Esempio n. 1
0
File: cir.py Progetto: giserh/cvxpy
 def graph_implementation(arg_objs, size, data=None):
     # min 1-sqrt(2z-z^2)
     # s.t. x>=0, z<=1, z = x+s, s<=0
     x = arg_objs[0]
     z = lu.create_var(size)
     s = lu.create_var(size)
     zeros = lu.create_const(np.mat(np.zeros(size)),size)
     ones = lu.create_const(np.mat(np.ones(size)),size)
     z2, constr_square = power.graph_implementation([z],size, (2, (Fraction(1,2), Fraction(1,2))))
     two_z = lu.sum_expr([z,z])
     sub = lu.sub_expr(two_z, z2)
     sq, constr_sqrt = power.graph_implementation([sub],size, (Fraction(1,2), (Fraction(1,2), Fraction(1,2))))
     obj = lu.sub_expr(ones, sq)
     constr = [lu.create_eq(z, lu.sum_expr([x,s]))]+[lu.create_leq(zeros,x)]+[lu.create_leq(z, ones)]+[lu.create_leq(s,zeros)]+constr_square+constr_sqrt
     return (obj, constr)
Esempio n. 2
0
    def graph_implementation(arg_objs, size, data=None):
        """Reduces the atom to an affine expression and list of constraints.

        Parameters
        ----------
        arg_objs : list
            LinExpr for each argument.
        size : tuple
            The size of the resulting expression.
        data :
            Additional data required by the atom.

        Returns
        -------
        tuple
            (LinOp for objective, list of constraints)
        """
        # min sum_entries(t) + kq
        # s.t. x <= t + q
        #      0 <= t
        x = arg_objs[0]
        k = lu.create_const(data[0], (1, 1))
        q = lu.create_var((1, 1))
        t = lu.create_var(x.size)
        sum_t, constr = sum_entries.graph_implementation([t], (1, 1))
        obj = lu.sum_expr([sum_t, lu.mul_expr(k, q, (1, 1))])
        prom_q = lu.promote(q, x.size)
        constr.append(lu.create_leq(x, lu.sum_expr([t, prom_q])))
        constr.append(lu.create_geq(t))
        return (obj, constr)
Esempio n. 3
0
    def graph_implementation(arg_objs, size, data=None):
        """Reduces the atom to an affine expression and list of constraints.

        Parameters
        ----------
        arg_objs : list
            LinExpr for each argument.
        size : tuple
            The size of the resulting expression.
        data :
            Additional data required by the atom.

        Returns
        -------
        tuple
            (LinOp for objective, list of constraints)
        """
        x = arg_objs[0]
        t = lu.create_var((1, 1))
        # sum(exp(x - t))
        prom_t = lu.promote(t, x.size)
        expr = lu.sub_expr(x, prom_t)
        obj, constraints = exp.graph_implementation([expr], x.size)
        obj, constr = sum_entries.graph_implementation([obj], (1, 1))
        # obj <= 1
        one = lu.create_const(1, (1, 1))
        constraints += constr + [lu.create_leq(obj, one)]
        return (t, constraints)
Esempio n. 4
0
    def graph_implementation(arg_objs, size, data=None):
        """Reduces the atom to an affine expression and list of constraints.

        Parameters
        ----------
        arg_objs : list
            LinExpr for each argument.
        size : tuple
            The size of the resulting expression.
        data :
            Additional data required by the atom.

        Returns
        -------
        tuple
            (LinOp for objective, list of constraints)
        """
        x = arg_objs[0]
        t = lu.create_var(size)

        # log(1 + exp(x)) <= t <=> exp(-t) + exp(x - t) <= 1
        obj0, constr0 = exp.graph_implementation([lu.neg_expr(t)], size)
        obj1, constr1 = exp.graph_implementation([lu.sub_expr(x, t)], size)
        lhs = lu.sum_expr([obj0, obj1])
        ones = lu.create_const(np.mat(np.ones(size)), size)
        constr = constr0 + constr1 + [lu.create_leq(lhs, ones)]

        return (t, constr)
Esempio n. 5
0
    def graph_implementation(arg_objs, size, data=None):
        """Reduces the atom to an affine expression and list of constraints.

        Parameters
        ----------
        arg_objs : list
            LinExpr for each argument.
        size : tuple
            The size of the resulting expression.
        data :
            Additional data required by the atom.

        Returns
        -------
        tuple
            (LinOp for objective, list of constraints)
        """
        x = arg_objs[0]
        w = lu.create_var(size)
        v = lu.create_var(size)
        two = lu.create_const(2, (1, 1))
        # w**2 + 2*v
        obj, constraints = square.graph_implementation([w], size)
        obj = lu.sum_expr([obj, lu.mul_expr(two, v, size)])
        # x <= w + v
        constraints.append(lu.create_leq(x, lu.sum_expr([w, v])))
        # v >= 0
        constraints.append(lu.create_geq(v))
        return (obj, constraints)
Esempio n. 6
0
    def graph_implementation(arg_objs, size, data=None):
        """Reduces the atom to an affine expression and list of constraints.

        Parameters
        ----------
        arg_objs : list
            LinExpr for each argument.
        size : tuple
            The size of the resulting expression.
        data :
            Additional data required by the atom.

        Returns
        -------
        tuple
            (LinOp for objective, list of constraints)
        """
        t = lu.create_var(size)
        constraints = []
        for obj in arg_objs:
            # Promote obj.
            if obj.size != size:
                obj = lu.promote(obj, size)
            constraints.append(lu.create_leq(obj, t))
        return (t, constraints)
Esempio n. 7
0
    def graph_implementation(arg_objs, size, data=None):
        """Reduces the atom to an affine expression and list of constraints.

        Parameters
        ----------
        arg_objs : list
            LinExpr for each argument.
        size : tuple
            The size of the resulting expression.
        data :
            Additional data required by the atom.

        Returns
        -------
        tuple
            (LinOp for objective, list of constraints)
        """
        x = arg_objs[0]
        t = lu.create_var((1, 1))
        # sum(exp(x - t))
        prom_t = lu.promote(t, x.size)
        expr = lu.sub_expr(x, prom_t)
        obj, constraints = exp.graph_implementation([expr], x.size)
        obj, constr = sum_entries.graph_implementation([obj], (1, 1))
        # obj <= 1
        one = lu.create_const(1, (1, 1))
        constraints += constr + [lu.create_leq(obj, one)]
        return (t, constraints)
Esempio n. 8
0
    def graph_implementation(arg_objs, size, data=None):
        """Reduces the atom to an affine expression and list of constraints.

        Parameters
        ----------
        arg_objs : list
            LinExpr for each argument.
        size : tuple
            The size of the resulting expression.
        data :
            Additional data required by the atom.

        Returns
        -------
        tuple
            (LinOp for objective, list of constraints)
        """
        t = lu.create_var(size)
        constraints = []
        for obj in arg_objs:
            # Promote obj.
            if obj.size != size:
                obj = lu.promote(obj, size)
            constraints.append(lu.create_leq(obj, t))
        return (t, constraints)
Esempio n. 9
0
    def graph_implementation(arg_objs, size, data=None):
        """Reduces the atom to an affine expression and list of constraints.

        Parameters
        ----------
        arg_objs : list
            LinExpr for each argument.
        size : tuple
            The size of the resulting expression.
        data :
            Additional data required by the atom.

        Returns
        -------
        tuple
            (LinOp for objective, list of constraints)
        """
        x = arg_objs[0]
        w = lu.create_var(size)
        v = lu.create_var(size)
        two = lu.create_const(2, (1, 1))
        # w**2 + 2*v
        obj, constraints = square.graph_implementation([w], size)
        obj = lu.sum_expr([obj, lu.mul_expr(two, v, size)])
        # x <= w + v
        constraints.append(lu.create_leq(x, lu.sum_expr([w, v])))
        # v >= 0
        constraints.append(lu.create_geq(v))
        return (obj, constraints)
Esempio n. 10
0
    def graph_implementation(arg_objs, size, data=None):
        """Reduces the atom to an affine expression and list of constraints.

        Parameters
        ----------
        arg_objs : list
            LinExpr for each argument.
        size : tuple
            The size of the resulting expression.
        data :
            Additional data required by the atom.

        Returns
        -------
        tuple
            (LinOp for objective, list of constraints)
        """
        # min sum_entries(t) + kq
        # s.t. x <= t + q
        #      0 <= t
        x = arg_objs[0]
        k = lu.create_const(data[0], (1, 1))
        q = lu.create_var((1, 1))
        t = lu.create_var(x.size)
        sum_t, constr = sum_entries.graph_implementation([t], (1, 1))
        obj = lu.sum_expr([sum_t, lu.mul_expr(k, q, (1, 1))])
        prom_q = lu.promote(q, x.size)
        constr.append( lu.create_leq(x, lu.sum_expr([t, prom_q])) )
        constr.append( lu.create_geq(t) )
        return (obj, constr)
Esempio n. 11
0
    def graph_implementation(arg_objs, size, data=None):
        """Reduces the atom to an affine expression and list of constraints.

        Parameters
        ----------
        arg_objs : list
            LinExpr for each argument.
        size : tuple
            The size of the resulting expression.
        data :
            Additional data required by the atom.

        Returns
        -------
        tuple
            (LinOp for objective, list of constraints)
        """
        axis = data[0]
        if axis is None:
            t = lu.create_var((1, 1))
            promoted_t = lu.promote(t, arg_objs[0].size)
        elif axis == 0:
            t = lu.create_var((1, arg_objs[0].size[1]))
            const_size = (arg_objs[0].size[0], 1)
            ones = lu.create_const(np.ones(const_size), const_size)
            promoted_t = lu.mul_expr(ones, t, arg_objs[0].size)
        else:  # axis == 1
            t = lu.create_var((arg_objs[0].size[0], 1))
            const_size = (1, arg_objs[0].size[1])
            ones = lu.create_const(np.ones(const_size), const_size)
            promoted_t = lu.rmul_expr(t, ones, arg_objs[0].size)

        constraints = [lu.create_leq(arg_objs[0], promoted_t)]
        return (t, constraints)
Esempio n. 12
0
    def graph_implementation(arg_objs, size, data=None):
        """Reduces the atom to an affine expression and list of constraints.

        Parameters
        ----------
        arg_objs : list
            LinExpr for each argument.
        size : tuple
            The size of the resulting expression.
        data :
            Additional data required by the atom.

        Returns
        -------
        tuple
            (LinOp for objective, list of constraints)
        """
        x = arg_objs[0]
        t = lu.create_var(size)

        # log(1 + exp(x)) <= t <=> exp(-t) + exp(x - t) <= 1
        obj0, constr0 = exp.graph_implementation([lu.neg_expr(t)], size)
        obj1, constr1 = exp.graph_implementation([lu.sub_expr(x, t)], size)
        lhs = lu.sum_expr([obj0, obj1])
        ones = lu.create_const(np.mat(np.ones(size)), size)
        constr = constr0 + constr1 + [lu.create_leq(lhs, ones)]

        return (t, constr)
Esempio n. 13
0
    def graph_implementation(arg_objs, size, data=None):
        """Reduces the atom to an affine expression and list of constraints.

        Parameters
        ----------
        arg_objs : list
            LinExpr for each argument.
        size : tuple
            The size of the resulting expression.
        data :
            Additional data required by the atom.

        Returns
        -------
        tuple
            (LinOp for objective, list of constraints)
        """
        x = arg_objs[0]
        t = lu.create_var((1, 1))
        promoted_t = lu.promote(t, x.size)
        constraints = [
            lu.create_geq(lu.sum_expr([x, promoted_t])),
            lu.create_leq(x, promoted_t)
        ]
        return (t, constraints)
Esempio n. 14
0
 def graph_implementation(arg_objs, size, data=None):
     # min 1-sqrt(2z-z^2)
     # s.t. x>=0, z<=1, z = x+s, s<=0
     x = arg_objs[0]
     z = lu.create_var(size)
     s = lu.create_var(size)
     zeros = lu.create_const(np.mat(np.zeros(size)), size)
     ones = lu.create_const(np.mat(np.ones(size)), size)
     z2, constr_square = power.graph_implementation(
         [z], size, (2, (Fraction(1, 2), Fraction(1, 2))))
     two_z = lu.sum_expr([z, z])
     sub = lu.sub_expr(two_z, z2)
     sq, constr_sqrt = power.graph_implementation(
         [sub], size, (Fraction(1, 2), (Fraction(1, 2), Fraction(1, 2))))
     obj = lu.sub_expr(ones, sq)
     constr = [lu.create_eq(z, lu.sum_expr([x, s]))] + [
         lu.create_leq(zeros, x)
     ] + [lu.create_leq(z, ones)] + [lu.create_leq(s, zeros)
                                     ] + constr_square + constr_sqrt
     return (obj, constr)
Esempio n. 15
0
    def graph_implementation(arg_objs, size, data=None):
        """Reduces the atom to an affine expression and list of constraints.

        Parameters
        ----------
        arg_objs : list
            LinExpr for each argument.
        size : tuple
            The size of the resulting expression.
        data :
            Additional data required by the atom.

        Returns
        -------
        tuple
            (LinOp for objective, list of constraints)
        """
        x = arg_objs[0]
        axis = data[0]
        t = lu.create_var(size)

        # sum(exp(x - t)) <= 1
        if axis is None:
            prom_t = lu.promote(t, x.size)
            expr = lu.sub_expr(x, prom_t)
            obj, constraints = exp.graph_implementation([expr], x.size)
            obj = lu.sum_entries(obj)

        elif axis == 0:
            prom_size = (x.size[0], 1)
            ones = lu.create_const(np.ones(prom_size), prom_size)
            prom_t = lu.mul_expr(ones, t, x.size)
            expr = lu.sub_expr(x, prom_t)
            obj, constraints = exp.graph_implementation([expr], x.size)

            const_size = (1, x.size[0])
            ones = lu.create_const(np.ones(const_size), const_size)
            obj = lu.mul_expr(ones, obj, size)

        else:  # axis == 1
            prom_size = (1, x.size[1])
            ones = lu.create_const(np.ones(prom_size), prom_size)
            prom_t = lu.rmul_expr(t, ones, x.size)
            expr = lu.sub_expr(x, prom_t)
            obj, constraints = exp.graph_implementation([expr], x.size)

            const_size = (x.size[1], 1)
            ones = lu.create_const(np.ones(const_size), const_size)
            obj = lu.rmul_expr(obj, ones, size)

        ones = lu.create_const(np.ones(size), size)
        constraints += [lu.create_leq(obj, ones)]

        return (t, constraints)
Esempio n. 16
0
 def constraints(self):
     obj, constraints = super(BoolVar, self).canonicalize()
     one = lu.create_const(1, (1, 1))
     constraints += [lu.create_geq(obj),
                     lu.create_leq(obj, one)]
     for i in range(self.size[0]):
         row_sum = lu.sum_expr([self[i, j] for j in range(self.size[0])])
         col_sum = lu.sum_expr([self[j, i] for j in range(self.size[0])])
         constraints += [lu.create_eq(row_sum, one),
                         lu.create_eq(col_sum, one)]
     return constraints
Esempio n. 17
0
    def graph_implementation(arg_objs, size, data=None):
        """Reduces the atom to an affine expression and list of constraints.

        Parameters
        ----------
        arg_objs : list
            LinExpr for each argument.
        size : tuple
            The size of the resulting expression.
        data :
            Additional data required by the atom.

        Returns
        -------
        tuple
            (LinOp for objective, list of constraints)
        """
        x = arg_objs[0]
        axis = data[0]
        t = lu.create_var(size)

        # sum(exp(x - t)) <= 1
        if axis is None:
            prom_t = lu.promote(t, x.size)
            expr = lu.sub_expr(x, prom_t)
            obj, constraints = exp.graph_implementation([expr], x.size)
            obj = lu.sum_entries(obj)

        elif axis == 0:
            prom_size = (x.size[0], 1)
            ones = lu.create_const(np.ones(prom_size), prom_size)
            prom_t = lu.mul_expr(ones, t, x.size)
            expr = lu.sub_expr(x, prom_t)
            obj, constraints = exp.graph_implementation([expr], x.size)

            const_size = (1, x.size[0])
            ones = lu.create_const(np.ones(const_size), const_size)
            obj = lu.mul_expr(ones, obj, size)

        else:  # axis == 1
            prom_size = (1, x.size[1])
            ones = lu.create_const(np.ones(prom_size), prom_size)
            prom_t = lu.rmul_expr(t, ones, x.size)
            expr = lu.sub_expr(x, prom_t)
            obj, constraints = exp.graph_implementation([expr], x.size)

            const_size = (x.size[1], 1)
            ones = lu.create_const(np.ones(const_size), const_size)
            obj = lu.rmul_expr(obj, ones, size)

        ones = lu.create_const(np.ones(size), size)
        constraints += [lu.create_leq(obj, ones)]

        return (t, constraints)
Esempio n. 18
0
    def canonicalize(self):
        """Returns the graph implementation of the object.

        Marks the top level constraint as the dual_holder,
        so the dual value will be saved to the LeqConstraint.

        Returns:
            A tuple of (affine expression, [constraints]).
        """
        obj, constraints = self._expr.canonical_form
        dual_holder = lu.create_leq(obj, constr_id=self.id)
        return (None, constraints + [dual_holder])
Esempio n. 19
0
 def constr_func(aff_obj):
     G_aff = G.canonical_form[0]
     h_aff = h.canonical_form[0]
     Gx = lu.mul_expr(G_aff, aff_obj, h_aff.size)
     constraints = [lu.create_leq(Gx, h_aff)]
     if A is not None:
         A_const, b_const = map(self.cast_to_const, [A, b])
         A_aff = A_const.canonical_form[0]
         b_aff = b_const.canonical_form[0]
         Ax = lu.mul_expr(A_aff, aff_obj, b_aff.size)
         constraints += [lu.create_eq(Ax, b_aff)]
     return constraints
Esempio n. 20
0
    def canonicalize(self):
        """Returns the graph implementation of the object.

        Marks the top level constraint as the dual_holder,
        so the dual value will be saved to the LeqConstraint.

        Returns:
            A tuple of (affine expression, [constraints]).
        """
        obj, constraints = self._expr.canonical_form
        dual_holder = lu.create_leq(obj, constr_id=self.id)
        return (None, constraints + [dual_holder])
Esempio n. 21
0
 def constr_func(aff_obj):
     G_aff = G.canonical_form[0]
     h_aff = h.canonical_form[0]
     Gx = lu.mul_expr(G_aff, aff_obj, h_aff.size)
     constraints = [lu.create_leq(Gx, h_aff)]
     if A is not None:
         A_const, b_const = map(self.cast_to_const, [A, b])
         A_aff = A_const.canonical_form[0]
         b_aff = b_const.canonical_form[0]
         Ax = lu.mul_expr(A_aff, aff_obj, b_aff.size)
         constraints += [lu.create_eq(Ax, b_aff)]
     return constraints
Esempio n. 22
0
 def canonicalize(self):
     obj, constraints = super(Assign, self).canonicalize()
     shape = (self.size[1], 1)
     one_row_vec = lu.create_const(np.ones(shape), shape)
     shape = (1, self.size[0])
     one_col_vec = lu.create_const(np.ones(shape), shape)
     # Row sum <= 1
     row_sum = lu.rmul_expr(obj, one_row_vec, (self.size[0], 1))
     constraints += [lu.create_leq(row_sum, lu.transpose(one_col_vec))]
     # Col sum == 1.
     col_sum = lu.mul_expr(one_col_vec, obj, (1, self.size[1]))
     constraints += [lu.create_eq(col_sum, lu.transpose(one_row_vec))]
     return (obj, constraints)
Esempio n. 23
0
 def canonicalize(self):
     obj, constraints = super(Assign, self).canonicalize()
     shape = (self.size[1], 1)
     one_row_vec = lu.create_const(np.ones(shape), shape)
     shape = (1, self.size[0])
     one_col_vec = lu.create_const(np.ones(shape), shape)
     # Row sum <= 1
     row_sum = lu.rmul_expr(obj, one_row_vec, (self.size[0], 1))
     constraints += [lu.create_leq(row_sum, lu.transpose(one_col_vec))]
     # Col sum == 1.
     col_sum = lu.mul_expr(one_col_vec, obj, (1, self.size[1]))
     constraints += [lu.create_eq(col_sum, lu.transpose(one_row_vec))]
     return (obj, constraints)
Esempio n. 24
0
 def test_leq_constr(self):
     """Test creating a less than or equal constraint.
     """
     shape = (5, 5)
     x = create_var(shape)
     y = create_var(shape)
     lh_expr = sum_expr([x, y])
     value = np.ones(shape)
     rh_expr = create_const(value, shape)
     constr = create_leq(lh_expr, rh_expr)
     self.assertEqual(constr.shape, shape)
     vars_ = get_expr_vars(constr.expr)
     ref = [(x.data, shape), (y.data, shape)]
     self.assertCountEqual(vars_, ref)
Esempio n. 25
0
    def graph_implementation(arg_objs, size, data=None):
        """Reduces the atom to an affine expression and list of constraints.

        Parameters
        ----------
        arg_objs : list
            LinExpr for each argument.
        size : tuple
            The size of the resulting expression.
        data :
            Additional data required by the atom.

        Returns
        -------
        tuple
            (LinOp for objective, list of constraints)
        """
        x = arg_objs[0]
        t = lu.create_var(size)

        # log(1 + exp(x)) <= t <=> exp(-t) + exp(x - t) <= 1
        '''
        obj0, constr0 = exp.graph_implementation([lu.neg_expr(t)], size)
        obj1, constr1 = exp.graph_implementation([lu.sub_expr(x, t)], size)
        lhs = lu.sum_expr([obj0, obj1])
        ones = lu.create_const(np.mat(np.ones(size)), size)
        constr = constr0 + constr1 + [lu.create_leq(lhs, ones)]
        '''
        s = data[0]
        if isinstance(s, Parameter):
            s = lu.create_param(s, (1, 1))
        else: # M is constant.
            s = lu.create_const(s, (1, 1))

        #Wrong sign?
        obj0, constr0 = exp.graph_implementation([lu.neg_expr(t)], size)
        obj1, constr1 = exp.graph_implementation([lu.sub_expr(s, lu.sum_expr([t, x]))], size)
        obj2, constr2 = exp.graph_implementation([lu.sub_expr(lu.neg_expr(s), lu.sum_expr([t, x]))], size)
        obj3, constr3 = exp.graph_implementation([lu.sub_expr(lu.neg_expr(t), lu.mul_expr(2, x, size))], size)

        lhs = lu.sum_expr([obj0, obj1, obj2, obj3])
        ones = lu.create_const(np.mat(np.ones(size)), size)
        constr = constr0 + constr1 + constr2 + constr3 + [lu.create_leq(lhs, ones)]


        return (t, constr)
Esempio n. 26
0
    def graph_implementation(arg_objs,size,data=None):
        x = arg_objs[0]
        beta,x0 = data[0],data[1]
        beta_val,x0_val = beta.value,x0.value

        if isinstance(beta,Parameter):
            beta = lu.create_param(beta,(1,1))
        else:
            beta = lu.create_const(beta.value,(1,1))
        if isinstance(x0,Parameter):
            x0 = lu.create_param(x0,(1,1))
        else:
            x0 = lu.create_const(x0.value,(1,1))

        xi,psi = lu.create_var(size),lu.create_var(size)
        one = lu.create_const(1,(1,1))
        one_over_beta = lu.create_const(1/beta_val,(1,1))
        k = np.exp(-beta_val*x0_val)
        k = lu.create_const(k,(1,1))

        # 1/beta * (1 - exp(-beta*(xi+x0)))
        xi_plus_x0 = lu.sum_expr([xi,x0])
        minus_beta_times_xi_plus_x0  = lu.neg_expr(lu.mul_expr(beta,xi_plus_x0,size))
        exp_xi,constr_exp = exp.graph_implementation([minus_beta_times_xi_plus_x0],size)
        minus_exp_minus_etc = lu.neg_expr(exp_xi)
        left_branch = lu.mul_expr(one_over_beta, lu.sum_expr([one,minus_exp_minus_etc]),size)

        # psi*exp(-beta*r0)
        right_branch = lu.mul_expr(k,psi,size)

        obj = lu.sum_expr([left_branch,right_branch])

        #x-x0 == xi + psi, xi >= 0, psi <= 0
        zero = lu.create_const(0,size)
        constraints = constr_exp
        prom_x0 = lu.promote(x0, size)
        constraints.append(lu.create_eq(x,lu.sum_expr([prom_x0,xi,psi])))
        constraints.append(lu.create_geq(xi,zero))
        constraints.append(lu.create_leq(psi,zero))

        return (obj, constraints)
Esempio n. 27
0
    def graph_implementation(arg_objs, size, data=None):
        """Reduces the atom to an affine expression and list of constraints.

        Parameters
        ----------
        arg_objs : list
            LinExpr for each argument.
        size : tuple
            The size of the resulting expression.
        data :
            Additional data required by the atom.

        Returns
        -------
        tuple
            (LinOp for objective, list of constraints)
        """
        x = arg_objs[0]
        t = lu.create_var(size)
        obj, constraints = square.graph_implementation([t], size)
        return (t, constraints + [lu.create_leq(obj, x)])
Esempio n. 28
0
    def graph_implementation(arg_objs, size, data=None):
        """Reduces the atom to an affine expression and list of constraints.

        Parameters
        ----------
        arg_objs : list
            LinExpr for each argument.
        size : tuple
            The size of the resulting expression.
        data :
            Additional data required by the atom.

        Returns
        -------
        tuple
            (LinOp for objective, list of constraints)
        """
        x = arg_objs[0]
        t = lu.create_var(size)
        obj, constraints = square.graph_implementation([t], size)
        return (t, constraints + [lu.create_leq(obj, x)])
Esempio n. 29
0
    def graph_implementation(arg_objs, size, data=None):
        """Reduces the atom to an affine expression and list of constraints.

        Parameters
        ----------
        arg_objs : list
            LinExpr for each argument.
        size : tuple
            The size of the resulting expression.
        data :
            Additional data required by the atom.

        Returns
        -------
        tuple
            (LinOp for objective, list of constraints)
        """
        x = arg_objs[0]
        t = lu.create_var((1, 1))
        promoted_t = lu.promote(t, x.size)
        constraints = [lu.create_leq(x, promoted_t)]
        return (t, constraints)
Esempio n. 30
0
 def constr_func(aff_obj):
     G_aff = G.canonical_form[0]
     h_aff = h.canonical_form[0]
     Gx = lu.mul_expr(G_aff, aff_obj, h_aff.size)
     constraints = [lu.create_leq(Gx, h_aff)]
     return constraints
Esempio n. 31
0
    def graph_implementation(arg_objs, size, data=None):
        r"""Reduces the atom to an affine expression and list of constraints.

        Parameters
        ----------
        arg_objs : list
            LinExpr for each argument.
        size : tuple
            The size of the resulting expression.
        data :
            Additional data required by the atom.

        Returns
        -------
        tuple
            (LinOp for objective, list of constraints)

        Notes
        -----

        Implementation notes.

        For general ``p``, the p-norm is equivalent to the following convex inequalities:

        .. math::

            x_i &\leq r_i\\
            -x_i &\leq r_i\\
            r_i &\leq s_i^{1/p} t^{1 - 1/p}\\
            \sum_i s_i &\leq t,

        where :math:`p \geq 1`.

        These inequalities are also correct for :math:`p = +\infty` if we interpret :math:`1/\infty` as :math:`0`.


        Although the inequalities above are correct, for a few special cases, we can represent the p-norm
        more efficiently and with fewer variables and inequalities.

        - For :math:`p = 1`, we use the representation

            .. math::

                x_i &\leq r_i\\
                -x_i &\leq r_i\\
                \sum_i r_i &\leq t

        - For :math:`p = \infty`, we use the representation

            .. math::

                x_i &\leq t\\
                -x_i &\leq t

          Note that we don't need the :math:`s` variables or the sum inequality.

        - For :math:`p = 2`, we use the natural second-order cone representation

            .. math::

                \|x\|_2 \leq t

          Note that we could have used the set of inequalities given above if we wanted an alternate decomposition
          of a large second-order cone into into several smaller inequalities.

        """
        p, w = data
        x = arg_objs[0]
        t = None  # dummy value so linter won't complain about initialization
        if p != 1:
            t = lu.create_var((1, 1))

        if p == 2:
            return t, [SOC(t, [x])]

        if p == np.inf:
            r = lu.promote(t, x.size)
        else:
            r = lu.create_var(x.size)

        constraints = [lu.create_geq(lu.sum_expr([x, r])),
                       lu.create_leq(x, r)]

        if p == 1:
            return lu.sum_entries(r), constraints

        if p == np.inf:
            return t, constraints

        # otherwise do case of general p
        s = lu.create_var(x.size)
        # todo: no need to run gm_constr to form the tree each time. we only need to form the tree once
        constraints += gm_constrs(r, [s, lu.promote(t, x.size)], w)
        constraints += [lu.create_leq(lu.sum_entries(s), t)]
        return t, constraints
Esempio n. 32
0
 def constr_func(aff_obj):
     G_aff = X.canonical_form[0]
     h_aff = r.canonical_form[0]
     Gx = cvx.norm(G_aff - aff_obj, 2, axis=1)
     constraints = [lu.create_leq(Gx, h_aff)]
     return constraints
Esempio n. 33
0
 def canonicalize(self):
     obj, constraints = super(BoolVar, self).canonicalize()
     one = lu.create_const(1, (1, 1))
     constraints += [lu.create_geq(obj),
                     lu.create_leq(obj, one)]
     return (obj, constraints)
Esempio n. 34
0
 def canonicalize(self):
     obj, constraints = super(BoolVar, self).canonicalize()
     one = lu.create_const(1, (1, 1))
     constraints += [lu.create_geq(obj), lu.create_leq(obj, one)]
     return (obj, constraints)
Esempio n. 35
0
    def graph_implementation(arg_objs, size, data=None):
        r"""Reduces the atom to an affine expression and list of constraints.

        Parameters
        ----------
        arg_objs : list
            LinExpr for each argument.
        size : tuple
            The size of the resulting expression.
        data :
            Additional data required by the atom.

        Returns
        -------
        tuple
            (LinOp for objective, list of constraints)

        Notes
        -----

        Implementation notes.

        - For general :math:`p \geq 1`, the inequality :math:`\|x\|_p \leq t`
          is equivalent to the following convex inequalities:

          .. math::

              |x_i| &\leq r_i^{1/p} t^{1 - 1/p}\\
              \sum_i r_i &= t.

          These inequalities happen to also be correct for :math:`p = +\infty`,
          if we interpret :math:`1/\infty` as :math:`0`.

        - For general :math:`0 < p < 1`, the inequality :math:`\|x\|_p \geq t`
          is equivalent to the following convex inequalities:

          .. math::

              r_i &\leq x_i^{p} t^{1 - p}\\
              \sum_i r_i &= t.

        - For general :math:`p < 0`, the inequality :math:`\|x\|_p \geq t`
          is equivalent to the following convex inequalities:

          .. math::

              t &\leq x_i^{-p/(1-p)} r_i^{1/(1 - p)}\\
              \sum_i r_i &= t.




        Although the inequalities above are correct, for a few special cases, we can represent the p-norm
        more efficiently and with fewer variables and inequalities.

        - For :math:`p = 1`, we use the representation

            .. math::

                x_i &\leq r_i\\
                -x_i &\leq r_i\\
                \sum_i r_i &= t

        - For :math:`p = \infty`, we use the representation

            .. math::

                x_i &\leq t\\
                -x_i &\leq t

          Note that we don't need the :math:`r` variable or the sum inequality.

        - For :math:`p = 2`, we use the natural second-order cone representation

            .. math::

                \|x\|_2 \leq t

          Note that we could have used the set of inequalities given above if we wanted an alternate decomposition
          of a large second-order cone into into several smaller inequalities.

        """
        p = data[0]
        x = arg_objs[0]
        t = lu.create_var((1, 1))
        constraints = []

        # first, take care of the special cases of p = 2, inf, and 1
        if p == 2:
            return t, [SOC(t, [x])]

        if p == np.inf:
            t_ = lu.promote(t, x.size)
            return t, [lu.create_leq(x, t_), lu.create_geq(lu.sum_expr([x, t_]))]

        # we need an absolute value constraint for the symmetric convex branches (p >= 1)
        # we alias |x| as x from this point forward to make the code pretty :)
        if p >= 1:
            absx = lu.create_var(x.size)
            constraints += [lu.create_leq(x, absx), lu.create_geq(lu.sum_expr([x, absx]))]
            x = absx

        if p == 1:
            return lu.sum_entries(x), constraints

        # now, we take care of the remaining convex and concave branches
        # to create the rational powers, we need a new variable, r, and
        # the constraint sum(r) == t
        r = lu.create_var(x.size)
        t_ = lu.promote(t, x.size)
        constraints += [lu.create_eq(lu.sum_entries(r), t)]

        # make p a fraction so that the input weight to gm_constrs
        # is a nice tuple of fractions.
        p = Fraction(p)
        if p < 0:
            constraints += gm_constrs(t_, [x, r], (-p / (1 - p), 1 / (1 - p)))
        if 0 < p < 1:
            constraints += gm_constrs(r, [x, t_], (p, 1 - p))
        if p > 1:
            constraints += gm_constrs(x, [r, t_], (1 / p, 1 - 1 / p))

        return t, constraints
Esempio n. 36
0
 def canonicalize(self):
     obj, constraints = super(Boolean, self).canonicalize()
     one = lu.create_const(np.ones(self.size), self.size)
     constraints += [lu.create_geq(obj), lu.create_leq(obj, one)]
     return (obj, constraints)
Esempio n. 37
0
 def canonicalize(self):
     obj, constraints = super(Boolean, self).canonicalize()
     one = lu.create_const(np.ones(self.size), self.size)
     constraints += [lu.create_geq(obj),
                     lu.create_leq(obj, one)]
     return (obj, constraints)