Advanced Features¶
This section of the tutorial covers features of CVXPY intended for users with advanced knowledge of convex optimization. We recommend Convex Optimization by Boyd and Vandenberghe as a reference for any terms you are unfamiliar with.
Dual variables¶
You can use CVXPY to find the optimal dual variables for a problem. When you call prob.solve()
each dual variable in the solution is stored in the dual_value
field of the constraint it corresponds to.
import cvxpy as cp
# Create two scalar optimization variables.
x = cp.Variable()
y = cp.Variable()
# Create two constraints.
constraints = [x + y == 1,
x  y >= 1]
# Form objective.
obj = cp.Minimize((x  y)**2)
# Form and solve problem.
prob = cp.Problem(obj, constraints)
prob.solve()
# The optimal dual variable (Lagrange multiplier) for
# a constraint is stored in constraint.dual_value.
print("optimal (x + y == 1) dual variable", constraints[0].dual_value)
print("optimal (x  y >= 1) dual variable", constraints[1].dual_value)
print("x  y value:", (x  y).value)
optimal (x + y == 1) dual variable 6.47610300459e18
optimal (x  y >= 1) dual variable 2.00025244976
x  y value: 0.999999986374
The dual variable for x  y >= 1
is 2. By complementarity this implies that x  y
is 1, which we can see is true. The fact that the dual variable is nonzero also tells us that if we tighten x  y >= 1
, (i.e., increase the righthand side), the optimal value of the problem will increase.
Attributes¶
Variables and parameters can be created with attributes specifying additional properties.
For example, Variable(nonneg=True)
is a scalar variable constrained to be nonnegative.
Similarly, Parameter(nonpos=True)
is a scalar parameter constrained to be nonpositive.
The full constructor for Leaf
(the parent class
of Variable
and
Parameter
) is given below.

Leaf
(shape=None, name=None, value=None, nonneg=False, nonpos=False, symmetric=False, diag=False, PSD=False, NSD=False, boolean=False, integer=False)¶ Creates a Leaf object (e.g., Variable or Parameter). Only one attribute can be active (set to True).
Parameters:  shape (tuple or int) – The variable dimensions (0D by default). Cannot be more than 2D.
 name (str) – The variable name.
 value (numeric type) – A value to assign to the variable.
 nonneg (bool) – Is the variable constrained to be nonnegative?
 nonpos (bool) – Is the variable constrained to be nonpositive?
 symmetric (bool) – Is the variable constrained to be symmetric?
 hermitian (bool) – Is the variable constrained to be Hermitian?
 diag (bool) – Is the variable constrained to be diagonal?
 complex (bool) – Is the variable complex valued?
 imag (bool) – Is the variable purely imaginary?
 PSD (bool) – Is the variable constrained to be symmetric positive semidefinite?
 NSD (bool) – Is the variable constrained to be symmetric negative semidefinite?
 boolean (bool or list of tuple) – Is the variable boolean (i.e., 0 or 1)? True, which constrains the entire variable to be boolean, False, or a list of indices which should be constrained as boolean, where each index is a tuple of length exactly equal to the length of shape.
 integer (bool or list of tuple) – Is the variable integer? The semantics are the same as the boolean argument.
The value
field of Variables and Parameters can be assigned a value after construction,
but the assigned value must satisfy the object attributes.
A Euclidean projection onto the set defined by the attributes is given by the
project
method.
p = Parameter(nonneg=True)
try:
p.value = 1
except Exception as e:
print(e)
print("Projection:", p.project(1))
Parameter value must be nonnegative.
Projection: 0.0
A sensible idiom for assigning values to leaves is
leaf.value = leaf.project(val)
,
ensuring that the assigned value satisfies the leaf’s properties.
A slightly more efficient variant is
leaf.project_and_assign(val)
,
which projects and assigns the value directly, without additionally checking
that the value satisfies the leaf’s properties. In most cases project
and
checking that a value satisfies a leaf’s properties are cheap operations (i.e.,
\(O(n)\)), but for symmetric positive semidefinite or negative semidefinite
leaves, the operations compute an eigenvalue decomposition.
Many attributes, such as nonnegativity and symmetry, can be easily specified with constraints.
What is the advantage then of specifying attributes in a variable?
The main benefit is that specifying attributes enables more finegrained DCP analysis.
For example, creating a variable x
via x = Variable(nonpos=True)
informs the DCP analyzer that x
is nonpositive.
Creating the variable x
via x = Variable()
and adding the constraint x >= 0
separately does not provide any information
about the sign of x
to the DCP analyzer.
Semidefinite matrices¶
Many convex optimization problems involve constraining matrices to be positive or negative semidefinite (e.g., SDPs).
You can do this in CVXPY in two ways.
The first way is to use
Variable((n, n), PSD=True)
to create an n
by n
variable constrained to be symmetric and positive semidefinite. For example,
# Creates a 100 by 100 positive semidefinite variable.
X = cp.Variable((100, 100), PSD=True)
# You can use X anywhere you would use
# a normal CVXPY variable.
obj = cp.Minimize(cp.norm(X) + cp.sum(X))
The second way is to create a positive semidefinite cone constraint using the >>
or <<
operator.
If X
and Y
are n
by n
variables,
the constraint X >> Y
means that \(z^T(X  Y)z \geq 0\), for all \(z \in \mathcal{R}^n\).
In other words, \(X + X^T\) is positive semidefinite.
The constraint does not require that X
and Y
be symmetric.
Both sides of a postive semidefinite cone constraint must be square matrices and affine.
The following code shows how to constrain matrix expressions to be positive or negative semidefinite (but not necessarily symmetric).
# expr1 must be positive semidefinite.
constr1 = (expr1 >> 0)
# expr2 must be negative semidefinite.
constr2 = (expr2 << 0)
To constrain a matrix expression to be symmetric, simply write
# expr must be symmetric.
constr = (expr == expr.T)
You can also use Variable((n, n), symmetric=True)
to create an n
by n
variable constrained to be symmetric.
The difference between specifying that a variable is symmetric via attributes and adding the constraint X == X.T
is that
attributes are parsed for DCP information and a symmetric variable is defined over the (lower dimensional) vector space of symmetric matrices.
Mixedinteger programs¶
In mixedinteger programs, certain variables are constrained to be boolean (i.e., 0 or 1) or integer valued. You can construct mixedinteger programs by creating variables with the attribute that they have only boolean or integer valued entries:
# Creates a 10vector constrained to have boolean valued entries.
x = cp.Variable(10, boolean=True)
# expr1 must be boolean valued.
constr1 = (expr1 == x)
# Creates a 5 by 7 matrix constrained to have integer valued entries.
Z = cp.Variable((5, 7), integer=True)
# expr2 must be integer valued.
constr2 = (expr2 == Z)
Complex valued expressions¶
By default variables and parameters are real valued.
Complex valued variables and parameters can be created by setting the attribute complex=True
.
Similarly, purely imaginary variables and parameters can be created by setting the attributes imag=True
.
Expressions containing complex variables, parameters, or constants may be complex valued.
The functions is_real
, is_complex
, and is_imag
return whether an expression is purely real, complex, or purely imaginary, respectively.
# A complex valued variable.
x = cp.Variable(complex=True)
# A purely imaginary parameter.
p = cp.Parameter(imag=True)
print("p.is_imag() = ", p.is_imag())
print("(x + 2).is_real() = ", (x + 2).is_real())
p.is_imag() = True
(x + 2).is_real() = False
The toplevel expressions in the problem objective and inequality constraints must be real valued,
but subexpressions may be complex.
Arithmetic and all linear atoms are defined for complex expressions.
The nonlinear atoms abs
and all norms except norm(X, p)
for p < 1
are also defined for complex expressions.
All atoms whose domain is symmetric matrices are defined for Hermitian matrices.
Similarly, the atoms quad_form(x, P)
and matrix_frac(x, P)
are defined for complex x
and Hermitian P
.
Lastly, equality and postive semidefinite constraints are defined for complex expressions.
The following additional atoms are provided for working with complex expressions:
real(expr)
gives the real part ofexpr
.imag(expr)
gives the imaginary part ofexpr
(i.e.,expr = real(expr) + 1j*imag(expr)
).conj(expr)
gives the complex conjugate ofexpr
.expr.H
gives the Hermitian (conjugate) transpose ofexpr
.
Transforms¶
Transforms provide additional ways of manipulating CVXPY objects
beyond the atomic functions. For example, the indicator
transform converts a list of constraints into an
expression representing the convex function that takes value 0 when the
constraints hold and \(\infty\) when they are violated.
x = cp.Variable()
constraints = [0 <= x, x <= 1]
expr = cp.indicator(constraints)
x.value = .5
print("expr.value = ", expr.value)
x.value = 2
print("expr.value = ", expr.value)
expr.value = 0.0
expr.value = inf
The full set of transforms available is discussed in Transforms.
Problem arithmetic¶
For convenience, arithmetic operations have been overloaded for problems and objectives. Problem arithmetic is useful because it allows you to write a problem as a sum of smaller problems. The rules for adding, subtracting, and multiplying objectives are given below.
# Addition and subtraction.
Minimize(expr1) + Minimize(expr2) == Minimize(expr1 + expr2)
Maximize(expr1) + Maximize(expr2) == Maximize(expr1 + expr2)
Minimize(expr1) + Maximize(expr2) # Not allowed.
Minimize(expr1)  Maximize(expr2) == Minimize(expr1  expr2)
# Multiplication (alpha is a positive scalar).
alpha*Minimize(expr) == Minimize(alpha*expr)
alpha*Maximize(expr) == Maximize(alpha*expr)
alpha*Minimize(expr) == Maximize(alpha*expr)
alpha*Maximize(expr) == Minimize(alpha*expr)
The rules for adding and multiplying problems are equally straightforward:
# Addition and subtraction.
prob1 + prob2 == Problem(prob1.objective + prob2.objective,
prob1.constraints + prob2.constraints)
prob1  prob2 == Problem(prob1.objective  prob2.objective,
prob1.constraints + prob2.constraints)
# Multiplication (alpha is any scalar).
alpha*prob == Problem(alpha*prob.objective, prob.constraints)
Note that the +
operator concatenates lists of constraints,
since this is the default behavior for Python lists.
The inplace operators +=
, =
, and *=
are also supported for
objectives and problems and follow the same rules as above.
Solve method options¶
The solve
method takes optional arguments that let you change how CVXPY solves the problem. Here is the signature for the solve
method:

solve
(solver=None, verbose=False, gp=False, **kwargs)¶ Solves a DCP compliant optimization problem.
Parameters:  solver (str, optional) – The solver to use.
 verbose (bool, optional) – Overrides the default of hiding solver output.
 gp (bool, optional) – If True, parses the problem as a disciplined geometric program instead of a disciplined convex program.
 kwargs – Additional keyword arguments specifying solver specific options.
Returns: The optimal value for the problem, or a string indicating why the problem could not be solved.
We will discuss the optional arguments in detail below.
Choosing a solver¶
CVXPY is distributed with the open source solvers ECOS, ECOS_BB, OSQP, and SCS. Many other solvers can be called by CVXPY if installed separately. The table below shows the types of problems the supported solvers can handle.
LP  QP  SOCP  SDP  EXP  MIP  

CBC  X  X  
GLPK  X  
GLPK_MI  X  X  
OSQP  X  X  
CPLEX  X  X  X  X  
Elemental  X  X  X  
ECOS  X  X  X  X  
ECOS_BB  X  X  X  X  X  
GUROBI  X  X  X  X  
MOSEK  X  X  X  X  
CVXOPT  X  X  X  X  X  
SCS  X  X  X  X  X 
Here EXP refers to problems with exponential cone constraints. The exponential cone is defined as
\(\{(x,y,z) \mid y > 0, y\exp(x/y) \leq z \} \cup \{ (x,y,z) \mid x \leq 0, y = 0, z \geq 0\}\).
You cannot specify cone constraints explicitly in CVXPY, but cone constraints are added when CVXPY converts the problem into standard form.
By default CVXPY calls the solver most specialized to the problem type. For example, ECOS is called for SOCPs. SCS can both handle all problems (except mixedinteger programs). ECOS_BB is called for mixedinteger LPs and SOCPs. If the problem is a QP, CVXPY will use OSQP.
You can change the solver called by CVXPY using the solver
keyword argument. If the solver you choose cannot solve the problem, CVXPY will raise an exception. Here’s example code solving the same problem with different solvers.
# Solving a problem with different solvers.
x = cp.Variable(2)
obj = cp.Minimize(x[0] + cp.norm(x, 1))
constraints = [x >= 2]
prob = cp.Problem(obj, constraints)
# Solve with OSQP.
prob.solve(solver=cp.OSQP)
print("optimal value with OSQP:", prob.value)
# Solve with ECOS.
prob.solve(solver=cp.ECOS)
print("optimal value with ECOS:", prob.value)
# Solve with ECOS_BB.
prob.solve(solver=cp.ECOS_BB)
print("optimal value with ECOS_BB:", prob.value)
# Solve with CVXOPT.
prob.solve(solver=cp.CVXOPT)
print("optimal value with CVXOPT:", prob.value)
# Solve with SCS.
prob.solve(solver=cp.SCS)
print("optimal value with SCS:", prob.value)
# Solve with GLPK.
prob.solve(solver=cp.GLPK)
print("optimal value with GLPK:", prob.value)
# Solve with GLPK_MI.
prob.solve(solver=cp.GLPK_MI)
print("optimal value with GLPK_MI:", prob.value)
# Solve with GUROBI.
prob.solve(solver=cp.GUROBI)
print("optimal value with GUROBI:", prob.value)
# Solve with MOSEK.
prob.solve(solver=cp.MOSEK)
print("optimal value with MOSEK:", prob.value)
# Solve with Elemental.
prob.solve(solver=cp.ELEMENTAL)
print("optimal value with Elemental:", prob.value)
# Solve with CBC.
prob.solve(solver=cp.CBC)
print("optimal value with CBC:", prob.value)
# Solve with CPLEX.
prob.solve(solver=cp.CPLEX)
print "optimal value with CPLEX:", prob.value
optimal value with OSQP: 6.0
optimal value with ECOS: 5.99999999551
optimal value with ECOS_BB: 5.99999999551
optimal value with CVXOPT: 6.00000000512
optimal value with SCS: 6.00046055789
optimal value with GLPK: 6.0
optimal value with GLPK_MI: 6.0
optimal value with GUROBI: 6.0
optimal value with MOSEK: 6.0
optimal value with Elemental: 6.0000044085242727
optimal value with CBC: 6.0
optimal value with CPLEX: 6.0
Use the installed_solvers
utility function to get a list of the solvers your installation of CVXPY supports.
print installed_solvers()
['CBC', 'CVXOPT', 'MOSEK', 'GLPK', 'GLPK_MI', 'ECOS_BB', 'ECOS', 'SCS', 'GUROBI', 'ELEMENTAL', 'OSQP', 'CPLEX']
Viewing solver output¶
All the solvers can print out information about their progress while solving the problem. This information can be useful in debugging a solver error. To see the output from the solvers, set verbose=True
in the solve method.
# Solve with ECOS and display output.
prob.solve(solver=cp.ECOS, verbose=True)
print "optimal value with ECOS:", prob.value
ECOS 1.0.3  (c) A. Domahidi, Automatic Control Laboratory, ETH Zurich, 20122014.
It pcost dcost gap pres dres k/t mu step IR
0 +0.000e+00 +4.000e+00 +2e+01 2e+00 1e+00 1e+00 3e+00 N/A 1 1 
1 +6.451e+00 +8.125e+00 +5e+00 7e01 5e01 7e01 7e01 0.7857 1 1 1
2 +6.788e+00 +6.839e+00 +9e02 1e02 8e03 3e02 2e02 0.9829 1 1 1
3 +6.828e+00 +6.829e+00 +1e03 1e04 8e05 3e04 2e04 0.9899 1 1 1
4 +6.828e+00 +6.828e+00 +1e05 1e06 8e07 3e06 2e06 0.9899 2 1 1
5 +6.828e+00 +6.828e+00 +1e07 1e08 8e09 4e08 2e08 0.9899 2 1 1
OPTIMAL (within feastol=1.3e08, reltol=1.5e08, abstol=1.0e07).
Runtime: 0.000121 seconds.
optimal value with ECOS: 6.82842708233
Solving disciplined geometric programs¶
TODO see XXX
Solver stats¶
When the solve
method is called on a problem object and a solver is invoked,
the problem object records the optimal value, the values of the primal and dual variables,
and several solver statistics.
We have already discussed how to view the optimal value and variable values.
The solver statistics are accessed via the problem.solver_stats
attribute,
which returns a SolverStats
object.
For example, problem.solver_stats.solve_time
gives the time it took the solver to solve the problem.
Warm start¶
When solving the same problem for multiple values of a parameter, many solvers can exploit work from previous solves (i.e., warm start).
For example, the solver might use the previous solution as an initial point or reuse cached matrix factorizations.
Warm start is enabled by default and controlled with the warm_start
solver option.
The code below shows how warm start can accelerate solving a sequence of related leastsquares problems.
import cvxpy as cp
import numpy
# Problem data.
m = 2000
n = 1000
numpy.random.seed(1)
A = numpy.random.randn(m, n)
b = cp.Parameter(m)
# Construct the problem.
x = cp.Variable(n)
prob = cp.Problem(cp.Minimize(cp.sum_squares(A*x  b)),
[x >= 0])
b.value = numpy.random.randn(m)
prob.solve()
print("First solve time:", prob.solve_time)
b.value = numpy.random.randn(m)
prob.solve(warm_start=True)
print("Second solve time:", prob.solve_time)
First solve time: 11.14
Second solve time: 2.95
The speed up in this case comes from caching the KKT matrix factorization.
If A
were a parameter, factorization caching would not be possible and the benefit of
warm start would only be a good initial point.
Setting solver options¶
The OSQP, ECOS, ECOS_BB, MOSEK, CBC, CVXOPT, and SCS Python interfaces allow you to set solver options such as the maximum number of iterations. You can pass these options along through CVXPY as keyword arguments.
For example, here we tell SCS to use an indirect method for solving linear equations rather than a direct method.
# Solve with SCS, use sparseindirect method.
prob.solve(solver=cp.SCS, verbose=True, use_indirect=True)
print "optimal value with SCS:", prob.value

SCS v1.0.5  Splitting Conic Solver
(c) Brendan O'Donoghue, Stanford University, 2012

Linsys: sparseindirect, nnz in A = 13, CG tol ~ 1/iter^(2.00)
EPS = 1.00e03, ALPHA = 1.80, MAX_ITERS = 2500, NORMALIZE = 1, SCALE = 5.00
Variables n = 5, constraints m = 9
Cones: linear vars: 6
soc vars: 3, soc blks: 1
Setup time: 2.78e04s

Iter  pri res  dua res  rel gap  pri obj  dua obj  kap/tau  time (s)

0 4.60e+00 5.78e01 nan inf inf inf 3.86e05
60 3.92e05 1.12e04 6.64e06 6.83e+00 6.83e+00 1.41e17 9.51e05

Status: Solved
Timing: Total solve time: 9.76e05s
Linsys: avg # CG iterations: 1.00, avg solve time: 2.24e07s
Cones: avg projection time: 4.90e08s

Error metrics:
Ax + s  b_2 / (1 + b_2) = 3.9223e05
A'y + c_2 / (1 + c_2) = 1.1168e04
c'x + b'y / (1 + c'x + b'y) = 6.6446e06
dist(s, K) = 0, dist(y, K*) = 0, s'y = 0

c'x = 6.8284, b'y = 6.8285
============================================================================
optimal value with SCS: 6.82837896975
Here is the complete list of solver options.
OSQP options:
See OSQP documentation.
ECOS options:
'max_iters'
 maximum number of iterations (default: 100).
'abstol'
 absolute accuracy (default: 1e7).
'reltol'
 relative accuracy (default: 1e6).
'feastol'
 tolerance for feasibility conditions (default: 1e7).
'abstol_inacc'
 absolute accuracy for inaccurate solution (default: 5e5).
'reltol_inacc'
 relative accuracy for inaccurate solution (default: 5e5).
'feastol_inacc'
 tolerance for feasibility condition for inaccurate solution (default: 1e4).
ECOS_BB options:
'mi_max_iters'
 maximum number of branch and bound iterations (default: 1000)
'mi_abs_eps'
 absolute tolerance between upper and lower bounds (default: 1e6)
'mi_rel_eps'
 relative tolerance, (UL)/L, between upper and lower bounds (default: 1e3)
MOSEK options:
'mosek_params'
 A dictionary of MOSEK parameters. Refer to MOSEK’s Python or C API for
details. Note that if parameters are given as stringvalue pairs, parameter
names must be of the form
'MSK_DPAR_BASIS_TOL_X'
as in the C API. Alternatively, Python enum options like'mosek.dparam.basis_tol_x'
are also supported. 'save_file'
 The name of a file where MOSEK will save the problem just before optimization. Refer to MOSEK documentation for a list of supported file formats. File format is chosen based on the extension.
CVXOPT options:
'max_iters'
 maximum number of iterations (default: 100).
'abstol'
 absolute accuracy (default: 1e7).
'reltol'
 relative accuracy (default: 1e6).
'feastol'
 tolerance for feasibility conditions (default: 1e7).
'refinement'
 number of iterative refinement steps after solving KKT system (default: 1).
'kktsolver'
 The KKT solver used. The default, “chol”, does a Cholesky factorization with preprocessing to make A and [A; G] full rank. The “robust” solver does an LDL factorization without preprocessing. It is slower, but more robust.
SCS options:
'max_iters'
 maximum number of iterations (default: 2500).
'eps'
 convergence tolerance (default: 1e3).
'alpha'
 relaxation parameter (default: 1.8).
'scale'
 balance between minimizing primal and dual residual (default: 5.0).
'normalize'
 whether to precondition data matrices (default: True).
'use_indirect'
 whether to use indirect solver for KKT sytem (instead of direct) (default: True).
CBC options:
Cutgeneration through CGL
 General remarks:
 some of these cutgenerators seem to be buggy (observed problems with AllDifferentCuts, RedSplitCuts, LandPCuts, PreProcessCuts)
 a few of these cutgenerators will generate noisy output even if
'verbose=False'
 The following cutgenerators are available:
GomoryCuts
,MIRCuts
,MIRCuts2
,TwoMIRCuts
,ResidualCapacityCuts
,KnapsackCuts
FlowCoverCuts
,CliqueCuts
,LiftProjectCuts
,AllDifferentCuts
,OddHoleCuts
,RedSplitCuts
,LandPCuts
,PreProcessCuts
,ProbingCuts
,SimpleRoundingCuts
.'CutGenName'
 if cutgenerator is activated (e.g.
'GomoryCuts=True'
)
CPLEX options:
'cplex_params'
 a dictionary where the keyvalue pairs are composed of parameter names (as used in the CPLEX Python API) and parameter values. For example, to set the advance start switch parameter (i.e., CPX_PARAM_ADVIND), use “advance” for the parameter name. For the data consistency checking and modeling assistance parameter (i.e., CPX_PARAM_DATACHECK), use “read.datacheck” for the parameter name, and so on.
'cplex_filename'
 a string specifying the filename to which the problem will be written. For example, use “model.lp”, “model.sav”, or “model.mps” to export to the LP, SAV, and MPS formats, respectively.
Getting the standard form¶
If you are interested in getting the standard form that CVXPY produces for a problem, you can use the get_problem_data
method. Calling get_problem_data(solver)
on a problem object returns a dict of the arguments that CVXPY would pass to that solver. If the solver you choose cannot solve the problem, CVXPY will raise an exception.
# Get OSQP arguments.
data = prob.get_problem_data(cvx.OSQP)
# Get ECOS arguments.
data = prob.get_problem_data(cvx.ECOS)
# Get ECOS_BB arguments.
data = prob.get_problem_data(cvx.ECOS_BB)
# Get CVXOPT arguments.
data = prob.get_problem_data(cvx.CVXOPT)
# Get SCS arguments.
data = prob.get_problem_data(cvx.SCS)
After you solve the standard conic form problem returned by get_problem_data
, you can unpack the raw solver output using the unpack_results
method. Calling unpack_results(solver, solver_output)
on a problem will update the values of all primal and dual variables as well as the problem value and status.
For example, the following code is equivalent to solving the problem directly with CVXPY:
# Get ECOS arguments.
data = prob.get_problem_data(cvx.ECOS)
# Call ECOS solver.
solver_output = ecos.solve(data["c"], data["G"], data["h"],
data["dims"], data["A"], data["b"])
# Unpack raw solver output.
prob.unpack_results(cvx.ECOS, solver_output)
Reductions¶
CVXPY uses a system of reductions to rewrite problems from the form provided by the user into the standard form that a solver will accept. A reduction is a transformation from one problem to an equivalent problem. Two problems are equivalent if a solution of one can be converted efficiently to a solution of the other. Reductions take a CVXPY Problem as input and output a CVXPY Problem. The full set of reductions available is discussed in Reductions.