Managing Expressions
Creating a String Representation of an Expression
There are several ways that string representations can be created
from an expression, but the expression_to_string
function provides
the most flexible mechanism for generating a string representation.
The options to this function control distinct aspects of the string
representation.
Algebraic vs. Nested Functional Form
The default string representation is an algebraic form, which closely
mimics the Python operations used to construct an expression. The
verbose
flag can be set to True
to generate a
string representation that is a nested functional form. For example:
from pyomo.core.expr import current as EXPR
M = ConcreteModel()
M.x = Var()
e = sin(M.x) + 2 * M.x
# sin(x) + 2*x
print(EXPR.expression_to_string(e))
# sum(sin(x), prod(2, x))
print(EXPR.expression_to_string(e, verbose=True))
Labeler and Symbol Map
The string representation used for variables in expression can be customized to
define different label formats. If the labeler
option is specified, then this
function (or class functor) is used to generate a string label used to represent the variable. Pyomo
defines a variety of labelers in the pyomo.core.base.label module. For example, the
NumericLabeler
defines a functor that can be used to sequentially generate
simple labels with a prefix followed by the variable count:
from pyomo.core.expr import current as EXPR
M = ConcreteModel()
M.x = Var()
M.y = Var()
e = sin(M.x) + 2 * M.y
# sin(x1) + 2*x2
print(EXPR.expression_to_string(e, labeler=NumericLabeler('x')))
The smap
option is used to specify a symbol map object
(SymbolMap
), which
caches the variable label data. This option is normally specified
in contexts where the string representations for many expressions
are being generated. In that context, a symbol map ensures that
variables in different expressions have a consistent label in their
associated string representations.
Standardized String Representations
The standardize
option can be used to reorder the string
representation to print polynomial terms before nonlinear terms. By
default, standardize
is False
, and the string
representation reflects the order in which terms were combined to
form the expression. Pyomo does not guarantee that the string
representation exactly matches the Python expression order, since
some simplification and reordering of terms is done automatically to
improve the efficiency of expression generation. But in most cases
the string representation will closely correspond to the
Python expression order.
If standardize
is True
, then the pyomo expression
is processed to identify polynomial terms, and the string representation
consists of the constant and linear terms followed by
an expression that contains other nonlinear terms. For example:
from pyomo.core.expr import current as EXPR
M = ConcreteModel()
M.x = Var()
M.y = Var()
e = sin(M.x) + 2 * M.y + M.x * M.y  3
# 3 + 2*y + sin(x) + x*y
print(EXPR.expression_to_string(e, standardize=True))
Other Ways to Generate String Representations
There are two other standard ways to generate string representations:
Call the
__str__()
magic method (e.g. using the Pythonstr()
function. This callsexpression_to_string
with the optionstandardize
equal toTrue
(see below).Call the
to_string()
method on theExpressionBase
class. This defaults to callingexpression_to_string
with the optionstandardize
equal toFalse
(see below).
In practice, we expect at the __str__()
magic method will be
used by most users, and the standardization of the output provides
a consistent ordering of terms that should make it easier to interpret
expressions.
Cloning Expressions
Expressions are automatically cloned only during certain expression
transformations. Since this can be an expensive operation, the
clone_counter
context
manager object is provided to track the number of times the
clone_expression
function is executed.
For example:
from pyomo.core.expr import current as EXPR
M = ConcreteModel()
M.x = Var()
with EXPR.clone_counter() as counter:
start = counter.count
e1 = sin(M.x)
e2 = e1.clone()
total = counter.count  start
assert total == 1
Evaluating Expressions
Expressions can be evaluated when all variables and parameters in
the expression have a value. The value
function can be used to walk the expression tree and compute the
value of an expression. For example:
M = ConcreteModel()
M.x = Var()
M.x.value = math.pi / 2.0
val = value(M.x)
assert isclose(val, math.pi / 2.0)
Additionally, expressions define the __call__()
method, so the
following is another way to compute the value of an expression:
val = M.x()
assert isclose(val, math.pi / 2.0)
If a parameter or variable is undefined, then the value
function and __call__()
method will
raise an exception. This exception can be suppressed using the
exception
option. For example:
M = ConcreteModel()
M.x = Var()
val = value(M.x, exception=False)
assert val is None
This option is useful in contexts where adding a try block is inconvenient in your modeling script.
Note
Both the value
function and
__call__()
method call the evaluate_expression
function. In
practice, this function will be slightly faster, but the
difference is only meaningful when expressions are evaluated
many times.
Identifying Components and Variables
Expression transformations sometimes need to find all nodes in an
expression tree that are of a given type. Pyomo contains two utility
functions that support this functionality. First, the
identify_components
function is a generator function that walks the expression tree and yields all
nodes whose type is in a specified set of node types. For example:
from pyomo.core.expr import current as EXPR
M = ConcreteModel()
M.x = Var()
M.p = Param(mutable=True)
e = M.p + M.x
s = set([type(M.p)])
assert list(EXPR.identify_components(e, s)) == [M.p]
The identify_variables
function is a generator function that yields all nodes that are
variables. Pyomo uses several different classes to represent variables,
but this set of variable types does not need to be specified by the user.
However, the include_fixed
flag can be specified to omit fixed
variables. For example:
from pyomo.core.expr import current as EXPR
M = ConcreteModel()
M.x = Var()
M.y = Var()
e = M.x + M.y
M.y.value = 1
M.y.fixed = True
assert set(id(v) for v in EXPR.identify_variables(e)) == set([id(M.x), id(M.y)])
assert set(id(v) for v in EXPR.identify_variables(e, include_fixed=False)) == set(
[id(M.x)]
)
Walking an Expression Tree with a Visitor Class
Many of the utility functions defined above are implemented by walking an expression tree and performing an operation at nodes in the tree. For example, evaluating an expression is performed using a postorder depthfirst search process where the value of a node is computed using the values of its children.
Walking an expression tree can be tricky, and the code requires intimate knowledge of the design of the expression system. Pyomo includes several classes that define socalled visitor patterns for walking expression tree:
SimpleExpressionVisitor
A
visitor()
method is called for each node in the tree, and the visitor class collects information about the tree.ExpressionValueVisitor
When the
visitor()
method is called on each node in the tree, the values of its children have been computed. The value of the node is returned fromvisitor()
.ExpressionReplacementVisitor
When the
visitor()
method is called on each node in the tree, it may clone or otherwise replace the node using objects for its children (which themselves may be clones or replacements from the original child objects). The new node object is returned fromvisitor()
.
These classes define a variety of suitable tree search methods:

xbfs: breadthfirst search where leaf nodes are immediately visited
xbfs_yield_leaves: breadthfirst search where leaf nodes are immediately visited, and the visit method yields a value

dfs_postorder_stack: postorder depthfirst search using a stack

dfs_postorder_stack: postorder depthfirst search using a stack
Note
The PyUtilib visitor classes define several other search methods that could be used with Pyomo expressions. But these are the only search methods currently used within Pyomo.
To implement a visitor object, a user creates a subclass of one of these classes. Only one of a few methods will need to be defined to implement the visitor:
visitor()
Defines the operation that is performed when a node is visited. In the
ExpressionValueVisitor
andExpressionReplacementVisitor
visitor classes, this method returns a value that is used by its parent node.visiting_potential_leaf()
Checks if the search should terminate with this node. If no, then this method returns the tuple
(False, None)
. If yes, then this method returns(False, value)
, where value is computed by this method. This method is not used in theSimpleExpressionVisitor
visitor class.finalize()
This method defines the final value that is returned from the visitor. This is not normally redefined.
Detailed documentation of the APIs for these methods is provided with the class documentation for these visitors.
SimpleExpressionVisitor Example
In this example, we describe an visitor class that counts the number of nodes in an expression (including leaf nodes). Consider the following class:
from pyomo.core.expr import current as EXPR
class SizeofVisitor(EXPR.SimpleExpressionVisitor):
def __init__(self):
self.counter = 0
def visit(self, node):
self.counter += 1
def finalize(self):
return self.counter
The class constructor creates a counter, and the visit()
method
increments this counter for every node that is visited. The finalize()
method returns the value of this counter after the tree has been walked. The
following function illustrates this use of this visitor class:
def sizeof_expression(expr):
#
# Create the visitor object
#
visitor = SizeofVisitor()
#
# Compute the value using the :func:`xbfs` search method.
#
return visitor.xbfs(expr)
ExpressionValueVisitor Example
In this example, we describe an visitor class that clones the expression tree (including leaf nodes). Consider the following class:
from pyomo.core.expr import current as EXPR
class CloneVisitor(EXPR.ExpressionValueVisitor):
def __init__(self):
self.memo = {'__block_scope__': {id(None): False}}
def visit(self, node, values):
#
# Clone the interior node
#
return node.construct_clone(tuple(values), self.memo)
def visiting_potential_leaf(self, node):
#
# Clone leaf nodes in the expression tree
#
if (
node.__class__ in native_numeric_types
or node.__class__ not in pyomo5_expression_types
):
return True, copy.deepcopy(node, self.memo)
return False, None
The visit()
method creates a new expression node with children
specified by values
. The visiting_potential_leaf()
method performs a deepcopy()
on leaf nodes, which are native
Python types or nonexpression objects.
def clone_expression(expr):
#
# Create the visitor object
#
visitor = CloneVisitor()
#
# Clone the expression using the :func:`dfs_postorder_stack`
# search method.
#
return visitor.dfs_postorder_stack(expr)
ExpressionReplacementVisitor Example
In this example, we describe an visitor class that replaces variables with scaled variables, using a mutable parameter that can be modified later. the following class:
from pyomo.core.expr import current as EXPR
class ScalingVisitor(EXPR.ExpressionReplacementVisitor):
def __init__(self, scale):
super(ScalingVisitor, self).__init__()
self.scale = scale
def visiting_potential_leaf(self, node):
#
# Clone leaf nodes in the expression tree
#
if node.__class__ in native_numeric_types:
return True, node
if node.is_variable_type():
return True, self.scale[id(node)] * node
if isinstance(node, EXPR.LinearExpression):
node_ = copy.deepcopy(node)
node_.constant = node.constant
node_.linear_vars = copy.copy(node.linear_vars)
node_.linear_coefs = []
for i, v in enumerate(node.linear_vars):
node_.linear_coefs.append(node.linear_coefs[i] * self.scale[id(v)])
return True, node_
return False, None
No visit()
method needs to be defined. The
visiting_potential_leaf()
function identifies variable nodes
and returns a product expression that contains a mutable parameter.
The _LinearExpression
class has a different representation
that embeds variables. Hence, this class must be handled
in a separate condition that explicitly transforms this subexpression.
def scale_expression(expr, scale):
#
# Create the visitor object
#
visitor = ScalingVisitor(scale)
#
# Scale the expression using the :func:`dfs_postorder_stack`
# search method.
#
return visitor.dfs_postorder_stack(expr)
The scale_expression()
function is called with an expression and
a dictionary, scale
, that maps variable ID to model parameter. For example:
M = ConcreteModel()
M.x = Var(range(5))
M.p = Param(range(5), mutable=True)
scale = {}
for i in M.x:
scale[id(M.x[i])] = M.p[i]
e = quicksum(M.x[i] for i in M.x)
f = scale_expression(e, scale)
# p[0]*x[0] + p[1]*x[1] + p[2]*x[2] + p[3]*x[3] + p[4]*x[4]
print(f)