4  constraints

"""
from pyomo.environ import *


class Nonconvex4(ConcreteModel):
    def __init__(self, *args, **kwargs):
        """Create the problem."""
        kwargs.setdefault('name', 'Nonconvex4')
        super(Nonconvex4, self).__init__(*args, **kwargs)
        m = self

        m.x1 = Var(within=Reals, bounds=(1, 10))
        m.x2 = Var(within=Reals, bounds=(1, 6))
        m.y1 = Var(within=Binary, bounds=(0, 1), initialize=0)
        m.y2 = Var(within=Binary, bounds=(0, 1), initialize=0)
        m.y3 = Var(within=Binary, bounds=(0, 1), initialize=0)

        m.objective = Objective(expr=-5 * m.x1 + 3 * m.x2, sense=minimize)

        m.c1 = Constraint(expr=2 * (m.x2 ** 2) - 2 * (m.x2 ** 0.5) -
                          2 * (m.x1 ** 0.5) * (m.x2 ** 2) + 11 * m.x2 + 8 * m.x1 <= 39)
        m.c2 = Constraint(expr=m.x1 - m.x2 <= 3)
        m.c3 = Constraint(expr=3 * m.x1 + 2 * m.x2 <= 24)
        m.c4 = Constraint(expr=-m.x1 + m.y1 + 2 * m.y2 + 4 * m.y3 == 0)