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.

from cvxpy import *

# Create two scalar optimization variables.
x = Variable()
y = Variable()

# Create two constraints.
constraints = [x + y == 1,
               x - y >= 1]

# Form objective.
obj = Minimize(square(x - y))

# Form and solve problem.
prob = 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 6.47610300459e-18
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 non-zero also tells us that if we tighten x - y >= 1, (i.e., increase the right-hand side), the optimal value of the problem will increase.

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 Semidef(n) 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 = Semidef(100)

# You can use X anywhere you would use
# a normal CVXPY variable.
obj = Minimize(norm(X) + sum_entries(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 \(z \in \mathcal{R}^n\). 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 Symmetric(n) to create an n by n variable constrained to be symmetric.

Mixed-integer programs¶

In mixed-integer programs, certain variables are constrained to be boolean or integer valued. You can construct mixed-integer programs using the Bool and Int constructors. These take the same arguments as the Variable constructor, and they return a variable constrained to have only boolean or integer valued entries.

The following code shows the Bool and Int constructors in action:

# Creates a 10-vector constrained to have boolean valued entries.
x = Bool(10)

# expr1 must be boolean valued.
constr1 = (expr1 == x)

# Creates a 5 by 7 matrix constrained to have integer valued entries.
Z = Int(5, 7)

# expr2 must be integer valued.
constr2 = (expr2 == Z)

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 in-place 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, **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. 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, CVXOPT, and SCS. CVXPY also supports GLPK and GLPK_MI via the CVXOPT GLPK interface, CBC, MOSEK, GUROBI, and Elemental. The table below shows the types of problems the solvers can handle.

	LP	SOCP	SDP	EXP	MIP
CBC	X				X
GLPK	X
GLPK_MI	X				X
Elemental	X	X
ECOS	X	X		X
ECOS_BB	X	X		X	X
GUROBI	X	X			X
MOSEK	X	X	X
CVXOPT	X	X	X	X
SCS	X	X	X	X

A special solver `LS`_ is also available. It is unable to solve any of the problem types in the table above, but it recognizes and solves linearly constrained least squares problems very quickly.

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 and CVXOPT can both handle all problems (except mixed-integer programs). CVXOPT is preferred by default. For many problems SCS will be faster, though less accurate. ECOS_BB is called for mixed-integer LPs and SOCPs. If the problem has a quadratic objective function and equality constraints only, CVXPY will use `LS`_.

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 = Variable(2)
obj = Minimize(x[0] + norm(x, 1))
constraints = [x >= 2]
prob = Problem(obj, constraints)

# Solve with ECOS.
prob.solve(solver=ECOS)
print "optimal value with ECOS:", prob.value

# Solve with ECOS_BB.
prob.solve(solver=ECOS_BB)
print "optimal value with ECOS_BB:", prob.value

# Solve with CVXOPT.
prob.solve(solver=CVXOPT)
print "optimal value with CVXOPT:", prob.value

# Solve with SCS.
prob.solve(solver=SCS)
print "optimal value with SCS:", prob.value

# Solve with GLPK.
prob.solve(solver=GLPK)
print "optimal value with GLPK:", prob.value

# Solve with GLPK_MI.
prob.solve(solver=GLPK_MI)
print "optimal value with GLPK_MI:", prob.value

# Solve with GUROBI.
prob.solve(solver=GUROBI)
print "optimal value with GUROBI:", prob.value

# Solve with MOSEK.
prob.solve(solver=MOSEK)
print "optimal value with MOSEK:", prob.value

# Solve with Elemental.
prob.solve(solver=ELEMENTAL)
print "optimal value with Elemental:", prob.value

# Solve with CBC.
prob.solve(solver=CBC)
print "optimal value with CBC:", prob.value

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

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', 'LS']

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=ECOS, verbose=True)
print "optimal value with ECOS:", prob.value

ECOS 1.0.3 - (c) A. Domahidi, Automatic Control Laboratory, ETH Zurich, 2012-2014.

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   7e-01   5e-01   7e-01   7e-01   0.7857   1 1 1
 2   +6.788e+00   +6.839e+00   +9e-02   1e-02   8e-03   3e-02   2e-02   0.9829   1 1 1
 3   +6.828e+00   +6.829e+00   +1e-03   1e-04   8e-05   3e-04   2e-04   0.9899   1 1 1
 4   +6.828e+00   +6.828e+00   +1e-05   1e-06   8e-07   3e-06   2e-06   0.9899   2 1 1
 5   +6.828e+00   +6.828e+00   +1e-07   1e-08   8e-09   4e-08   2e-08   0.9899   2 1 1

OPTIMAL (within feastol=1.3e-08, reltol=1.5e-08, abstol=1.0e-07).
Runtime: 0.000121 seconds.

optimal value with ECOS: 6.82842708233

Setting solver options¶

The 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 sparse-indirect method.
prob.solve(solver=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
----------------------------------------------------------------------------
Lin-sys: sparse-indirect, nnz in A = 13, CG tol ~ 1/iter^(2.00)
EPS = 1.00e-03, 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: 2.78e-04s
----------------------------------------------------------------------------
 Iter | pri res | dua res | rel gap | pri obj | dua obj | kap/tau | time (s)
----------------------------------------------------------------------------
     0| 4.60e+00  5.78e-01       nan      -inf       inf       inf  3.86e-05
    60| 3.92e-05  1.12e-04  6.64e-06  6.83e+00  6.83e+00  1.41e-17  9.51e-05
----------------------------------------------------------------------------
Status: Solved
Timing: Total solve time: 9.76e-05s
    Lin-sys: avg # CG iterations: 1.00, avg solve time: 2.24e-07s
    Cones: avg projection time: 4.90e-08s
----------------------------------------------------------------------------
Error metrics:
|Ax + s - b|_2 / (1 + |b|_2) = 3.9223e-05
|A'y + c|_2 / (1 + |c|_2) = 1.1168e-04
|c'x + b'y| / (1 + |c'x| + |b'y|) = 6.6446e-06
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’s the complete list of solver options.

ECOS options:

'max_iters': maximum number of iterations (default: 100).
'abstol': absolute accuracy (default: 1e-7).
'reltol': relative accuracy (default: 1e-6).
'feastol': tolerance for feasibility conditions (default: 1e-7).
'abstol_inacc': absolute accuracy for inaccurate solution (default: 5e-5).
'reltol_inacc': relative accuracy for inaccurate solution (default: 5e-5).
'feastol_inacc': tolerance for feasibility condition for inaccurate solution (default: 1e-4).

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: 1e-6)
'mi_rel_eps': relative tolerance, (U-L)/L, between upper and lower bounds (default: 1e-3)

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 string-value 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.

CVXOPT options:

'max_iters': maximum number of iterations (default: 100).
'abstol': absolute accuracy (default: 1e-7).
'reltol': relative accuracy (default: 1e-6).
'feastol': tolerance for feasibility conditions (default: 1e-7).
'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: 1e-3).
'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).
'warm_start': whether to initialize the solver with the previous solution (default: False). The use case for warm start is solving the same problem for multiple values of a parameter.

CBC options:

Cut-generation through CGL

General remarks:

some of these cut-generators seem to be buggy (observed problems with AllDifferentCuts, RedSplitCuts, LandPCuts, PreProcessCuts)
a few of these cut-generators will generate noisy output even if 'verbose=False'

The following cut-generators are available:

GomoryCuts, MIRCuts, MIRCuts2, TwoMIRCuts, ResidualCapacityCuts, KnapsackCuts FlowCoverCuts, CliqueCuts, LiftProjectCuts, AllDifferentCuts, OddHoleCuts, RedSplitCuts, LandPCuts, PreProcessCuts, ProbingCuts, SimpleRoundingCuts.

'CutGenName'

if cut-generator is activated (e.g. 'GomoryCuts=True')

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 ECOS arguments.
data = prob.get_problem_data(ECOS)

# Get ECOS_BB arguments.
data = prob.get_problem_data(ECOS_BB)

# Get CVXOPT arguments.
data = prob.get_problem_data(CVXOPT)

# Get SCS arguments.
data = prob.get_problem_data(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(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(ECOS, solver_output)