DGP fundamentals
================
This notebook will introduce you to the fundamentals of **disciplined
geometric programming** (**DGP**), which lets you formulate and solve
*log-log convex programs* (LLCPs) in CVXPY.
LLCPs are problems that become convex after the variables, objective
functions, and constraint functions are replaced with their logs, an
operation that we refer to as a log-log transformation. LLCPs generalize
geometric programming.
.. code:: python
import cvxpy as cp
1. Log-log curvature
====================
Just as every Expression in CVXPY has a curvature (constant, affine,
convex, concave, or unknown), every Expression also has a log-log
curvature.
A function :math:`f : D \subseteq \mathbf{R}^{n}_{++} \to \mathbf{R}` is
said to be **log-log convex** if the function :math:`F(u)=\log f(e^u)`,
with domain :math:`\{u \in \mathbf{R}^n : e^u \in D\}`, is convex (where
:math:`\mathbf{R}^{n}_{++}` denotes the set of positive reals and the
logarithm and exponential are meant elementwise); the function :math:`F`
is called the log-log transformation of :math:`f`. The function
:math:`f` is **log-log concave** if :math:`F` is concave, and it is
**log-log affine** if :math:`F` is affine.
Notice that if a function has log-log curvature, then its domain and
range can only include positive numbers.
The log-log curvature of an ``Expression`` can be accessed via its
``.log_log_curvature`` attribute. For an ``Expression`` to have known
log-log curvature, all of the ``Constant``\ s, ``Variable``\ s, and
``Parameter``\ s it refers to must be elementwise positive.
.. code:: python
# Only elementwise positive constants are allowed in DGP.
c = cp.Constant(1.0)
print(c, c.log_log_curvature)
c = cp.Constant([1.0, 2.0])
print(c, c.log_log_curvature)
c = cp.Constant([1.0, 0.0])
print(c, c.log_log_curvature)
c = cp.Constant(-2.0)
print(c, c.log_log_curvature)
.. parsed-literal::
1.0 LOG-LOG CONSTANT
[1. 2.] LOG-LOG CONSTANT
[1. 0.] UNKNOWN
-2.0 UNKNOWN
.. code:: python
# Variables and parameters must be positive, ie, they must be constructed with the option `pos=True`
v = cp.Variable(pos=True)
print(v, v.log_log_curvature)
v = cp.Variable()
print(v, v.log_log_curvature)
p = cp.Parameter(pos=True)
print(p, p.log_log_curvature)
p = cp.Parameter()
print(p, p.log_log_curvature)
.. parsed-literal::
var0 LOG-LOG AFFINE
var1 UNKNOWN
param2 LOG-LOG CONSTANT
param3 UNKNOWN
Functions from geometric programming
------------------------------------
A function :math:`f(x)` is log-log affine if and only if it is given by
.. math::
f(x) = cx_1^{a_1}x_2^{a_2} \ldots x_n^{a_n},
where :math:`c > 0` and the :math:`a_i` are real numbers. In the context
of geometric programming, such a function is called a monomial.
.. code:: python
x = cp.Variable(shape=(3,), pos=True, name="x")
c = 2.0
a = [0.5, 2.0, 1.8]
monomial = c * x[0] ** a[0] * x[1] ** a[1] * x[2] ** a[2]
# Monomials are not convex.
assert not monomial.is_convex()
# They are, however, log-log affine.
print(monomial, ":", monomial.log_log_curvature)
assert monomial.is_log_log_affine()
.. parsed-literal::
2.0 * power(x[0], 1/2) * power(x[1], 2) * power(x[2], 9/5) : LOG-LOG AFFINE
A sum of monomial functions is log-log convex; in the context of
geometric programming, such a function is called a posynomial. There are
functions that are not posynomials that are still log-log convex.
.. code:: python
x = cp.Variable(pos=True, name="x")
y = cp.Variable(pos=True, name="y")
constant = cp.Constant(2.0)
monomial = constant * x * y
posynomial = monomial + (x ** 1.5) * (y ** -1)
reciprocal = posynomial ** -1
unknown = reciprocal + posynomial
print(constant, ":", constant.log_log_curvature)
print(monomial, ":", monomial.log_log_curvature)
print(posynomial, ":", posynomial.log_log_curvature)
print(reciprocal, ":", reciprocal.log_log_curvature)
print(unknown, ":", unknown.log_log_curvature)
.. parsed-literal::
2.0 : LOG-LOG CONSTANT
2.0 * x * y : LOG-LOG AFFINE
2.0 * x * y + power(x, 3/2) * power(y, -1) : LOG-LOG CONVEX
power(2.0 * x * y + power(x, 3/2) * power(y, -1), -1) : LOG-LOG CONCAVE
power(2.0 * x * y + power(x, 3/2) * power(y, -1), -1) + 2.0 * x * y + power(x, 3/2) * power(y, -1) : UNKNOWN
2. Log-log curvature ruleset
============================
CVXPY has a library of atomic functions with known log-log curvature and
monotonicty. It uses this information to tag every ``Expression``, i.e.,
every composition of atomic functions, with a log-log curvature. In
particular,
A function :math:`f(expr_1,expr_2,...,expr_n)` is log-log convex if
:math:`f` is a log-log convex function and for each expri one of the
following conditions holds:
:math:`f` is increasing in argument i and :math:`expr_i` is log-log
convex. :math:`f` is decreasing in argument :math:`i` and :math:`expr_i`
is log-log concave. :math:`expr_i` is log-log affine. A function
:math:`f(expr_1,expr_2,...,expr_n)` is log-log concave if :math:`f` is a
log-log concave function and for each :math:`expr_i` one of the
following conditions holds:
:math:`f` is increasing in argument :math:`i` and :math:`expr_i` is
log-log concave. :math:`f` is decreasing in argument :math:`i` and
:math:`expr_i` is log-log convex. :math:`expr_i` is log-log affine. A
function :math:`f(expr_1,expr_2,...,expr_n)` is log-log affine if
:math:`f` is an log-log affine function and each :math:`expr_i` is
log-log affine.
If none of the three rules apply, the expression
:math:`f(expr_1,expr_2,...,expr_n)` is marked as having unknown
curvature.
If an Expression satisfies the composition rule, we colloquially say
that the ``Expression`` “is DGP.” You can check whether an
``Expression`` is DGP by calling the method ``is_dgp()``.
.. code:: python
x = cp.Variable(pos=True, name="x")
y = cp.Variable(pos=True, name="y")
monomial = 2.0 * x * y
posynomial = monomial + (x ** 1.5) * (y ** -1)
print(monomial, "is dgp?", monomial.is_dgp())
print(posynomial, "is dgp?", posynomial.is_dgp())
.. parsed-literal::
2.0 * x * y is dgp? True
2.0 * x * y + power(x, 3/2) * power(y, -1) is dgp? True
3. DGP problems
===============
An LLCP is an optimization problem of the form
.. math::
\begin{equation}
\begin{array}{ll}
\mbox{minimize} & f_0(x) \\
\mbox{subject to} & f_i(x) \leq \tilde{f_i}, \quad i=1, \ldots, m\\
& g_i(x) = \tilde{g_i}, \quad i=1, \ldots, p,
\end{array}
\end{equation}
where the functions :math:`f_i` are log-log convex, :math:`\tilde{f_i}`
are log-log concave, and the functions :math:`g_i` and
:math:`\tilde{g_i}` are log-log affine. An optimization problem with
constraints of the above form in which the goal is to maximize a log-log
concave function is also an LLCP.
A problem is DGP if additionally all the functions are DGP. You can
check whether a CVXPY ``Problem`` is DGP by calling its ``.is_dgp()``
method.
.. code:: python
x = cp.Variable(pos=True, name="x")
y = cp.Variable(pos=True, name="y")
z = cp.Variable(pos=True, name="z")
objective_fn = x * y * z
constraints = [
4 * x * y * z + 2 * x * z <= 10, x <= 2*y, y <= 2*x, z >= 1]
assert objective_fn.is_log_log_concave()
assert all(constraint.is_dgp() for constraint in constraints)
problem = cp.Problem(cp.Maximize(objective_fn), constraints)
print(problem)
print("Is this problem DGP?", problem.is_dgp())
.. parsed-literal::
maximize x * y * z
subject to 4.0 * x * y * z + 2.0 * x * z <= 10.0
x <= 2.0 * y
y <= 2.0 * x
1.0 <= z
Is this problem DGP? True
Solving DGP problems
--------------------
You can solve a DGP ``Problem`` by calling its ``solve`` method with
``gp=True``.
.. code:: python
problem.solve(gp=True)
print("Optimal value:", problem.value)
print(x, ":", x.value)
print(y, ":", y.value)
print(z, ":", z.value)
print("Dual values: ", list(c.dual_value for c in constraints))
.. parsed-literal::
Optimal value: 1.9999999926890524
x : 0.9999999989968756
y : 1.9999999529045318
z : 1.000000020895385
Dual values: [1.1111111199586956, 1.94877846244994e-09, 0.1111111217156332, 0.11111112214962586]
If you forget to supply ``gp=True``, an error will be raised.
.. code:: python
try:
problem.solve()
except cp.DCPError as e:
print(e)
.. parsed-literal::
Problem does not follow DCP rules. However, the problem does follow DGP rules. Consider calling this function with `gp=True`.
4. Next steps
=============
Atoms
-----
CVXPY has a large library of log-log convex functions, including common
functions like :math:`\exp`, :math:`\log`, and the difference between
two numbers. Check out the tutorial on our website for the full list of
atoms: https://www.cvxpy.org/tutorial/dgp/index.html
References
----------
For a reference on DGP, consult the following paper:
https://web.stanford.edu/~boyd/papers/dgp.html