## Derived Variables¶

### Evaluated Fixed Variables¶

Some fixed variables may be derived from the values of other fixed variables. For example, air density, viscosity, and temperature are functions of altitude. These can be represented by a substitution or value that is a one-argument function accepting model.substitutions (for details, see Substitutions below).

# code from t_GPSubs.test_calcconst in tests/t_sub.py
x = Variable("x", "hours")
t_day = Variable("t_{day}", 12, "hours")
t_night = Variable("t_{night}", lambda c: 24 - c[t_day], "hours")
# note that t_night has a function as its value
m = Model(x, [x >= t_day, x >= t_night])
sol = m.solve(verbosity=0)
self.assertAlmostEqual(sol(t_night)/gpkit.ureg.hours, 12)
m.substitutions.update({t_day: ("sweep", [8, 12, 16])})
sol = m.solve(verbosity=0)
self.assertEqual(len(sol["cost"]), 3)
npt.assert_allclose(sol(t_day) + sol(t_night), 24)


### Evaluated Free Variables¶

Some free variables may be evaluated from the values of other (non-evaluated) free variables after the optimization is performed. For example, if the efficiency $$\nu$$ of a motor is not a GP-compatible variable, but $$(1-\nu)$$ is a valid GP variable, then $$\nu$$ can be calculated after solving. These evaluated free variables can be represented by a Variable with evalfn metadata. Note that this variable should not be used in constructing your model!

# code from t_constraints.test_evalfn in tests/t_sub.py
x = Variable("x")
x2 = Variable("x^2", evalfn=lambda v: v[x]**2)
m = Model(x, [x >= 2])
m.unique_varkeys = set([x2.key])
sol = m.solve(verbosity=0)
self.assertAlmostEqual(sol(x2), sol(x)**2)


For evaluated variables that can be used during a solution, see externalfn under Sequential Geometric Programs.

## Sweeps¶

Sweeps are useful for analyzing tradeoff surfaces. A sweep “value” is an Iterable of numbers, e.g. [1, 2, 3]. The simplest way to sweep a model is to call model.sweep({sweepvar: sweepvalues}), which will return a solution array but not change the model’s substitutions dictionary. If multiple sweepvars are given, the method will run them all as independent one-dimensional sweeps and return a list of one solution per sweep. The method model.autosweep({sweepvar: (start, end)}, tol=0.01) behaves very similarly, except that only the bounds of the sweep need be specified and the region in betwen will be swept to a maximum possible error of tol in the log of the cost. For details see 1D Autosweeps below.

### Sweep Substitutions¶

Alternatively, or to sweep a higher-dimensional grid, Variables can swept with a substitution value takes the form ('sweep', Iterable), such as ('sweep', np.linspace(1e6, 1e7, 100)). During variable declaration, giving an Iterable value for a Variable is assumed to be giving it a sweep value: for example, x = Variable("x", [1, 2, 3]) will sweep x over three values.

Vector variables may also be substituted for: y = VectorVariable(3, "y", ("sweep" ,[[1, 2], [1, 2], [1, 2]]) will sweep $$y\ \forall~y_i\in\left\{1,2\right\}$$.

A Model with sweep substitutions will solve for all possible combinations: e.g., if there’s a variable x with value ('sweep', [1, 3]) and a variable y with value ('sweep', [14, 17]) then the gp will be solved four times, for $$(x,y)\in\left\{(1, 14),\ (1, 17),\ (3, 14),\ (3, 17)\right\}$$. The returned solutions will be a one-dimensional array (or 2-D for vector variables), accessed in the usual way.

### Parallel Sweeps¶

During a normal sweep, each result is independent, so they can be run in parallel. To use this feature, run \$ ipcluster start at a terminal: it will automatically start a number of iPython parallel computing engines equal to the number of cores on your machine, and when you next import gpkit you should see a note like Using parallel execution of sweeps on 4 clients. If you do, then all sweeps performed with that import of gpkit will be parallelized.

This parallelization sets the stage for gpkit solves to be outsourced to a server, which may be valuable for faster results; alternately, it could allow the use of gpkit without installing a solver.

### 1D Autosweeps¶

If you’re only sweeping over a single variable, autosweeping lets you specify a tolerance for cost error instead of a number of exact positions to solve at. GPkit will then search the sweep segment for a locally optimal number of sweeps that can guarantee a max absolute error on the log of the cost.

Accessing variable and cost values from an autosweep is slightly different, as can be seen in this example:

"Show autosweep_1d functionality"
import numpy as np
import gpkit
from gpkit import units, Variable, Model
from gpkit.tools.autosweep import autosweep_1d
from gpkit.small_scripts import mag

A = Variable("A", "m**2")
l = Variable("l", "m")

m1 = Model(A**2, [A >= l**2 + units.m**2])
tol1 = 1e-3
bst1 = autosweep_1d(m1, tol1, l, [1, 10], verbosity=0)
print "Solved after %2i passes, cost logtol +/-%.3g" % (bst1.nsols, bst1.tol)
# autosweep solution accessing
l_vals = np.linspace(1, 10, 10)
sol1 = bst1.sample_at(l_vals)
print "values of l:", l_vals
print "values of A:", sol1("A")
cost_estimate = sol1["cost"]
cost_lb, cost_ub = sol1.cost_lb(), sol1.cost_ub()
print "cost lower bound:", cost_lb
print "cost estimate:   ", cost_estimate
print "cost upper bound:", cost_ub
# you can evaluate arbitrary posynomials
np.testing.assert_allclose(mag(2*sol1(A)), mag(sol1(2*A)))
assert (sol1["cost"] == sol1(A**2)).all()
# the cost estimate is the logspace mean of its upper and lower bounds
np.testing.assert_allclose((np.log(mag(cost_lb)) + np.log(mag(cost_ub)))/2,
np.log(mag(cost_estimate)))

# this problem is two intersecting lines in logspace
m2 = Model(A**2, [A >= (l/3)**2, A >= (l/3)**0.5 * units.m**1.5])
tol2 = {"mosek": 1e-12, "cvxopt": 1e-7,
"mosek_cli": 1e-6}[gpkit.settings["default_solver"]]
bst2 = autosweep_1d(m2, tol2, l, [1, 10], verbosity=0)
print "Solved after %2i passes, cost logtol +/-%.3g" % (bst2.nsols, bst2.tol)
print "Table of solutions used in the autosweep:"
print bst2.solarray.table()


If you need access to the raw solutions arrays, the smallest simplex tree containing any given point can be gotten with min_bst = bst.min_bst(val), the extents of that tree with bst.bounds and solutions of that tree with bst.sols. More information is in help(bst).

## Tight ConstraintSets¶

Tight ConstraintSets will warn if any inequalities they contain are not tight (that is, the right side does not equal the left side) after solving. This is useful when you know that a constraint _should_ be tight for a given model, but reprenting it as an equality would be non-convex.

from gpkit import Variable, Model
from gpkit.constraints.tight import Tight

Tight.reltol = 1e-2  # set the global tolerance of Tight
x = Variable('x')
x_min = Variable('x_{min}', 2)
m = Model(x, [Tight([x >= 1], reltol=1e-3),  # set the specific tolerance
x >= x_min])
m.solve(verbosity=0)  # prints warning


## Substitutions¶

Substitutions are a general-purpose way to change every instance of one variable into either a number or another variable.

### Substituting into Posynomials, NomialArrays, and GPs¶

The examples below all use Posynomials and NomialArrays, but the syntax is identical for GPs (except when it comes to sweep variables).

# adapted from t_sub.py / t_NomialSubs / test_Basic
from gpkit import Variable
x = Variable("x")
p = x**2
assert p.sub(x, 3) == 9
assert p.sub(x.varkeys["x"], 3) == 9
assert p.sub("x", 3) == 9


Here the variable x is being replaced with 3 in three ways: first by substituting for x directly, then by substituting for the VarKey("x"), then by substituting the string “x”. In all cases the substitution is understood as being with the VarKey: when a variable is passed in the VarKey is pulled out of it, and when a string is passed in it is used as an argument to the Posynomial’s varkeys dictionary.

### Substituting multiple values¶

# adapted from t_sub.py / t_NomialSubs / test_Vector
from gpkit import Variable, VectorVariable
x = Variable("x")
y = Variable("y")
z = VectorVariable(2, "z")
p = x*y*z
assert all(p.sub({x: 1, "y": 2}) == 2*z)
assert all(p.sub({x: 1, y: 2, "z": [1, 2]}) == z.sub(z, [2, 4]))


To substitute in multiple variables, pass them in as a dictionary where the keys are what will be replaced and values are what it will be replaced with. Note that you can also substitute for VectorVariables by their name or by their NomialArray.

### Substituting with nonnumeric values¶

You can also substitute in sweep variables (see Sweeps), strings, and monomials:

# adapted from t_sub.py / t_NomialSubs
from gpkit import Variable
from gpkit.small_scripts import mag

x = Variable("x", "m")
xvk = x.varkeys.values()[0]
descr_before = x.exp.keys()[0].descr
y = Variable("y", "km")
yvk = y.varkeys.values()[0]
for x_ in ["x", xvk, x]:
for y_ in ["y", yvk, y]:
if not isinstance(y_, str) and type(xvk.units) != str:
expected = 0.001
else:
expected = 1.0
assert abs(expected - mag(x.sub(x_, y_).c)) < 1e-6
if type(xvk.units) != str:
# this means units are enabled
z = Variable("z", "s")
# y.sub(y, z) will raise ValueError due to unit mismatch


Note that units are preserved, and that the value can be either a string (in which case it just renames the variable), a varkey (in which case it changes its description, including the name) or a Monomial (in which case it substitutes for the variable with a new monomial).

### Substituting with replacement¶

Any of the substitutions above can be run with p.sub(*args, replace=True) to clobber any previously-substituted values.

### Fixed Variables¶

When a Model is created, any fixed Variables are used to form a dictionary: {var: var.descr["value"] for var in self.varlocs if "value" in var.descr}. This dictionary in then substituted into the Model’s cost and constraints before the substitutions argument is (and hence values are supplanted by any later substitutions).

solution.subinto(p) will substitute the solution(s) for variables into the posynomial p, returning a NomialArray. For a non-swept solution, this is equivalent to p.sub(solution["variables"]).

You can also substitute by just calling the solution, i.e. solution(p). This returns a numpy array of just the coefficients (c) of the posynomial after substitution, and will raise a ValueError if some of the variables in p were not found in solution.

### Freeing Fixed Variables¶

After creating a Model, it may be useful to “free” a fixed variable and resolve. This can be done using the command del m.substitutions["x"], where m is a Model. An example of how to do this is shown below.

from gpkit import Variable, Model
x = Variable("x")
y = Variable("y", 3)  # fix value to 3
m = Model(x, [x >= 1 + y, y >= 1])
_ = m.solve()  # optimal cost is 4; y appears in sol["constants"]

del m.substitutions["y"]
_ = m.solve()  # optimal cost is 2; y appears in Free Variables


Note that del m.substitutions["y"] affects m but not y.key. y.value will still be 3, and if y is used in a new model, it will still carry the value of 3.

## Composite Objectives¶

Given $$n$$ posynomial objectives $$g_i$$, you can sweep out the problem’s Pareto frontier with the composite objective:

$$g_0 w_0 \prod_{i\not=0} v_i + g_1 w_1 \prod_{i\not=1} v_i + ... + g_n \prod_i v_i$$

where $$i \in 0 ... n-1$$ and $$v_i = 1- w_i$$ and $$w_i \in [0, 1]$$

GPkit has the helper function composite_objective for constructing these.

import numpy as np
import gpkit

L, W = gpkit.Variable("L"), gpkit.Variable("W")

eqns = [L >= 1, W >= 1, L*W == 10]

co_sweep = [0] + np.logspace(-6, 0, 10).tolist()

obj = gpkit.tools.composite_objective(L+W, W**-1 * L**-3,
normsub={L:10, W: 10},
sweep=co_sweep)

m = gpkit.Model(obj, eqns)
m.solve()


The normsub argument specifies an expected value for your solution to normalize the different $$g_i$$ (you can also do this by hand). The feasibility of the problem should not depend on the normalization, but the spacing of the sweep will.

The sweep argument specifies what points between 0 and 1 you wish to sample the weights at. If you want different resolutions or spacings for different weights, the sweeps argument accepts a list of sweep arrays.