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