from __future__ import annotations
from typing import TYPE_CHECKING
import gamspy._symbols.implicits as implicits
import gamspy.math
import gamspy.utils as utils
from gamspy._algebra.number import Number
from gamspy._container import Container
from gamspy._symbols.equation import Equation
from gamspy._symbols.variable import Variable
from gamspy.exceptions import ValidationError
from gamspy.formulations.result import FormulationResult
from gamspy.math.matrix import next_alias
if TYPE_CHECKING:
from gamspy._algebra.expression import Expression
from gamspy._algebra.operation import Operation
from gamspy._symbols.parameter import Parameter
def _get_random_name(prefix: str) -> str:
return f"{prefix}_{utils._get_unique_name()}"
def _get_lb(x: Variable, default_lb: float):
lb = x.container.addParameter(
_get_random_name("lb"),
domain=x.domain,
)
lb[...] = x.lo[...]
lb[...].where[lb[...] == "-inf"] = default_lb
return lb
def _get_ub(x: Variable, default_ub: float):
ub = x.container.addParameter(
_get_random_name("ub"),
domain=x.domain,
)
ub[...] = x.up[...]
ub[...].where[ub[...] == "inf"] = default_ub
return ub
def tanh(x: Variable) -> tuple[Variable, list[Equation]]:
"""
Convenience wrapper that uses gamspy.math.tanh. Unlike gamspy.math.tanh,
this function creates a new variable and the equation that
sets it to follow formulations structure.
Parameters
----------
x : Variable
Returns
-------
tuple[Variable, list[Equation]]
Examples
--------
>>> import gamspy as gp
>>> m = gp.Container()
>>> v1 = gp.Variable(m)
>>> v2, eqs = gp.math.activation.tanh(v1)
"""
y = x.container.addVariable(domain=x.domain)
set_y = x.container.addEquation(domain=x.domain)
set_y[...] = y == gamspy.math.tanh(x)
y.lo[...] = gamspy.math.tanh(x.lo[...])
y.up[...] = gamspy.math.tanh(x.up[...])
return y, [set_y]
[docs]
def relu_with_sos1_var(
x: (
Parameter
| Variable
| implicits.ImplicitParameter
| implicits.ImplicitVariable
| Expression
| Operation
),
return_slack_var: bool = False,
):
"""
Implements the ReLU activation function using
`SOS1 <https://www.gams.com/47/docs/UG_LanguageFeatures.html?search=sos#UG_LanguageFeatures_SpecialOrderSetsOfType1-SOS1>`_ variables.
The ReLU function is defined as ReLU(x) = max(x, 0). This implementation
**generates** one SOS1 type variable which is necessary to represent the
mathematical relationship and one equation. The SOS1 variable contains the
activation variable and the slack variable.
Unlike ``relu_with_binary_var``, this function does not require lower and
upper bounds for the formulation. It is claimed that when providing tight
bounds is not straightforward, using ``relu_with_sos1_var`` might perform
better than ``relu_with_binary_var``, as the relaxation of
``relu_with_binary_var`` can be weak.
Usage of SOS1 variables require MIP or MINLP and a solver that supports SOS1
variables. Main intended use case of this function is embedding the trained
neural network into MIP models, we do not suggest using it in training since
you would need a MINLP solver that support SOS1 variables.
Returns the activation variable and the equation list if ``return_slack_var`` is
False, otherwise returns activation, slack variable and the equation list in
order. Since activation variable and slack variable are the same variable
only separated by the last domain this function returns ImplicitVariable
instead of Variable.
Based on paper:
`PySCIPOpt-ML: Embedding trained machine learning models into mixed-integer programs. <https://arxiv.org/pdf/2312.08074>`_
Parameters
----------
x : Parameter | Variable | implicits.ImplicitParameter | implicits.ImplicitVariable | Expression | Operation
return_slack_var: bool
Returns
-------
tuple[implicits.ImplicitVariable, list[Equation]] | tuple[implicits.ImplicitVariable, implicits.ImplicitVariable, list[Equation]]
Examples
--------
>>> from gamspy import Container, Variable, Set
>>> from gamspy.math.activation import relu_with_sos1_var
>>> m = Container()
>>> i = Set(m, "i", records=range(3))
>>> x = Variable(m, "x", domain=[i])
>>> y, eqs = relu_with_sos1_var(x)
>>> y, s, eqs = relu_with_sos1_var(x, return_slack_var=True)
>>> y.domain # implicit activation variable has the same domain
[Set(name='i', domain=['*'])]
>>> s.domain # implicit slack variable has the same domain as well
[Set(name='i', domain=['*'])]
>>> y.name == s.name # In the background that y and s are parts of the same variable
True
>>> id(y.parent) == id(s.parent)
True
"""
domain = x.domain
assert isinstance(x.container, Container)
last_dim = gamspy.math._generate_dims(x.container, [2])[0]
y = x.container.addVariable(
_get_random_name("sos1"), type="sos1", domain=[*domain, last_dim]
)
eq = x.container.addEquation(
_get_random_name("eq"),
domain=domain,
)
activation_domain = [*domain, "0"]
sos1_var_domain = [*domain, "1"]
eq[...] = y[activation_domain] == x + y[sos1_var_domain]
if return_slack_var:
return y[activation_domain], y[sos1_var_domain], [eq]
return y[activation_domain], [eq]
[docs]
def leaky_relu_with_binary_var(
x: (
Parameter
| Variable
| implicits.ImplicitParameter
| implicits.ImplicitVariable
| Expression
| Operation
),
negative_slope: float,
default_lb: float = -(10**6),
default_ub: float = 10**6,
*,
return_binary_var: bool = False,
):
"""
Implements the LeakyReLU activation function using binary variables. The LeakyReLU
function is defined as LeakyReLU(x, negative_slope) = max(x, 0) + negative_slope * min(0, x).
This implementation **generates** one binary variable, one output variable and four
equations. The binary variable is necessary to represent the mathematical relationship,
while the output variable serves as the activation variable. Both the binary and
ouput variables share the same domain as the input.
The formulation of this function requires having lower and upper bounds
for the input ``x``. This function utilizes the bounds from the variables
if provided. If not, it defaults to the bounds defined by ``default_lb``
and ``default_ub``. Providing tighter and **correct** bounds can enhance
the quality of linear relaxations.
Returns the activation variable and the equation list if ``return_binary_var`` is False,
otherwise returns activation, binary variable and equation list in order.
Parameters
----------
x : Parameter | Variable | implicits.ImplicitParameter | implicits.ImplicitVariable | Expression | Operation
negative_slope: float
default_ub : float
default_lb : float
return_binary_var: bool
Returns
-------
tuple[Variable, list[Equation]] | tuple[Variable, Variable, list[Equation]]
Examples
--------
>>> from gamspy import Container, Variable, Set
>>> from gamspy.math.activation import leaky_relu_with_binary_var
>>> m = Container()
>>> i = Set(m, "i", records=range(3))
>>> x = Variable(m, "x", domain=[i])
>>> y, eqs = leaky_relu_with_binary_var(x, 0.01)
>>> len(eqs)
4
>>> y, b, eqs = leaky_relu_with_binary_var(x, 0.01, return_binary_var=True)
>>> b.type
'binary'
>>> y.domain # i many activation variables
[Set(name='i', domain=['*'])]
>>> b.domain # i many binary variables
[Set(name='i', domain=['*'])]
"""
assert isinstance(x.container, Container)
if negative_slope <= 0 or negative_slope >= 1:
raise ValidationError("negative_slope must be in the range (0, 1).")
domain = x.domain
sigma = Variable._constructor_bypass(
x.container,
_get_random_name("bin"),
type="binary",
domain=domain,
)
y = Variable._constructor_bypass(
x.container,
_get_random_name("y"),
domain=domain,
type="free",
)
eq = [
Equation._constructor_bypass(
x.container,
_get_random_name("eq"),
domain=domain,
)
for _ in range(4)
]
eq[0][...] = y >= x
eq[1][...] = y >= negative_slope * x
if isinstance(x, Variable):
eq[2][...] = y <= x - (1 - sigma) * (1 - negative_slope) * _get_lb(
x, default_lb
)
eq[3][...] = y <= negative_slope * x + sigma * (1 - negative_slope) * _get_ub(
x, default_ub
)
y.lo[...] = x.lo[...] * negative_slope
y.up[...] = gamspy.math.Max(0, x.up[...])
else:
eq[2][...] = y <= x - (1 - sigma) * (1 - negative_slope) * default_lb
eq[3][...] = y <= negative_slope * x + sigma * default_ub * (1 - negative_slope)
if return_binary_var:
return y, sigma, eq
else:
return y, eq
[docs]
def relu_with_binary_var(
x: (
Parameter
| Variable
| implicits.ImplicitParameter
| implicits.ImplicitVariable
| Expression
| Operation
),
default_lb: float = -(10**6),
default_ub: float = 10**6,
return_binary_var: bool = False,
) -> FormulationResult:
"""
Implements the ReLU activation function using binary variables. The ReLU
function is defined as ReLU(x) = max(x, 0). This implementation **generates**
one binary variable, one positive variable and three equations. The binary
variable is necessary to represent the mathematical relationship, while the
positive variable serves as the activation variable. Both the binary and
positive variables share the same domain as the input.
The formulation of this function requires having lower and upper bounds
for the input ``x``. This function utilizes the bounds from the variables
if provided. If not, it defaults to the bounds defined by ``default_lb``
and ``default_ub``. Providing tighter and **correct** bounds can enhance
the quality of linear relaxations.
Returns the activation variable and the equation list if ``return_binary_var`` is False,
otherwise returns activation, binary variable and equation list in order.
Adapted from `OMLT <https://github.com/cog-imperial/OMLT/blob/e60563859a66ac5dd3348bf1763de57eec95171e/src/omlt/neuralnet/activations/relu.py>`_
FormulationResult:
- variables_created: ["output", "binary"]
- equations_created: ["y_gte_x", "y_lte_x_1", "y_lte_x_2"]
Parameters
----------
x : Parameter | Variable | implicits.ImplicitParameter | implicits.ImplicitVariable | Expression | Operation
default_ub : float
default_lb : float
return_binary_var: bool
Returns
-------
FormulationResult
Examples
--------
>>> from gamspy import Container, Variable, Set
>>> from gamspy.math.activation import relu_with_binary_var
>>> m = Container()
>>> i = Set(m, "i", records=range(3))
>>> x = Variable(m, "x", domain=[i])
>>> y, eqs = relu_with_binary_var(x) # FormulationResult can be unpacked for backwards compat
>>> y.type
'positive'
>>> len(eqs)
3
>>> y, b, eqs = relu_with_binary_var(x, return_binary_var=True)
>>> b.type
'binary'
>>> y.domain # i many activation variables
[Set(name='i', domain=['*'])]
>>> b.domain # i many binary variables
[Set(name='i', domain=['*'])]
>>> output = relu_with_binary_var(x) # You can use FormulationResult too
>>> binary_var = output.variables_created["binary"]
>>> y = output.result
"""
assert isinstance(x.container, Container)
domain = x.domain
sigma = x.container.addVariable(
_get_random_name("bin"),
type="binary",
domain=domain,
)
y = x.container.addVariable(
_get_random_name("y"),
type="positive",
domain=domain,
)
eq = [
x.container.addEquation(
_get_random_name("eq"),
domain=domain,
)
for _ in range(3)
]
# y >= 0 implied by positive variable
eq[0][...] = y >= x
if isinstance(x, Variable):
eq[1][...] = y <= x - (1 - sigma) * _get_lb(x, default_lb)
eq[2][...] = y <= sigma * _get_ub(x, default_ub)
y.lo[...] = gamspy.math.Max(0, x.lo[...])
y.up[...] = gamspy.math.Max(0, x.up[...])
else:
eq[1][...] = y <= x - (1 - sigma) * default_lb
eq[2][...] = y <= sigma * default_ub
y.lo[...] = 0
result = FormulationResult(
y,
{
"y_gte_x": eq[0],
"y_lte_x_1": eq[1],
"y_lte_x_2": eq[2],
},
)
result.variables_created = {"binary": sigma, "output": y}
result.extra_return = sigma if return_binary_var else None
return result
[docs]
def relu_with_equilibrium(
x: (
Parameter
| Variable
| implicits.ImplicitParameter
| implicits.ImplicitVariable
| Expression
| Operation
),
) -> FormulationResult:
"""
Implements the ReLU activation function using Equilibrium Constraints.
This implementation is suitable for models of type Mathematical Program with
Equilibrium Constraints (MPEC) or Mixed Complementarity Problem (MCP).
One positive variable is **generated**, which serves as the activation
variable and no equations. The activation variable shares the same domain as
the input. Lower and upper bounds are not required for this formulation.
Returns FormulationResult which can be unpacked as activation variable,
matches dictionary and the equation list (empty).
FormulationResult:
variables_created: ["output"]
equations_created: []
matches: {(output - x) : output}
extra_return: yes (matches)
or if the provided input was not a Variable, this formulation assigns it to a
new variable and uses the new variable instead
FormulationResult:
variables_created: ["output", "new_input"]
equations_created: ["set_new_input"]
matches: {(output - new_input) : output}
extra_return: yes (matches)
Parameters
----------
x : Variable
Returns
-------
FormulationResult
Examples
--------
>>> from gamspy import Container, Variable, Set
>>> from gamspy.math.activation import relu_with_equilibrium
>>> m = Container()
>>> i = Set(m, "i", records=range(3))
>>> x = Variable(m, "x", domain=[i])
>>> y, matches, eqs = relu_with_equilibrium(x)
>>> y.type
'positive'
>>> len(eqs)
0
>>> len(matches)
1
>>> result = relu_with_equilibrium(x)
>>> type(result)
<class 'gamspy.formulations.result.FormulationResult'>
>>> result = relu_with_equilibrium(x - 5)
>>> new_input = result.variables_created["new_input"]
"""
assert isinstance(x.container, Container)
domain = x.domain
y = x.container.addVariable(
type="positive",
domain=domain,
)
new_input = None
set_new_input = None
if not isinstance(x, Variable):
new_input = Variable._constructor_bypass(
x.container,
_get_random_name("new_input"),
domain=domain,
)
set_new_input = Equation._constructor_bypass(
x.container,
_get_random_name("set_new_input"),
domain=domain,
)
set_new_input[...] = new_input == x
else:
new_input = x
eq = Equation._constructor_bypass(
x.container,
_get_random_name("matches_eq"),
domain=domain,
)
eq[...] = y - new_input >= Number(0)
result = FormulationResult(y, {})
result.variables_created["output"] = y
if set_new_input is not None:
result.variables_created["new_input"] = new_input
result.equations_created["set_new_input"] = set_new_input
result.extra_return = {eq: y}
result.matches = {eq: y}
return result
[docs]
def relu_with_complementarity_var(
x: (
Parameter
| Variable
| implicits.ImplicitParameter
| implicits.ImplicitVariable
| Expression
| Operation
),
):
"""
Implements the ReLU activation function using complementarity conditions.
The ReLU function is defined as ReLU(x) = max(x, 0). This implementation
**generates** one positive variable, which serves as the activation variable
and two equations. The activation variable shares the same domain as the
input. Unlike ``relu_with_binary_var``, this function does not require lower
and upper bounds for the formulation.
Returns the activation variable and the equation list.
Adapted from `OMLT <https://github.com/cog-imperial/OMLT/blob/e60563859a66ac5dd3348bf1763de57eec95171e/src/omlt/neuralnet/activations/relu.py>`_
Parameters
----------
x : Parameter | Variable | implicits.ImplicitParameter | implicits.ImplicitVariable | Expression | Operation
Returns
-------
tuple[Variable, list[Equation]]
Examples
--------
>>> from gamspy import Container, Variable, Set
>>> from gamspy.math.activation import relu_with_complementarity_var
>>> m = Container()
>>> i = Set(m, "i", records=range(3))
>>> x = Variable(m, "x", domain=[i])
>>> y, eqs = relu_with_complementarity_var(x)
>>> y.type
'positive'
>>> len(eqs)
2
"""
assert isinstance(x.container, Container)
domain = x.domain
y = x.container.addVariable(
_get_random_name("y"),
type="positive",
domain=domain,
)
eq = [
x.container.addEquation(
_get_random_name("eq"),
domain=domain,
)
for _ in range(2)
]
# y >= 0 implied by positive variable
eq[0][...] = y * (y - x) == 0
eq[1][...] = y - x >= 0
if isinstance(x, Variable):
y.lo[...] = gamspy.math.Max(0, x.lo[...])
y.up[...] = gamspy.math.Max(0, x.up[...])
else:
y.lo[...] = 0
return y, eq
[docs]
def log_softmax(x: Variable, dim: int = -1, skip_intrinsic: bool = False):
"""
Implements the log_softmax activation function. This function strictly
requires a GAMSPy Variable, `y = log_softmax(x)`. The ``dim`` parameter
specifies the **index** of the softmax dimension. If not provided, it
calculates log_softmax for the last dimension. This function is preferred
over the :meth:`softmax <gamspy.math.softmax>` function because, when
the softmax dimension has 20 or fewer elements, it uses the
:meth:`lse_max <gamspy.math.lse_max>` (log-sum-exp) intrinsic function for
improved numerical stability which usually leads to faster solve times.
Some solvers do not support :meth:`lse_max <gamspy.math.lse_max>`, in that
case you can set ``skip_intrinsic`` parameter to True to not use intrinsic
functions even when possible.
To learn more about `Log-Sum-Exp trick <https://gregorygundersen.com/blog/2020/02/09/log-sum-exp/>`_ .
This function is usually combined with Negative Log Likelihood loss for
classification problems.
Returns the activation variable and the equation list.
Parameters
----------
x : Variable
dim : int
skip_intrinsic: bool
Returns
-------
Variable, list[Equation]
Examples
--------
>>> from gamspy import Container, Variable
>>> from gamspy.math import dim
>>> from gamspy.math.activation import log_softmax
>>> m = Container()
>>> x = Variable(m, "x", domain=dim([500, 10]))
>>> y, eqs1 = log_softmax(x) # uses LSE because 10 <= 20
>>> y.domain
[Set(name='DenseDim500_1', domain=['*']), Set(name='DenseDim10_1', domain=['*'])]
>>> y2, eqs2 = log_softmax(x, dim=0) # cannot use LSE because 500 > 20
>>> y3, eqs3 = log_softmax(x, skip_intrinsic=True) # don't use LSE because of skip_intrinsic
"""
if not isinstance(x, Variable):
raise ValidationError("log_softmax expects a variable")
if dim < 0:
dim = len(x.domain) + dim
sum_domain = next_alias(x.domain[dim])
y = x.container.addVariable(
_get_random_name("y"),
domain=x.domain,
)
eq = x.container.addEquation(
_get_random_name("eq"),
domain=x.domain,
)
if not skip_intrinsic and len(sum_domain) != 0 and len(sum_domain) <= 20:
# Use built-in LSE if possible
scalars = list(sum_domain.records["uni"])
variables = []
for scalar in scalars:
expr_domain = [*x.domain[:dim], scalar, *x.domain[dim + 1 :]]
variables.append(x[expr_domain])
log_sum_exp = gamspy.math.lse_max(*variables)
eq[...] = y[...] == x - log_sum_exp
else:
expr_domain = [d if i != dim else sum_domain for (i, d) in enumerate(x.domain)]
sum_expr = gamspy.Sum(sum_domain, gamspy.math.exp(x[expr_domain]))
eq[...] = y[...] == x - gamspy.math.log(sum_expr)
return y, [eq]
[docs]
def softplus(
x: Variable,
beta=1.0,
*,
skip_intrinsic: bool = False,
) -> tuple[Variable, list[Equation]]:
"""
Implements the softplus activation function. This function uses the
:meth:`lse_max_sc <gamspy.math.lse_max_sc>` (log-sum-exp) intrinsic function for
improved numerical stability which usually leads to faster solve times. Some solvers
do not support :meth:`lse_max_sc <gamspy.math.lse_max_sc>`, in that case you can set
``skip_intrinsic`` parameter to True to not use intrinsic functions.
``beta`` value controls the smoothness and slope of the function, by default equals
to 1.
``skip_intrinsic`` (Default `False`)
Parameters
----------
x : Variable
beta : float
skip_intrinsic: bool
Returns
-------
tuple[Variable, list[Equation]]
Examples
--------
>>> from gamspy import Container, Variable
>>> from gamspy.math import dim
>>> from gamspy.math.activation import softplus
>>> m = Container()
>>> x = Variable(m, "x", domain=dim([500, 10]))
>>> y, eqs1 = softplus(x)
>>> y.domain
[Set(name='DenseDim500_1', domain=['*']), Set(name='DenseDim10_1', domain=['*'])]
>>> y2, eqs2 = softplus(x, skip_intrinsic=True) # don't use LSE because of skip_intrinsic
"""
y = Variable._constructor_bypass(
x.container,
_get_random_name("y"),
domain=x.domain,
)
eq = Equation._constructor_bypass(
x.container,
_get_random_name("eq"),
domain=x.domain,
)
if skip_intrinsic:
eq[...] = y == (1 / beta) * gamspy.math.log(1 + gamspy.math.exp(beta * x))
else:
eq[...] = y == gamspy.math.lse_max_sc(beta, 0, x)
return y, [eq]
[docs]
def softmax(x: Variable, dim: int = -1):
"""
Implements the softmax activation function. This function strictly requires
a GAMSPy Variable, `y = softmax(x)`. The ``dim`` parameter specifies the
index of the softmax dimension. If not provided, the softmax is calculated
for the last dimension. This function is implemented for completeness;
however, in many cases, you can use :meth:`log_softmax <gamspy.math.log_softmax>`
for better numerical stability.
Use :meth:`log_softmax <gamspy.math.log_softmax>` if you need to take the
logarithm of the softmax function.
Returns the activation variable and the equation list.
Parameters
----------
x : Variable
dim : int
Returns
-------
tuple[Variable, list[Equation]]
Examples
--------
>>> from gamspy import Container, Variable
>>> from gamspy.math import dim
>>> from gamspy.math.activation import softmax
>>> m = Container()
>>> x = Variable(m, "x", domain=dim([500, 10]))
>>> y, eqs = softmax(x)
>>> y.domain
[Set(name='DenseDim500_1', domain=['*']), Set(name='DenseDim10_1', domain=['*'])]
"""
if not isinstance(x, Variable):
raise ValidationError("softmax expects a variable")
if dim < 0:
dim = len(x.domain) + dim
sum_domain = next_alias(x.domain[dim])
expr_domain = [d if i != dim else sum_domain for (i, d) in enumerate(x.domain)]
y = x.container.addVariable(
_get_random_name("y"),
domain=x.domain,
)
eq = x.container.addEquation(
_get_random_name("eq"),
domain=x.domain,
)
sum_expr = gamspy.Sum(sum_domain, gamspy.math.exp(x[expr_domain]))
eq[...] = y[...] == gamspy.math.exp(x) / sum_expr
return y, [eq]
