"""
Copyright 2013 Steven Diamond

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

    http://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
"""

import cvxpy as cp
from cvxpy.atoms.affine.add_expr import AddExpression
from cvxpy.expressions.variable import Variable
from cvxpy.expressions.constants import Constant
from cvxpy.expressions.constants import Parameter
from cvxpy import Problem, Minimize
import cvxpy.interface.matrix_utilities as intf
import cvxpy.settings as s
from cvxpy.tests.base_test import BaseTest
import numpy as np
import scipy.sparse as sp
import warnings


class TestExpressions(BaseTest):
    """ Unit tests for the expression/expression module. """

    def setUp(self):
        self.a = Variable(name='a')

        self.x = Variable(2, name='x')
        self.y = Variable(3, name='y')
        self.z = Variable(2, name='z')

        self.A = Variable((2, 2), name='A')
        self.B = Variable((2, 2), name='B')
        self.C = Variable((3, 2), name='C')
        self.intf = intf.DEFAULT_INTF

    # Test the Variable class.
    def test_variable(self):
        x = Variable(2)
        y = Variable(2)
        assert y.name() != x.name()

        x = Variable(2, name='x')
        y = Variable()
        self.assertEqual(x.name(), 'x')
        self.assertEqual(x.shape, (2,))
        self.assertEqual(y.shape, tuple())
        self.assertEqual(x.curvature, s.AFFINE)
        # self.assertEqual(x.canonical_form[0].shape, (2, 1))
        # self.assertEqual(x.canonical_form[1], [])

        self.assertEqual(repr(self.x), "Variable((2,))")
        self.assertEqual(repr(self.A), "Variable((2, 2))")

        # # Scalar variable
        # coeff = self.a.coefficients()
        # self.assertEqual(coeff[self.a.id], [1])

        # # Vector variable.
        # coeffs = x.coefficients()
        # self.assertItemsEqual(coeffs.keys(), [x.id])
        # vec = coeffs[x.id][0]
        # self.assertEqual(vec.shape, (2,2))
        # self.assertEqual(vec[0,0], 1)

        # # Matrix variable.
        # coeffs = self.A.coefficients()
        # self.assertItemsEqual(coeffs.keys(), [self.A.id])
        # self.assertEqual(len(coeffs[self.A.id]), 2) or 0 in self.shape
        # mat = coeffs[self.A.id][1]
        # self.assertEqual(mat.shape, (2,4))
        # self.assertEqual(mat[0,2], 1)

        with self.assertRaises(Exception) as cm:
            Variable((2, 2), diag=True, symmetric=True)
        self.assertEqual(str(cm.exception),
                         "Cannot set more than one special attribute in Variable.")

        with self.assertRaises(Exception) as cm:
            Variable((2, 0))
        self.assertEqual(str(cm.exception), "Invalid dimensions (2, 0).")

        with self.assertRaises(Exception) as cm:
            Variable((2, .5))
        self.assertEqual(str(cm.exception), "Invalid dimensions (2, 0.5).")

        with self.assertRaises(Exception) as cm:
            Variable(2, 1)
        self.assertEqual(str(cm.exception),
                         "Variable name 1 must be a string.")

    def test_assign_var_value(self):
        """Test assigning a value to a variable.
        """
        # Scalar variable.
        a = Variable()
        a.value = 1
        self.assertEqual(a.value, 1)
        with self.assertRaises(Exception) as cm:
            a.value = [2, 1]
        self.assertEqual(str(cm.exception), "Invalid dimensions (2,) for Variable value.")

        # Test assigning None.
        a.value = 1
        a.value = None
        assert a.value is None

        # Vector variable.
        x = Variable(2)
        x.value = [2, 1]
        self.assertItemsAlmostEqual(x.value, [2, 1])
        # Matrix variable.
        A = Variable((3, 2))
        A.value = np.ones((3, 2))
        self.assertItemsAlmostEqual(A.value, np.ones((3, 2)))

        # Test assigning negative val to nonnegative variable.
        x = Variable(nonneg=True)
        with self.assertRaises(Exception) as cm:
            x.value = -2
        self.assertEqual(str(cm.exception), "Variable value must be nonnegative.")

    # Test tranposing variables.
    def test_transpose_variable(self):
        var = self.a.T
        self.assertEqual(var.name(), "a")
        self.assertEqual(var.shape, tuple())

        self.a.save_value(2)
        self.assertEqual(var.value, 2)

        var = self.x
        self.assertEqual(var.name(), "x")
        self.assertEqual(var.shape, (2,))

        x = Variable((2, 1), name='x')
        var = x.T
        self.assertEqual(var.name(), "x.T")
        self.assertEqual(var.shape, (1, 2))

        x.save_value(np.array([[1, 2]]).T)
        self.assertEqual(var.value[0, 0], 1)
        self.assertEqual(var.value[0, 1], 2)

        var = self.C.T
        self.assertEqual(var.name(), "C.T")
        self.assertEqual(var.shape, (2, 3))

        # coeffs = var.canonical_form[0].coefficients()
        # mat = coeffs.values()[0][0]
        # self.assertEqual(mat.shape, (2,6))
        # self.assertEqual(mat[1,3], 1)

        index = var[1, 0]
        self.assertEqual(index.name(), "C.T[1, 0]")
        self.assertEqual(index.shape, tuple())

        var = x.T.T
        self.assertEqual(var.name(), "x.T.T")
        self.assertEqual(var.shape, (2, 1))

    # Test the Constant class.
    def test_constants(self):
        c = Constant(2.0)
        self.assertEqual(c.name(), str(2.0))

        c = Constant(2)
        self.assertEqual(c.value, 2)
        self.assertEqual(c.shape, tuple())
        self.assertEqual(c.curvature, s.CONSTANT)
        self.assertEqual(c.sign, s.NONNEG)
        self.assertEqual(Constant(-2).sign, s.NONPOS)
        self.assertEqual(Constant(0).sign, s.ZERO)
        # self.assertEqual(c.canonical_form[0].shape, (1, 1))
        # self.assertEqual(c.canonical_form[1], [])

        # coeffs = c.coefficients()
        # self.assertEqual(coeffs.keys(), [s.CONSTANT])
        # self.assertEqual(coeffs[s.CONSTANT], [2])

        # Test the sign.
        c = Constant([[2], [2]])
        self.assertEqual(c.shape, (1, 2))
        self.assertEqual(c.sign, s.NONNEG)
        self.assertEqual((-c).sign, s.NONPOS)
        self.assertEqual((0*c).sign, s.ZERO)
        c = Constant([[2], [-2]])
        self.assertEqual(c.sign, s.UNKNOWN)

        c = Constant(np.zeros((2, 1)))
        self.assertEqual(c.shape, (2, 1))

        # Test sign of a complex expression.
        c = Constant([1, 2])
        self.assertEqual(c.shape, (2,))
        A = Constant([[1, 1], [1, 1]])
        exp = c.T @ A @ c
        self.assertEqual(exp.sign, s.NONNEG)
        self.assertEqual((c.T @ c).sign, s.NONNEG)
        exp = c.T.T
        self.assertEqual(exp.sign, s.NONNEG)
        exp = c.T @ self.A
        self.assertEqual(exp.sign, s.UNKNOWN)

        # Test repr.
        self.assertEqual(repr(c), "Constant(CONSTANT, NONNEGATIVE, (2,))")

    def test_1D_array(self):
        """Test NumPy 1D arrays as constants.
        """
        c = np.array([1, 2])
        p = Parameter(2)
        p.value = [1, 1]
        self.assertEqual((c @ p).value, 3)
        self.assertEqual((c @ self.x).shape, tuple())

    # Test Parameter class on good inputs.
    def test_parameters_successes(self):
        # Parameter names and dimensions
        p = Parameter(name='p')
        self.assertEqual(p.name(), "p")
        self.assertEqual(p.shape, tuple())

        # Entry-wise constraints on parameter values.
        val = -np.ones((4, 3))
        val[0, 0] = 2
        p = Parameter((4, 3))
        p.value = val

        # Initialize a parameter with a value; later, set it to None.
        p = Parameter(value=10)
        self.assertEqual(p.value, 10)
        p.value = 10
        p.value = None
        self.assertEqual(p.value, None)

        # Test parameter representation.
        p = Parameter((4, 3), nonpos=True)
        self.assertEqual(repr(p), 'Parameter((4, 3), nonpos=True)')

        # Test valid diagonal parameter.
        p = Parameter((2, 2), diag=True)
        p.value = sp.csc_matrix(np.eye(2))
        self.assertItemsAlmostEqual(p.value.todense(), np.eye(2), places=10)

    def test_psd_nsd_parameters(self):
        # Test valid rank-deficeint PSD parameter.
        np.random.seed(42)
        a = np.random.normal(size=(100, 95))
        a2 = a.dot(a.T)  # This must be a PSD matrix.
        p = Parameter((100, 100), PSD=True)
        p.value = a2
        self.assertItemsAlmostEqual(p.value, a2, places=10)

        # Test positive definite matrix with non-distinct eigenvalues
        m, n = 10, 5
        A = np.random.randn(m, n) + 1j * np.random.randn(m, n)  # a random complex matrix
        A = np.dot(A.T.conj(), A)  # a random Hermitian positive definite matrix
        A = np.vstack([np.hstack([np.real(A), -np.imag(A)]),
                       np.hstack([np.imag(A), np.real(A)])])

        p = Parameter(shape=(2*n, 2*n), PSD=True)
        p.value = A
        self.assertItemsAlmostEqual(p.value, A)

        # Test arithmetic.
        p = Parameter(shape=(2, 2), PSD=True)
        self.assertTrue((2*p).is_psd())
        self.assertTrue((p + p).is_psd())
        self.assertTrue((-p).is_nsd())
        self.assertTrue(((-2)*(-p)).is_psd())

        # Test invalid PSD and NSD parameters
        n = 5
        P = Parameter(shape=(n, n), PSD=True)
        N = Parameter(shape=(n, n), NSD=True)
        np.random.randn(0)
        U = np.random.randn(n, n)
        U = U @ U.T
        (evals, U) = np.linalg.eigh(U)  # U is now an orthogonal matrix
        v1 = np.array([3, 2, 1, 1e-8, -1])
        v2 = np.array([3, 2, 2, 1e-6, -1])
        v3 = np.array([3, 2, 2, 1e-4, -1e-6])
        vs = [v1, v2, v3]
        for vi in vs:
            with self.assertRaises(Exception) as cm:
                P.value = U @ np.diag(vi) @ U.T
            self.assertEqual(str(cm.exception), "Parameter value must be positive semidefinite.")
            with self.assertRaises(Exception) as cm:
                N.value = -U @ np.diag(vi) @ U.T
            self.assertEqual(str(cm.exception), "Parameter value must be negative semidefinite.")

    # Test the Parameter class on bad inputs.
    def test_parameters_failures(self):
        p = Parameter(name='p')
        self.assertEqual(p.name(), "p")
        self.assertEqual(p.shape, tuple())

        p = Parameter((4, 3), nonneg=True)
        with self.assertRaises(Exception) as cm:
            p.value = 1
        self.assertEqual(str(cm.exception), "Invalid dimensions () for Parameter value.")

        val = -np.ones((4, 3))
        val[0, 0] = 2

        p = Parameter((4, 3), nonneg=True)
        with self.assertRaises(Exception) as cm:
            p.value = val
        self.assertEqual(str(cm.exception), "Parameter value must be nonnegative.")

        p = Parameter((4, 3), nonpos=True)
        with self.assertRaises(Exception) as cm:
            p.value = val
        self.assertEqual(str(cm.exception), "Parameter value must be nonpositive.")

        with self.assertRaises(Exception) as cm:
            p = Parameter(2, 1, nonpos=True, value=[2, 1])
        self.assertEqual(str(cm.exception), "Parameter value must be nonpositive.")

        with self.assertRaises(Exception) as cm:
            p = Parameter((4, 3), nonneg=True, value=[1, 2])
        self.assertEqual(str(cm.exception), "Invalid dimensions (2,) for Parameter value.")

        with self.assertRaises(Exception) as cm:
            p = Parameter((2, 2), diag=True, symmetric=True)
        self.assertEqual(str(cm.exception),
                         "Cannot set more than one special attribute in Parameter.")

        # Boolean
        with self.assertRaises(Exception) as cm:
            p = Parameter((2, 2), boolean=True, value=[[1, 1], [1, -1]])
        self.assertEqual(str(cm.exception), "Parameter value must be boolean.")

        # Integer
        with self.assertRaises(Exception) as cm:
            p = Parameter((2, 2), integer=True, value=[[1, 1.5], [1, -1]])
        self.assertEqual(str(cm.exception), "Parameter value must be integer.")

        # Diag.
        with self.assertRaises(Exception) as cm:
            p = Parameter((2, 2), diag=True, value=[[1, 1], [1, -1]])
        self.assertEqual(str(cm.exception), "Parameter value must be diagonal.")

        # Symmetric.
        with self.assertRaises(Exception) as cm:
            p = Parameter((2, 2), symmetric=True, value=[[1, 1], [-1, -1]])
        self.assertEqual(str(cm.exception), "Parameter value must be symmetric.")

    def test_symmetric(self):
        """Test symmetric variables.
        """
        with self.assertRaises(Exception) as cm:
            v = Variable((4, 3), symmetric=True)
        self.assertEqual(str(cm.exception), "Invalid dimensions (4, 3). Must be a square matrix.")

        v = Variable((2, 2), symmetric=True)
        assert v.is_symmetric()
        v = Variable((2, 2), PSD=True)
        assert v.is_symmetric()
        v = Variable((2, 2), NSD=True)
        assert v.is_symmetric()
        v = Variable((2, 2), diag=True)
        assert v.is_symmetric()
        assert self.a.is_symmetric()
        assert not self.A.is_symmetric()

        v = Variable((2, 2), symmetric=True)
        expr = v + v
        assert expr.is_symmetric()
        expr = -v
        assert expr.is_symmetric()
        expr = v.T
        assert expr.is_symmetric()
        expr = cp.real(v)
        assert expr.is_symmetric()
        expr = cp.imag(v)
        assert expr.is_symmetric()
        expr = cp.conj(v)
        assert expr.is_symmetric()
        expr = cp.promote(Variable(), (2, 2))
        assert expr.is_symmetric()

    def test_hermitian(self):
        """Test Hermitian variables.
        """
        with self.assertRaises(Exception) as cm:
            v = Variable((4, 3), hermitian=True)
        self.assertEqual(str(cm.exception), "Invalid dimensions (4, 3). Must be a square matrix.")

        v = Variable((2, 2), hermitian=True)
        assert v.is_hermitian()
        # v = Variable((2,2), PSD=True)
        # assert v.is_symmetric()
        # v = Variable((2,2), NSD=True)
        # assert v.is_symmetric()
        v = Variable((2, 2), diag=True)
        assert v.is_hermitian()

        v = Variable((2, 2), hermitian=True)
        expr = v + v
        assert expr.is_hermitian()
        expr = -v
        assert expr.is_hermitian()
        expr = v.T
        assert expr.is_hermitian()
        expr = cp.real(v)
        assert expr.is_hermitian()
        expr = cp.imag(v)
        assert expr.is_hermitian()
        expr = cp.conj(v)
        assert expr.is_hermitian()
        expr = cp.promote(Variable(), (2, 2))
        assert expr.is_hermitian()

    def test_round_attr(self):
        """Test rounding for attributes.
        """
        # Nonpos
        v = Variable(1, nonpos=True)
        self.assertAlmostEqual(v.project(1), 0)
        v = Variable(2, nonpos=True)
        self.assertItemsAlmostEqual(v.project(np.array([1, -1])), [0, -1])

        # Nonneg
        v = Variable(1, nonneg=True)
        self.assertAlmostEqual(v.project(-1), 0)
        v = Variable(2, nonneg=True)
        self.assertItemsAlmostEqual(v.project(np.array([1, -1])), [1, 0])

        # Boolean
        v = Variable((2, 2), boolean=True)
        self.assertItemsAlmostEqual(v.project(np.array([[1, -1], [1, 0]]).T),
                                    [1, 0, 1, 0])

        # Integer
        v = Variable((2, 2), integer=True)
        self.assertItemsAlmostEqual(v.project(np.array([[1, -1.6], [1, 0]]).T),
                                    [1, -2, 1, 0])

        # Symmetric
        v = Variable((2, 2), symmetric=True)
        self.assertItemsAlmostEqual(v.project(np.array([[1, -1], [1, 0]])),
                                    [1, 0, 0, 0])

        # PSD
        v = Variable((2, 2), PSD=True)
        self.assertItemsAlmostEqual(v.project(np.array([[1, -1], [1, -1]])),
                                    [1, 0, 0, 0])

        # NSD
        v = Variable((2, 2), NSD=True)
        self.assertItemsAlmostEqual(v.project(np.array([[1, -1], [1, -1]])),
                                    [0, 0, 0, -1])

        # diag
        v = Variable((2, 2), diag=True)
        self.assertItemsAlmostEqual(v.project(np.array([[1, -1], [1, 0]])).todense(),
                                    [1, 0, 0, 0])

        # Hermitian
        v = Variable((2, 2), hermitian=True)
        self.assertItemsAlmostEqual(v.project(np.array([[1, -1j], [1, 0]])),
                                    [1, 0.5+0.5j, 0.5-0.5j, 0])

        A = Constant(np.array([[1.0]]))
        self.assertEqual(A.is_psd(), True)
        self.assertEqual(A.is_nsd(), False)
        A = Constant(np.array([[-1.0]]))
        self.assertEqual(A.is_psd(), False)
        self.assertEqual(A.is_nsd(), True)
        A = Constant(np.array([[0.0]]))
        self.assertEqual(A.is_psd(), True)
        self.assertEqual(A.is_nsd(), True)

    # Test the AddExpresion class.
    def test_add_expression(self):
        # Vectors
        c = Constant([2, 2])
        exp = self.x + c
        self.assertEqual(exp.curvature, s.AFFINE)
        self.assertEqual(exp.sign, s.UNKNOWN)
        # self.assertEqual(exp.canonical_form[0].shape, (2, 1))
        # self.assertEqual(exp.canonical_form[1], [])
        # self.assertEqual(exp.name(), self.x.name() + " + " + c.name())
        self.assertEqual(exp.shape, (2,))

        z = Variable(2, name='z')
        exp = exp + z + self.x

        # Incompatible dimensions
        with self.assertRaises(ValueError):
            (self.x + self.y)

        # Matrices
        exp = self.A + self.B
        self.assertEqual(exp.curvature, s.AFFINE)
        self.assertEqual(exp.shape, (2, 2))

        # Incompatible dimensions
        with self.assertRaises(ValueError):
            (self.A + self.C)

        # Incompatible dimensions
        with self.assertRaises(ValueError):
            AddExpression([self.A, self.C])

        # Test that sum is flattened.
        exp = self.x + c + self.x
        self.assertEqual(len(exp.args), 3)

        # Test repr.
        self.assertEqual(repr(exp), "Expression(AFFINE, UNKNOWN, (2,))")

    # Test the SubExpresion class.
    def test_sub_expression(self):
        # Vectors
        c = Constant([2, 2])
        exp = self.x - c
        self.assertEqual(exp.curvature, s.AFFINE)
        self.assertEqual(exp.sign, s.UNKNOWN)
        # self.assertEqual(exp.canonical_form[0].shape, (2, 1))
        # self.assertEqual(exp.canonical_form[1], [])
        # self.assertEqual(exp.name(), self.x.name() + " - " + Constant([2,2]).name())
        self.assertEqual(exp.shape, (2,))

        z = Variable(2, name='z')
        exp = exp - z - self.x

        # Incompatible dimensions
        with self.assertRaises(ValueError):
            (self.x - self.y)

        # Matrices
        exp = self.A - self.B
        self.assertEqual(exp.curvature, s.AFFINE)
        self.assertEqual(exp.shape, (2, 2))

        # Incompatible dimensions
        with self.assertRaises(ValueError):
            (self.A - self.C)

        # Test repr.
        self.assertEqual(repr(self.x - c), "Expression(AFFINE, UNKNOWN, (2,))")

    # Test the MulExpresion class.
    def test_mul_expression(self):
        # Vectors
        c = Constant([[2], [2]])
        exp = c @ self.x
        self.assertEqual(exp.curvature, s.AFFINE)
        self.assertEqual((c[0] @ self.x).sign, s.UNKNOWN)
        # self.assertEqual(exp.canonical_form[0].shape, (1, 1))
        # self.assertEqual(exp.canonical_form[1], [])
        # self.assertEqual(exp.name(), c.name() + " * " + self.x.name())
        self.assertEqual(exp.shape, (1,))

        # Incompatible dimensions
        with self.assertRaises(ValueError):
            ([2, 2, 3] @ self.x)

        # Matrices: Incompatible dimensions
        with self.assertRaises(ValueError):
            Constant([[2, 1], [2, 2]]) @ self.C

        # Affine times affine is okay
        with warnings.catch_warnings():
            warnings.simplefilter("ignore")
            q = self.A @ self.B
            self.assertTrue(q.is_quadratic())

        # Constant expressions
        T = Constant([[1, 2, 3], [3, 5, 5]])
        exp = (T + T) @ self.B
        self.assertEqual(exp.curvature, s.AFFINE)
        self.assertEqual(exp.shape, (3, 2))

        # Expression that would break sign multiplication without promotion.
        c = Constant([[2], [2], [-2]])
        exp = [[1], [2]] + c @ self.C
        self.assertEqual(exp.sign, s.UNKNOWN)

        # By default, warnings are raised if we access matmul from *.
        c = Constant([[2], [2]])
        with warnings.catch_warnings(record=True) as w:
            c * self.x
            self.assertEqual(2, len(w))
            self.assertEqual(w[0].category, UserWarning)
            self.assertEqual(w[1].category, DeprecationWarning)
            # repeat, to make sure warnings continue to be displayed
            c * self.x
            self.assertEqual(4, len(w))
            self.assertEqual(w[2].category, UserWarning)
            self.assertEqual(w[3].category, DeprecationWarning)
            # suppress one of the two warnings
            warnings.simplefilter('ignore', DeprecationWarning)
            c * self.x
            self.assertEqual(5, len(w))
            # suppress both warnings
            warnings.simplefilter('ignore', UserWarning)
            c * self.x
            self.assertEqual(len(w), 5)
            # verify that an error can be raised.
            warnings.simplefilter("error", UserWarning)
            with self.assertRaises(UserWarning):
                c * self.x

    def test_matmul_expression(self):
        """Test matmul function, corresponding to .__matmul__( operator.
        """
        # Vectors
        c = Constant([[2], [2]])
        exp = c.__matmul__(self.x)
        self.assertEqual(exp.curvature, s.AFFINE)
        self.assertEqual(exp.sign, s.UNKNOWN)
        # self.assertEqual(exp.name(), c.name() + " .__matmul__( " + self.x.name())
        self.assertEqual(exp.shape, (1,))

        with self.assertRaises(Exception) as cm:
            self.x.__matmul__(2)
        self.assertEqual(str(cm.exception),
                         "Scalar operands are not allowed, use '*' instead")

        # Incompatible dimensions
        with self.assertRaises(ValueError) as cm:
            (self.x.__matmul__(np.array([2, 2, 3])))

        # Incompatible dimensions
        with self.assertRaises(Exception) as cm:
            Constant([[2, 1], [2, 2]]) .__matmul__(self.C)

        # Affine times affine is okay
        with warnings.catch_warnings():
            warnings.simplefilter("ignore")
            q = self.A .__matmul__(self.B)
            self.assertTrue(q.is_quadratic())

        # # Nonaffine times nonconstant raises error
        # with warnings.catch_warnings():
        #     warnings.simplefilter("ignore")
        #     with self.assertRaises(Exception) as cm:
        #         (self.A.__matmul__(self.B).__matmul__(self.A))
        #     self.assertEqual(str(cm.exception), "Cannot multiply UNKNOWN and AFFINE.")

        # Constant expressions
        T = Constant([[1, 2, 3], [3, 5, 5]])
        exp = (T + T) .__matmul__(self.B)
        self.assertEqual(exp.curvature, s.AFFINE)
        self.assertEqual(exp.shape, (3, 2))

        # Expression that would break sign multiplication without promotion.
        c = Constant([[2], [2], [-2]])
        exp = [[1], [2]] + c.__matmul__(self.C)
        self.assertEqual(exp.sign, s.UNKNOWN)

        # Testing shape.
        a = Parameter((1,))
        x = Variable(shape=(1,))
        expr = a.__matmul__(x)
        self.assertEqual(expr.shape, ())

        # Testing shape.
        a = Parameter((1,))
        x = Variable(shape=(1,))
        expr = a.__matmul__(x)
        self.assertEqual(expr.shape, ())

        A = Parameter((4, 4))
        z = Variable((4, 1))
        expr = A.__matmul__(z)
        self.assertEqual(expr.shape, (4, 1))

        v = Variable((1, 1))
        col_scalar = Parameter((1, 1))
        assert v.shape == col_scalar.shape == col_scalar.T.shape

    # Test the DivExpresion class.
    def test_div_expression(self):
        # Vectors
        exp = self.x/2
        self.assertEqual(exp.curvature, s.AFFINE)
        self.assertEqual(exp.sign, s.UNKNOWN)
        # self.assertEqual(exp.canonical_form[0].shape, (2, 1))
        # self.assertEqual(exp.canonical_form[1], [])
        # self.assertEqual(exp.name(), c.name() + " * " + self.x.name())
        self.assertEqual(exp.shape, (2,))

        with self.assertRaises(Exception) as cm:
            (self.x/[2, 2, 3])
        print(cm.exception)
        self.assertRegexpMatches(str(cm.exception),
                                 "Incompatible shapes for division.*")

        c = Constant([3.0, 4.0, 12.0])
        self.assertItemsAlmostEqual(
          (c / Constant([1.0, 2.0, 3.0])).value, np.array([3.0, 2.0, 4.0]))

        # Constant expressions.
        c = Constant(2)
        exp = c/(3 - 5)
        self.assertEqual(exp.curvature, s.CONSTANT)
        self.assertEqual(exp.shape, tuple())
        self.assertEqual(exp.sign, s.NONPOS)

        # Parameters.
        p = Parameter(nonneg=True)
        exp = 2/p
        p.value = 2
        self.assertEqual(exp.value, 1)

        rho = Parameter(nonneg=True)
        rho.value = 1

        self.assertEqual(rho.sign, s.NONNEG)
        self.assertEqual(Constant(2).sign, s.NONNEG)
        self.assertEqual((Constant(2)/Constant(2)).sign, s.NONNEG)
        self.assertEqual((Constant(2)*rho).sign, s.NONNEG)
        self.assertEqual((rho/2).sign, s.NONNEG)

        # Broadcasting.
        x = cp.Variable((3, 3))
        c = np.arange(1, 4)[:, None]
        expr = x / c
        self.assertEqual((3, 3), expr.shape)
        x.value = np.ones((3, 3))
        A = np.ones((3, 3)) / c
        self.assertItemsAlmostEqual(A, expr.value)

        with self.assertRaises(Exception) as cm:
            (x/c[:, 0])
        print(cm.exception)
        self.assertRegexpMatches(str(cm.exception),
                                 "Incompatible shapes for division.*")

    # Test the NegExpression class.
    def test_neg_expression(self):
        # Vectors
        exp = -self.x
        self.assertEqual(exp.curvature, s.AFFINE)
        self.assertEqual(exp.shape, (2,))
        assert exp.is_affine()
        self.assertEqual(exp.sign, s.UNKNOWN)
        assert not exp.is_nonneg()
        # self.assertEqual(exp.canonical_form[0].shape, (2, 1))
        # self.assertEqual(exp.canonical_form[1], [])
        # self.assertEqual(exp.name(), "-%s" % self.x.name())
        self.assertEqual(exp.shape, self.x.shape)

        # Matrices
        exp = -self.C
        self.assertEqual(exp.curvature, s.AFFINE)
        self.assertEqual(exp.shape, (3, 2))

    # Test promotion of scalar constants.
    def test_scalar_const_promotion(self):
        # Vectors
        exp = self.x + 2
        self.assertEqual(exp.curvature, s.AFFINE)
        assert exp.is_affine()
        self.assertEqual(exp.sign, s.UNKNOWN)
        assert not exp.is_nonpos()
        # self.assertEqual(exp.canonical_form[0].shape, (2, 1))
        # self.assertEqual(exp.canonical_form[1], [])
        # self.assertEqual(exp.name(), self.x.name() + " + " + Constant(2).name())
        self.assertEqual(exp.shape, (2,))

        self.assertEqual((4 - self.x).shape, (2,))
        self.assertEqual((4 * self.x).shape, (2,))
        self.assertEqual((4 <= self.x).shape, (2,))
        self.assertEqual((4 == self.x).shape, (2,))
        self.assertEqual((self.x >= 4).shape, (2,))

        # Matrices
        exp = (self.A + 2) + 4
        self.assertEqual(exp.curvature, s.AFFINE)
        self.assertEqual((3 * self.A).shape, (2, 2))

        self.assertEqual(exp.shape, (2, 2))

    # Test indexing expression.
    def test_index_expression(self):
        # Tuple of integers as key.
        exp = self.x[1]
        # self.assertEqual(exp.name(), "x[1,0]")
        self.assertEqual(exp.curvature, s.AFFINE)
        assert exp.is_affine()
        self.assertEqual(exp.shape, tuple())
        # coeff = exp.canonical_form[0].coefficients()[self.x][0]
        # self.assertEqual(coeff[0,1], 1)
        self.assertEqual(exp.value, None)

        exp = self.x[1].T
        # self.assertEqual(exp.name(), "x[1,0]")
        self.assertEqual(exp.curvature, s.AFFINE)
        self.assertEqual(exp.shape, tuple())

        with self.assertRaises(Exception) as cm:
            (self.x[2, 0])
        self.assertEqual(str(cm.exception), "Too many indices for expression.")

        with self.assertRaises(Exception) as cm:
            (self.x[2])
        self.assertEqual(str(cm.exception), "Index 2 is out of bounds for axis 0 with size 2.")

        # Slicing
        exp = self.C[0:2, 1]
        # self.assertEqual(exp.name(), "C[0:2,1]")
        self.assertEqual(exp.shape, (2,))
        exp = self.C[0:, 0:2]
        # self.assertEqual(exp.name(), "C[0:,0:2]")
        self.assertEqual(exp.shape, (3, 2))
        exp = self.C[0::2, 0::2]
        # self.assertEqual(exp.name(), "C[0::2,0::2]")
        self.assertEqual(exp.shape, (2, 1))
        exp = self.C[:3, :1:2]
        # self.assertEqual(exp.name(), "C[0:3,0]")
        self.assertEqual(exp.shape, (3, 1))
        exp = self.C[0:, 0]
        # self.assertEqual(exp.name(), "C[0:,0]")
        self.assertEqual(exp.shape, (3,))

        c = Constant([[1, -2], [0, 4]])
        exp = c[1, 1]
        self.assertEqual(exp.curvature, s.CONSTANT)
        self.assertEqual(exp.sign, s.UNKNOWN)
        self.assertEqual(c[0, 1].sign, s.UNKNOWN)
        self.assertEqual(c[1, 0].sign, s.UNKNOWN)
        self.assertEqual(exp.shape, tuple())
        self.assertEqual(exp.value, 4)

        c = Constant([[1, -2, 3], [0, 4, 5], [7, 8, 9]])
        exp = c[0:3, 0:4:2]
        self.assertEqual(exp.curvature, s.CONSTANT)
        assert exp.is_constant()
        self.assertEqual(exp.shape, (3, 2))
        self.assertEqual(exp[0, 1].value, 7)

        # Slice of transpose
        exp = self.C.T[0:2, 1:2]
        self.assertEqual(exp.shape, (2, 1))

        # Arithmetic expression indexing
        exp = (self.x + self.z)[1]
        # self.assertEqual(exp.name(), "x[1,0] + z[1,0]")
        self.assertEqual(exp.curvature, s.AFFINE)
        self.assertEqual(exp.sign, s.UNKNOWN)
        self.assertEqual(exp.shape, tuple())

        exp = (self.x + self.a)[1:2]
        # self.assertEqual(exp.name(), "x[1,0] + a")
        self.assertEqual(exp.curvature, s.AFFINE)
        self.assertEqual(exp.shape, (1,))

        exp = (self.x - self.z)[1:2]
        # self.assertEqual(exp.name(), "x[1,0] - z[1,0]")
        self.assertEqual(exp.curvature, s.AFFINE)
        self.assertEqual(exp.shape, (1,))

        exp = (self.x - self.a)[1]
        # self.assertEqual(exp.name(), "x[1,0] - a")
        self.assertEqual(exp.curvature, s.AFFINE)
        self.assertEqual(exp.shape, tuple())

        exp = (-self.x)[1]
        # self.assertEqual(exp.name(), "-x[1,0]")
        self.assertEqual(exp.curvature, s.AFFINE)
        self.assertEqual(exp.shape, tuple())

        c = Constant([[1, 2], [3, 4]])
        exp = (c @ self.x)[1]
        # self.assertEqual(exp.name(), "[[2], [4]] * x[0:,0]")
        self.assertEqual(exp.curvature, s.AFFINE)
        self.assertEqual(exp.shape, tuple())

        c = Constant([[1, 2], [3, 4]])
        exp = (c * self.a)[1, 0:1]
        # self.assertEqual(exp.name(), "2 * a")
        self.assertEqual(exp.curvature, s.AFFINE)
        self.assertEqual(exp.shape, (1,))

    def test_none_idx(self):
        """Test None as index.
        """
        expr = self.a[None, None]
        self.assertEqual(expr.shape, (1, 1))

        expr = self.x[:, None]
        self.assertEqual(expr.shape, (2, 1))

        expr = self.x[None, :]
        self.assertEqual(expr.shape, (1, 2))

        expr = Constant([1, 2])[None, :]
        self.assertEqual(expr.shape, (1, 2))
        self.assertItemsAlmostEqual(expr.value, [1, 2])

    def test_out_of_bounds(self):
        """Test out of bounds indices.
        """
        with self.assertRaises(Exception) as cm:
            self.x[100]
        self.assertEqual(str(cm.exception), "Index 100 is out of bounds for axis 0 with size 2.")

        with self.assertRaises(Exception) as cm:
            self.x[-100]
        self.assertEqual(str(cm.exception), "Index -100 is out of bounds for axis 0 with size 2.")

        exp = self.x[:-100]
        self.assertEqual(exp.size, (0,))
        self.assertItemsAlmostEqual(exp.value, np.array([]))

        exp = self.C[100:2]
        self.assertEqual(exp.shape, (0, 2))

        exp = self.C[:, -199:2]
        self.assertEqual(exp.shape, (3, 2))

        exp = self.C[:, -199:-3]
        self.assertEqual(exp.shape, (3, 0))

    def test_neg_indices(self):
        """Test negative indices.
        """
        c = Constant([[1, 2], [3, 4]])
        exp = c[-1, -1]
        self.assertEqual(exp.value, 4)
        self.assertEqual(exp.shape, tuple())
        self.assertEqual(exp.curvature, s.CONSTANT)

        c = Constant([1, 2, 3, 4])
        exp = c[1:-1]
        self.assertItemsAlmostEqual(exp.value, [2, 3])
        self.assertEqual(exp.shape, (2,))
        self.assertEqual(exp.curvature, s.CONSTANT)

        c = Constant([1, 2, 3, 4])
        exp = c[::-1]
        self.assertItemsAlmostEqual(exp.value, [4, 3, 2, 1])
        self.assertEqual(exp.shape, (4,))
        self.assertEqual(exp.curvature, s.CONSTANT)

        x = Variable(4)
        self.assertEqual(x[::-1].shape, (4,))
        Problem(Minimize(0), [x[::-1] == c]).solve()
        self.assertItemsAlmostEqual(x.value, [4, 3, 2, 1])

        x = Variable(2)
        self.assertEqual(x[::-1].shape, (2,))

        x = Variable(100, name="x")
        self.assertEqual("x[0:99]", str(x[:-1]))

        c = Constant([[1, 2], [3, 4]])
        expr = c[0, 2:0:-1]
        self.assertEqual(expr.shape, (1,))
        self.assertAlmostEqual(expr.value, 3)

        expr = c[0, 2::-1]
        self.assertEqual(expr.shape, (2,))
        self.assertItemsAlmostEqual(expr.value, [3, 1])

    def test_logical_indices(self):
        """Test indexing with boolean arrays.
        """
        A = np.array([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12]])
        C = Constant(A)

        # Boolean array.
        expr = C[A <= 2]
        self.assertEqual(expr.shape, (2,))
        self.assertEqual(expr.sign, s.NONNEG)
        self.assertItemsAlmostEqual(A[A <= 2], expr.value)

        expr = C[A % 2 == 0]
        self.assertEqual(expr.shape, (6,))
        self.assertEqual(expr.sign, s.NONNEG)
        self.assertItemsAlmostEqual(A[A % 2 == 0], expr.value)

        # Boolean array for rows, index for columns.
        expr = C[np.array([True, False, True]), 3]
        self.assertEqual(expr.shape, (2,))
        self.assertEqual(expr.sign, s.NONNEG)
        self.assertItemsAlmostEqual(A[np.array([True, False, True]), 3], expr.value)

        # Index for row, boolean array for columns.
        expr = C[1, np.array([True, False, False, True])]
        self.assertEqual(expr.shape, (2,))
        self.assertEqual(expr.sign, s.NONNEG)
        self.assertItemsAlmostEqual(A[1, np.array([True, False, False, True])],
                                    expr.value)

        # Boolean array for rows, slice for columns.
        expr = C[np.array([True, True, True]), 1:3]
        self.assertEqual(expr.shape, (3, 2))
        self.assertEqual(expr.sign, s.NONNEG)
        self.assertItemsAlmostEqual(A[np.array([True, True, True]), 1:3], expr.value)

        # Slice for row, boolean array for columns.
        expr = C[1:-1, np.array([True, False, True, True])]
        self.assertEqual(expr.shape, (1, 3))
        self.assertEqual(expr.sign, s.NONNEG)
        self.assertItemsAlmostEqual(A[1:-1, np.array([True, False, True, True])],
                                    expr.value)

        # Boolean arrays for rows and columns.
        # Not sure what this does.
        expr = C[np.array([True, True, True]),
                 np.array([True, False, True, True])]
        self.assertEqual(expr.shape, (3,))
        self.assertEqual(expr.sign, s.NONNEG)
        self.assertItemsAlmostEqual(A[np.array([True, True, True]),
                                      np.array([True, False, True, True])], expr.value)

    def test_selector_list_indices(self):
        """Test indexing with lists/ndarrays of indices.
        """
        A = np.array([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12]])
        C = Constant(A)

        # List for rows.
        expr = C[[1, 2]]
        self.assertEqual(expr.shape, (2, 4))
        self.assertEqual(expr.sign, s.NONNEG)
        self.assertItemsAlmostEqual(A[[1, 2]], expr.value)

        # List for rows, index for columns.
        expr = C[[0, 2], 3]
        self.assertEqual(expr.shape, (2,))
        self.assertEqual(expr.sign, s.NONNEG)
        self.assertItemsAlmostEqual(A[[0, 2], 3], expr.value)

        # Index for row, list for columns.
        expr = C[1, [0, 2]]
        self.assertEqual(expr.shape, (2,))
        self.assertEqual(expr.sign, s.NONNEG)
        self.assertItemsAlmostEqual(A[1, [0, 2]], expr.value)

        # List for rows, slice for columns.
        expr = C[[0, 2], 1:3]
        self.assertEqual(expr.shape, (2, 2))
        self.assertEqual(expr.sign, s.NONNEG)
        self.assertItemsAlmostEqual(A[[0, 2], 1:3], expr.value)

        # Slice for row, list for columns.
        expr = C[1:-1, [0, 2]]
        self.assertEqual(expr.shape, (1, 2))
        self.assertEqual(expr.sign, s.NONNEG)
        self.assertItemsAlmostEqual(A[1:-1, [0, 2]], expr.value)

        # Lists for rows and columns.
        expr = C[[0, 1], [1, 3]]
        self.assertEqual(expr.shape, (2,))
        self.assertEqual(expr.sign, s.NONNEG)
        self.assertItemsAlmostEqual(A[[0, 1], [1, 3]], expr.value)

        # Ndarray for rows, list for columns.
        expr = C[np.array([0, 1]), [1, 3]]
        self.assertEqual(expr.shape, (2,))
        self.assertEqual(expr.sign, s.NONNEG)
        self.assertItemsAlmostEqual(A[np.array([0, 1]), [1, 3]], expr.value)

        # Ndarrays for rows and columns.
        expr = C[np.array([0, 1]), np.array([1, 3])]
        self.assertEqual(expr.shape, (2,))
        self.assertEqual(expr.sign, s.NONNEG)
        self.assertItemsAlmostEqual(A[np.array([0, 1]), np.array([1, 3])], expr.value)

    def test_powers(self):
        exp = self.x**2
        self.assertEqual(exp.curvature, s.CONVEX)
        exp = self.x**0.5
        self.assertEqual(exp.curvature, s.CONCAVE)
        exp = self.x**-1
        self.assertEqual(exp.curvature, s.CONVEX)

    def test_sum(self):
        """Test cvxpy sum function.
        """
        self.a.value = 1
        expr = cp.sum(self.a)
        self.assertEqual(expr.value, 1)

        self.x.value = [1, 2]
        expr = cp.sum(self.x)
        self.assertEqual(expr.value, 3)

    def test_var_copy(self):
        """Test the copy function for variable types.
        """
        x = Variable((3, 4), name="x")
        y = x.copy()
        self.assertEqual(y.shape, (3, 4))
        self.assertEqual(y.name(), "x")

        x = Variable((5, 5), PSD=True, name="x")
        y = x.copy()
        self.assertEqual(y.shape, (5, 5))

    def test_param_copy(self):
        """Test the copy function for Parameters.
        """
        x = Parameter((3, 4), name="x", nonneg=True)
        y = x.copy()
        self.assertEqual(y.shape, (3, 4))
        self.assertEqual(y.name(), "x")
        self.assertEqual(y.sign, "NONNEGATIVE")

    def test_constant_copy(self):
        """Test the copy function for Constants.
        """
        x = Constant(2)
        y = x.copy()
        self.assertEqual(y.shape, tuple())
        self.assertEqual(y.value, 2)

    def test_is_pwl(self):
        """Test is_pwl()
        """
        A = np.ones((2, 3))
        b = np.ones(2)

        expr = A @ self.y - b
        self.assertEqual(expr.is_pwl(), True)

        expr = cp.maximum(1, 3 * self.y)
        self.assertEqual(expr.is_pwl(), True)

        expr = cp.abs(self.y)
        self.assertEqual(expr.is_pwl(), True)

        expr = cp.pnorm(3 * self.y, 1)
        self.assertEqual(expr.is_pwl(), True)

        expr = cp.pnorm(3 * self.y ** 2, 1)
        self.assertEqual(expr.is_pwl(), False)

    def test_broadcast_mul(self):
        """Test multiply broadcasting.
        """
        y = Parameter((3, 1))
        z = Variable((1, 3))
        y.value = np.arange(3)[:, None]
        z.value = (np.arange(3) - 1)[None, :]
        expr = cp.multiply(y, z)
        self.assertItemsAlmostEqual(expr.value, y.value * z.value)

        prob = cp.Problem(cp.Minimize(cp.sum(expr)), [z == z.value])
        prob.solve()
        self.assertItemsAlmostEqual(expr.value, y.value * z.value)

        np.random.seed(0)
        m, n = 3, 4
        A = np.random.rand(m, n)

        col_scale = Variable(n)

        with self.assertRaises(ValueError) as cm:
            cp.multiply(A, col_scale)
        self.assertEqual(str(cm.exception), "Cannot broadcast dimensions  (3, 4) (4,)")

        col_scale = Variable([1, n])
        C = cp.multiply(A, col_scale)
        self.assertEqual(C.shape, (m, n))

        row_scale = Variable([m, 1])
        R = cp.multiply(A, row_scale)
        self.assertEqual(R.shape, (m, n))

    def test_broadcast_add(self):
        """Test addition broadcasting.
        """
        y = Parameter((3, 1))
        z = Variable((1, 3))
        y.value = np.arange(3)[:, None]
        z.value = (np.arange(3) - 1)[None, :]
        expr = y + z
        self.assertItemsAlmostEqual(expr.value, y.value + z.value)

        prob = cp.Problem(cp.Minimize(cp.sum(expr)), [z == z.value])
        prob.solve()
        self.assertItemsAlmostEqual(expr.value, y.value + z.value)

        np.random.seed(0)
        m, n = 3, 4
        A = np.random.rand(m, n)

        col_scale = Variable(n)

        with self.assertRaises(ValueError) as cm:
            A + col_scale
        self.assertEqual(str(cm.exception), "Cannot broadcast dimensions  (3, 4) (4,)")

        col_scale = Variable([1, n])
        C = A + col_scale
        self.assertEqual(C.shape, (m, n))

        row_scale = Variable([m, 1])
        R = A + row_scale
        self.assertEqual(R.shape, (m, n))
