In this chapter we consider convex optimization problems of the form
The linear inequality is a generalized inequality with respect to a proper convex cone. It may include componentwise vector inequalities, second-order cone inequalities, and linear matrix inequalities.
The main solvers are conelp and coneqp, described in the sections Linear Cone Programs and Quadratic Cone Programs. The function conelp is restricted to problems with linear cost functions, and can detect primal and dual infeasibility. The function coneqp solves the general quadratic problem, but requires the problem to be strictly primal and dual feasible. For convenience (and backward compatibility), simpler interfaces to these function are also provided that handle pure linear programs, quadratic programs, second-order cone programs, and semidefinite programs. These are described in the sections Linear Programming, Quadratic Programming, Second-Order Cone Programming, Semidefinite Programming. In the section Exploiting Structure we explain how custom solvers can be implemented that exploit structure in cone programs. The last two sections describe optional interfaces to external solvers, and the algorithm parameters that control the cone programming solvers.
Solves a pair of primal and dual cone programs
The primal variables are and . The dual variables are , . The inequalities are interpreted as , , where is a cone defined as a Cartesian product of a nonnegative orthant, a number of second-order cones, and a number of positive semidefinite cones:
with
In this definition, denotes a symmetric matrix stored as a vector in column major order. The structure of is specified by dims. This argument is a dictionary with three fields.
The default value of dims is {'l': G.size[0], 'q': [], 's': []}, i.e., by default the inequality is interpreted as a componentwise vector inequality.
The arguments c, h, and b are real single-column dense matrices. G and A are real dense or sparse matrices. The number of rows of G and h is equal to
The columns of G and h are vectors in
where the last components represent symmetric matrices stored in column major order. The strictly upper triangular entries of these matrices are not accessed (i.e., the symmetric matrices are stored in the 'L'-type column major order used in the blas and lapack modules). The default values for A and b are matrices with zero rows, meaning that there are no equality constraints.
primalstart is a dictionary with keys 'x' and 's', used as an optional primal starting point. primalstart['x'] and primalstart['s'] are real dense matrices of size (, 1) and (, 1), respectively, where is the length of c. The vector primalstart['s'] must be strictly positive with respect to the cone .
dualstart is a dictionary with keys 'y' and 'z', used as an optional dual starting point. dualstart['y'] and dualstart['z'] are real dense matrices of size (, 1) and (, 1), respectively, where is the number of rows in A. The vector dualstart['s'] must be strictly positive with respect to the cone .
The role of the optional argument kktsolver is explained in the section Exploiting Structure.
conelp returns a dictionary that contains the result and information about the accuracy of the solution. The most important fields have keys 'status', 'x', 's', 'y', 'z'. The 'status' field is a string with possible values 'optimal', 'primal infeasible', 'dual infeasible', and 'unknown'. The meaning of the 'x', 's', 'y', 'z' fields depends on the value of 'status'.
In this case the 'x', 's', 'y', and 'z' entries contain the primal and dual solutions, which approximately satisfy
The other entries in the output dictionary summarize the accuracy with which these optimality conditions are satisfied. The fields 'primal objective', 'dual objective', and 'gap' give the primal objective , dual objective , and the gap . The field 'relative gap' is the relative gap, defined as
and None otherwise. The fields 'primal infeasibility' and 'dual infeasibility' are the residuals in the primal and dual equality constraints, defined as
respectively.
The 'x' and 's' entries are None, and the 'y', 'z' entries provide an approximate certificate of infeasibility, i.e., vectors that approximately satisfy
The field 'residual as primal infeasibility certificate' gives the residual
The 'y' and 'z' entries are None, and the 'x' and 's' entries contain an approximate certificate of dual infeasibility
The field 'residual as dual infeasibility certificate' gives the residual
This indicates that the algorithm terminated early due to numerical difficulties or because the maximum number of iterations was reached. The 'x', 's', 'y', 'z' entries contain the iterates when the algorithm terminated. Whether these entries are useful, as approximate solutions or certificates of primal and dual infeasibility, can be determined from the other fields in the dictionary.
The fields 'primal objective', 'dual objective', 'gap', 'relative gap', 'primal infeasibility', 'dual infeasibility' are defined as when 'status' is 'optimal'. The field 'residual as primal infeasibility certificate' is defined as
if , and None otherwise. A small value of this residual indicates that and , divided by , are an approximate proof of primal infeasibility. The field 'residual as dual infeasibility certificate' is defined as
if , and as None otherwise. A small value indicates that and , divided by are an approximate proof of dual infeasibility.
It is required that
where is the number or rows of and is the number of columns of and .
As an example we solve the problem
>>> from cvxopt import matrix, solvers
>>> c = matrix([-6., -4., -5.])
>>> G = matrix([[ 16., 7., 24., -8., 8., -1., 0., -1., 0., 0.,
7., -5., 1., -5., 1., -7., 1., -7., -4.],
[-14., 2., 7., -13., -18., 3., 0., 0., -1., 0.,
3., 13., -6., 13., 12., -10., -6., -10., -28.],
[ 5., 0., -15., 12., -6., 17., 0., 0., 0., -1.,
9., 6., -6., 6., -7., -7., -6., -7., -11.]])
>>> h = matrix( [ -3., 5., 12., -2., -14., -13., 10., 0., 0., 0.,
68., -30., -19., -30., 99., 23., -19., 23., 10.] )
>>> dims = {'l': 2, 'q': [4, 4], 's': [3]}
>>> sol = solvers.conelp(c, G, h, dims)
>>> sol['status']
'optimal'
>>> print(sol['x'])
[-1.22e+00]
[ 9.66e-02]
[ 3.58e+00]
>>> print(sol['z'])
[ 9.30e-02]
[ 2.04e-08]
[ 2.35e-01]
[ 1.33e-01]
[-4.74e-02]
[ 1.88e-01]
[ 2.79e-08]
[ 1.85e-09]
[-6.32e-10]
[-7.59e-09]
[ 1.26e-01]
[ 8.78e-02]
[-8.67e-02]
[ 8.78e-02]
[ 6.13e-02]
[-6.06e-02]
[-8.67e-02]
[-6.06e-02]
[ 5.98e-02]
Only the entries of G and h defining the lower triangular portions of the coefficients in the linear matrix inequalities are accessed. We obtain the same result if we define G and h as below.
>>> G = matrix([[ 16., 7., 24., -8., 8., -1., 0., -1., 0., 0.,
7., -5., 1., 0., 1., -7., 0., 0., -4.],
[-14., 2., 7., -13., -18., 3., 0., 0., -1., 0.,
3., 13., -6., 0., 12., -10., 0., 0., -28.],
[ 5., 0., -15., 12., -6., 17., 0., 0., 0., -1.,
9., 6., -6., 0., -7., -7., 0., 0., -11.]])
>>> h = matrix( [ -3., 5., 12., -2., -14., -13., 10., 0., 0., 0.,
68., -30., -19., 0., 99., 23., 0., 0., 10.] )
Solves a pair of primal and dual quadratic cone programs
and
The primal variables are and the slack variable . The dual variables are and . The inequalities are interpreted as , , where is a cone defined as a Cartesian product of a nonnegative orthant, a number of second-order cones, and a number of positive semidefinite cones:
with
In this definition, denotes a symmetric matrix stored as a vector in column major order. The structure of is specified by dims. This argument is a dictionary with three fields.
The default value of dims is {'l': G.size[0], 'q': [], 's': []}, i.e., by default the inequality is interpreted as a componentwise vector inequality.
P is a square dense or sparse real matrix, representing a positive semidefinite symmetric matrix in 'L' storage, i.e., only the lower triangular part of P is referenced. q is a real single-column dense matrix.
The arguments h and b are real single-column dense matrices. G and A are real dense or sparse matrices. The number of rows of G and h is equal to
The columns of G and h are vectors in
where the last components represent symmetric matrices stored in column major order. The strictly upper triangular entries of these matrices are not accessed (i.e., the symmetric matrices are stored in the 'L'-type column major order used in the blas and lapack modules). The default values for G, h, A, and b are matrices with zero rows, meaning that there are no inequality or equality constraints.
initvals is a dictionary with keys 'x', 's', 'y', 'z' used as an optional starting point. The vectors initvals['s'] and initvals['z'] must be strictly positive with respect to the cone . If the argument initvals or any the four entries in it are missing, default starting points are used for the corresponding variables.
The role of the optional argument kktsolver is explained in the section Exploiting Structure.
coneqp returns a dictionary that contains the result and information about the accuracy of the solution. The most important fields have keys 'status', 'x', 's', 'y', 'z'. The 'status' field is a string with possible values 'optimal' and 'unknown'.
In this case the 'x', 's', 'y', and 'z' entries contain primal and dual solutions, which approximately satisfy
The other entries in the output dictionary summarize the accuracy with which the optimality conditions are satisfied. The fields 'primal objective', 'dual objective', and 'gap' give the primal objective , the dual objective calculated as
and the gap . The field 'relative gap' is the relative gap, defined as
and None otherwise. The fields 'primal infeasibility' and 'dual infeasibility' are the residuals in the primal and dual equality constraints, defined as
respectively.
It is required that the problem is solvable and that
where is the number or rows of and is the number of columns of and .
As an example, we solve a constrained least-squares problem
with
>>> from cvxopt import matrix, solvers
>>> A = matrix([ [ .3, -.4, -.2, -.4, 1.3 ],
[ .6, 1.2, -1.7, .3, -.3 ],
[-.3, .0, .6, -1.2, -2.0 ] ])
>>> b = matrix([ 1.5, .0, -1.2, -.7, .0])
>>> m, n = A.size
>>> I = matrix(0.0, (n,n))
>>> I[::n+1] = 1.0
>>> G = matrix([-I, matrix(0.0, (1,n)), I])
>>> h = matrix(n*[0.0] + [1.0] + n*[0.0])
>>> dims = {'l': n, 'q': [n+1], 's': []}
>>> x = solvers.coneqp(A.T*A, -A.T*b, G, h, dims)['x']
>>> print(x)
[ 7.26e-01]
[ 6.18e-01]
[ 3.03e-01]
The function lp is an interface to conelp for linear programs. It also provides the option of using the linear programming solvers from GLPK or MOSEK.
Solves the pair of primal and dual linear programs
The inequalities are componentwise vector inequalities.
The solver argument is used to choose among three solvers. When it is omitted or None, the CVXOPT function conelp is used. The external solvers GLPK and MOSEK (if installed) can be selected by setting solver to 'glpk' or 'mosek'; see the section Optional Solvers. The meaning of the other arguments and the return value are the same as for conelp called with dims equal to {'l': G.size[0], 'q': [], 's': []}.
The initial values are ignored when solver is 'mosek' or 'glpk'. With the GLPK option, the solver does not return certificates of primal or dual infeasibility: if the status is 'primal infeasible' or 'dual infeasible', all entries of the output dictionary are None. If the GLPK or MOSEK solvers are used, and the code returns with status 'unknown', all the other fields in the output dictionary are None.
As a simple example we solve the LP
>>> from cvxopt import matrix, solvers
>>> c = matrix([-4., -5.])
>>> G = matrix([[2., 1., -1., 0.], [1., 2., 0., -1.]])
>>> h = matrix([3., 3., 0., 0.])
>>> sol = solvers.lp(c, G, h)
>>> print(sol['x'])
[ 1.00e+00]
[ 1.00e+00]
The function qp is an interface to coneqp for quadratic programs. It also provides the option of using the quadratic programming solver from MOSEK.
Solves the pair of primal and dual convex quadratic programs
and
The inequalities are componentwise vector inequalities.
The default CVXOPT solver is used when the solver argument is absent or None. The MOSEK solver (if installed) can be selected by setting solver to 'mosek'; see the section Optional Solvers. The meaning of the other arguments and the return value is the same as for coneqp called with dims equal to {'l': G.size[0], 'q': [], 's': []}.
When solver is 'mosek', the initial values are ignored, and the 'status' string in the solution dictionary can take four possible values: 'optimal', 'unknown'. 'primal infeasible', 'dual infeasible'.
This means that a certificate of primal infeasibility has been found. The 'x' and 's' entries are None, and the 'z' and 'y' entries are vectors that approximately satisfy
This means that a certificate of dual infeasibility has been found. The 'z' and 'y' entries are None, and the 'x' and 's' entries are vectors that approximately satisfy
As an example we compute the trade-off curve on page 187 of the book Convex Optimization, by solving the quadratic program
for a sequence of positive values of . The code below computes the trade-off curve and produces two figures using the Matplotlib package.
from math import sqrt
from cvxopt import matrix
from cvxopt.blas import dot
from cvxopt.solvers import qp
import pylab
# Problem data.
n = 4
S = matrix([[ 4e-2, 6e-3, -4e-3, 0.0 ],
[ 6e-3, 1e-2, 0.0, 0.0 ],
[-4e-3, 0.0, 2.5e-3, 0.0 ],
[ 0.0, 0.0, 0.0, 0.0 ]])
pbar = matrix([.12, .10, .07, .03])
G = matrix(0.0, (n,n))
G[::n+1] = -1.0
h = matrix(0.0, (n,1))
A = matrix(1.0, (1,n))
b = matrix(1.0)
# Compute trade-off.
N = 100
mus = [ 10**(5.0*t/N-1.0) for t in range(N) ]
portfolios = [ qp(mu*S, -pbar, G, h, A, b)['x'] for mu in mus ]
returns = [ dot(pbar,x) for x in portfolios ]
risks = [ sqrt(dot(x, S*x)) for x in portfolios ]
# Plot trade-off curve and optimal allocations.
pylab.figure(1, facecolor='w')
pylab.plot(risks, returns)
pylab.xlabel('standard deviation')
pylab.ylabel('expected return')
pylab.axis([0, 0.2, 0, 0.15])
pylab.title('Risk-return trade-off curve (fig 4.12)')
pylab.yticks([0.00, 0.05, 0.10, 0.15])
pylab.figure(2, facecolor='w')
c1 = [ x[0] for x in portfolios ]
c2 = [ x[0] + x[1] for x in portfolios ]
c3 = [ x[0] + x[1] + x[2] for x in portfolios ]
c4 = [ x[0] + x[1] + x[2] + x[3] for x in portfolios ]
pylab.fill(risks + [.20], c1 + [0.0], '#F0F0F0')
pylab.fill(risks[-1::-1] + risks, c2[-1::-1] + c1, facecolor = '#D0D0D0')
pylab.fill(risks[-1::-1] + risks, c3[-1::-1] + c2, facecolor = '#F0F0F0')
pylab.fill(risks[-1::-1] + risks, c4[-1::-1] + c3, facecolor = '#D0D0D0')
pylab.axis([0.0, 0.2, 0.0, 1.0])
pylab.xlabel('standard deviation')
pylab.ylabel('allocation')
pylab.text(.15,.5,'x1')
pylab.text(.10,.7,'x2')
pylab.text(.05,.7,'x3')
pylab.text(.01,.7,'x4')
pylab.title('Optimal allocations (fig 4.12)')
pylab.show()
The function socp is a simpler interface to conelp for cone programs with no linear matrix inequality constraints.
Solves the pair of primal and dual second-order cone programs
and
The inequalities
are componentwise vector inequalities. In the other inequalities, it is assumed that the variables are partitioned as
The input argument c is a real single-column dense matrix. The arguments Gl and hl are the coefficient matrix and the right-hand side of the componentwise inequalities. Gl is a real dense or sparse matrix; hl is a real single-column dense matrix. The default values for Gl and hl are matrices with zero rows.
The argument Gq is a list of dense or sparse matrices , ..., . The argument hq is a list of dense single-column matrices , ldots, . The elements of Gq and hq must have at least one row. The default values of Gq and hq are empty lists.
A is dense or sparse matrix and b is a single-column dense matrix. The default values for A and b are matrices with zero rows.
The solver argument is used to choose between two solvers: the CVXOPT conelp solver (used when solver is absent or equal to None and the external solver MOSEK (solver is 'mosek'); see the section Optional Solvers. With the 'mosek' option the code does not accept problems with equality constraints.
primalstart and dualstart are dictionaries with optional primal, respectively, dual starting points. primalstart has elements 'x', 'sl', 'sq'. primalstart['x'] and primalstart['sl'] are single-column dense matrices with the initial values of and ; primalstart['sq'] is a list of single-column matrices with the initial values of , ldots, . The initial values must satisfy the inequalities in the primal problem strictly, but not necessarily the equality constraints.
dualstart has elements 'y', 'zl', 'zq'. dualstart['y'] and dualstart['zl'] are single-column dense matrices with the initial values of and . dualstart['zq'] is a list of single-column matrices with the initial values of , ldots, . These values must satisfy the dual inequalities strictly, but not necessarily the equality constraint.
The arguments primalstart and dualstart are ignored when the MOSEK solver is used.
socp returns a dictionary that include entries with keys 'status', 'x', 'sl', 'sq', 'y', 'zl', 'zq'. The 'sl' and 'zl' fields are matrices with the primal slacks and dual variables associated with the componentwise linear inequalities. The 'sq' and 'zq' fields are lists with the primal slacks and dual variables associated with the second-order cone inequalities. The other entries in the output dictionary have the same meaning as in the output of conelp.
As an example, we solve the second-order cone program
>>> from cvxopt import matrix, solvers
>>> c = matrix([-2., 1., 5.])
>>> G = [ matrix( [[12., 13., 12.], [6., -3., -12.], [-5., -5., 6.]] ) ]
>>> G += [ matrix( [[3., 3., -1., 1.], [-6., -6., -9., 19.], [10., -2., -2., -3.]] ) ]
>>> h = [ matrix( [-12., -3., -2.] ), matrix( [27., 0., 3., -42.] ) ]
>>> sol = solvers.socp(c, Gq = G, hq = h)
>>> sol['status']
optimal
>>> print(sol['x'])
[-5.02e+00]
[-5.77e+00]
[-8.52e+00]
>>> print(sol['zq'][0])
[ 1.34e+00]
[-7.63e-02]
[-1.34e+00]
>>> print(sol['zq'][1])
[ 1.02e+00]
[ 4.02e-01]
[ 7.80e-01]
[-5.17e-01]
The function sdp is a simple interface to conelp for cone programs with no second-order cone constraints. It also provides the option of using the DSDP semidefinite programming solver.
Solves the pair of primal and dual semidefinite programs
and
The inequalities
are componentwise vector inequalities. The other inequalities are matrix inequalities (ie, the require the left-hand sides to be positive semidefinite). We use the notation to denote a symmetric matrix stored in column major order as a column vector.
The input argument c is a real single-column dense matrix. The arguments Gl and hl are the coefficient matrix and the right-hand side of the componentwise inequalities. Gl is a real dense or sparse matrix; hl is a real single-column dense matrix. The default values for Gl and hl are matrices with zero rows.
Gs and hs are lists of length that specify the linear matrix inequality constraints. Gs is a list of dense or sparse real matrices , ldots, . The columns of these matrices can be interpreted as symmetric matrices stored in column major order, using the BLAS 'L'-type storage (i.e., only the entries corresponding to lower triangular positions are accessed). hs is a list of dense symmetric matrices , ldots, . Only the lower triangular elements of these matrices are accessed. The default values for Gs and hs are empty lists.
A is a dense or sparse matrix and b is a single-column dense matrix. The default values for A and b are matrices with zero rows.
The solver argument is used to choose between two solvers: the CVXOPT conelp solver (used when solver is absent or equal to None) and the external solver DSDP5 (solver is 'dsdp'); see the section Optional Solvers. With the 'dsdp' option the code does not accept problems with equality constraints.
The optional argument primalstart is a dictionary with keys 'x', 'sl', and 'ss', used as an optional primal starting point. primalstart['x'] and primalstart['sl'] are single-column dense matrices with the initial values of and ; primalstart['ss'] is a list of square matrices with the initial values of , ldots, . The initial values must satisfy the inequalities in the primal problem strictly, but not necessarily the equality constraints.
dualstart is a dictionary with keys 'y', 'zl', 'zs', used as an optional dual starting point. dualstart['y'] and dualstart['zl'] are single-column dense matrices with the initial values of and . dualstart['zs'] is a list of square matrices with the initial values of , ldots, . These values must satisfy the dual inequalities strictly, but not necessarily the equality constraint.
The arguments primalstart and dualstart are ignored when the DSDP solver is used.
sdp returns a dictionary that includes entries with keys 'status', 'x', 'sl', 'ss', 'y', 'zl', 'ss'. The 'sl' and 'zl' fields are matrices with the primal slacks and dual variables associated with the componentwise linear inequalities. The 'ss' and 'zs' fields are lists with the primal slacks and dual variables associated with the second-order cone inequalities. The other entries in the output dictionary have the same meaning as in the output of conelp.
We illustrate the calling sequence with a small example.
>>> from cvxopt import matrix, solvers
>>> c = matrix([1.,-1.,1.])
>>> G = [ matrix([[-7., -11., -11., 3.],
[ 7., -18., -18., 8.],
[-2., -8., -8., 1.]]) ]
>>> G += [ matrix([[-21., -11., 0., -11., 10., 8., 0., 8., 5.],
[ 0., 10., 16., 10., -10., -10., 16., -10., 3.],
[ -5., 2., -17., 2., -6., 8., -17., -7., 6.]]) ]
>>> h = [ matrix([[33., -9.], [-9., 26.]]) ]
>>> h += [ matrix([[14., 9., 40.], [9., 91., 10.], [40., 10., 15.]]) ]
>>> sol = solvers.sdp(c, Gs=G, hs=h)
>>> print(sol['x'])
[-3.68e-01]
[ 1.90e+00]
[-8.88e-01]
>>> print(sol['zs'][0])
[ 3.96e-03 -4.34e-03]
[-4.34e-03 4.75e-03]
>>> print(sol['zs'][1])
[ 5.58e-02 -2.41e-03 2.42e-02]
[-2.41e-03 1.04e-04 -1.05e-03]
[ 2.42e-02 -1.05e-03 1.05e-02]
Only the entries in Gs and hs that correspond to lower triangular entries need to be provided, so in the example h and G may also be defined as follows.
>>> G = [ matrix([[-7., -11., 0., 3.],
[ 7., -18., 0., 8.],
[-2., -8., 0., 1.]]) ]
>>> G += [ matrix([[-21., -11., 0., 0., 10., 8., 0., 0., 5.],
[ 0., 10., 16., 0., -10., -10., 0., 0., 3.],
[ -5., 2., -17., 0., -6., 8., 0., 0., 6.]]) ]
>>> h = [ matrix([[33., -9.], [0., 26.]]) ]
>>> h += [ matrix([[14., 9., 40.], [0., 91., 10.], [0., 0., 15.]]) ]
By default, the functions conelp and coneqp exploit no problem structure except (to some limited extent) sparsity. Two mechanisms are provided for implementing customized solvers that take advantage of problem structure.
The most expensive step of each iteration of conelp or coneqp is the solution of a set of linear equations (KKT equations) of the form
(1)
(with in conelp). The matrix depends on the current iterates and is defined as follows. We use the notation of the sections Linear Cone Programs and Quadratic Cone Programs. Let
Then is a block-diagonal matrix,
with the following diagonal blocks.
The first block is a positive diagonal scaling with a vector :
This transformation is symmetric:
The next blocks are positive multiples of hyperbolic Householder transformations:
where
These transformations are also symmetric:
The last blocks are congruence transformations with nonsingular matrices:
In general, this operation is not symmetric:
It is often possible to exploit problem structure to solve (1) faster than by standard methods. The last argument kktsolver of conelp and coneqp allows the user to supply a Python function for solving the KKT equations. This function will be called as f = kktsolver(W), where W is a dictionary that contains the parameters of the scaling:
The function call f = kktsolver(W) should return a routine for solving the KKT system (1) defined by W. It will be called as f(bx, by, bz). On entry, bx, by, bz contain the right-hand side. On exit, they should contain the solution of the KKT system, with the last component scaled, i.e., on exit,
In other words, the function returns the solution of
In the default use of conelp and coneqp, the linear constraints and the quadratic term in the objective are parameterized by CVXOPT matrices G, A, P. It is possible to specify these parameters via Python functions that evaluate the corresponding matrix-vector products and their adjoints.
If the argument G of conelp or coneqp is a Python function, then G(x, y[, alpha = 1.0, beta = 0.0, trans = 'N']) should evaluate the matrix-vector products
Similarly, if the argument A is a Python function, then A(x, y[, alpha = 1.0, beta = 0.0, trans = 'N']) should evaluate the matrix-vector products
If the argument P of coneqp is a Python function, then P(x, y[, alpha = 1.0, beta = 0.0]) should evaluate the matrix-vector products
If G, A, or P are Python functions, then the argument kktsolver must also be provided.
We illustrate these features with three applications.
Example: 1-norm approximation
The optimization problem
can be formulated as a linear program
By exploiting the structure in the inequalities, the cost of an iteration of an interior-point method can be reduced to the cost of least-squares problem of the same dimensions. (See section 11.8.2 in the book Convex Optimization.) The code below takes advantage of this fact.
from cvxopt import blas, lapack, solvers, matrix, spmatrix, mul, div def l1(P, q): """ Returns the solution u, w of the l1 approximation problem (primal) minimize ||P*u - q||_1 (dual) maximize q'*w subject to P'*w = 0 ||w||_infty <= 1. """ m, n = P.size # Solve the equivalent LP # # minimize [0; 1]' * [u; v] # subject to [P, -I; -P, -I] * [u; v] <= [q; -q] # # maximize -[q; -q]' * z # subject to [P', -P']*z = 0 # [-I, -I]*z + 1 = 0 # z >= 0. c = matrix(n*[0.0] + m*[1.0]) def G(x, y, alpha = 1.0, beta = 0.0, trans = 'N'): if trans=='N': # y := alpha * [P, -I; -P, -I] * x + beta*y u = P*x[:n] y[:m] = alpha * ( u - x[n:]) + beta * y[:m] y[m:] = alpha * (-u - x[n:]) + beta * y[m:] else: # y := alpha * [P', -P'; -I, -I] * x + beta*y y[:n] = alpha * P.T * (x[:m] - x[m:]) + beta * y[:n] y[n:] = -alpha * (x[:m] + x[m:]) + beta * y[n:] h = matrix([q, -q]) dims = {'l': 2*m, 'q': [], 's': []} def F(W): """ Returns a function f(x, y, z) that solves [ 0 0 P' -P' ] [ x[:n] ] [ bx[:n] ] [ 0 0 -I -I ] [ x[n:] ] [ bx[n:] ] [ P -I -D1^{-1} 0 ] [ z[:m] ] = [ bz[:m] ] [-P -I 0 -D2^{-1} ] [ z[m:] ] [ bz[m:] ] where D1 = diag(di[:m])^2, D2 = diag(di[m:])^2 and di = W['di']. """ # Factor A = 4*P'*D*P where D = d1.*d2 ./(d1+d2) and # d1 = di[:m].^2, d2 = di[m:].^2. di = W['di'] d1, d2 = di[:m]**2, di[m:]**2 D = div( mul(d1,d2), d1+d2 ) A = P.T * spmatrix(4*D, range(m), range(m)) * P lapack.potrf(A) def f(x, y, z): """ On entry bx, bz are stored in x, z. On exit x, z contain the solution, with z scaled: z./di is returned instead of z. """" # Solve for x[:n]: # # A*x[:n] = bx[:n] + P' * ( ((D1-D2)*(D1+D2)^{-1})*bx[n:] # + (2*D1*D2*(D1+D2)^{-1}) * (bz[:m] - bz[m:]) ). x[:n] += P.T * ( mul(div(d1-d2, d1+d2), x[n:]) + mul(2*D, z[:m]-z[m:]) ) lapack.potrs(A, x) # x[n:] := (D1+D2)^{-1} * (bx[n:] - D1*bz[:m] - D2*bz[m:] + (D1-D2)*P*x[:n]) u = P*x[:n] x[n:] = div(x[n:] - mul(d1, z[:m]) - mul(d2, z[m:]) + mul(d1-d2, u), d1+d2) # z[:m] := d1[:m] .* ( P*x[:n] - x[n:] - bz[:m]) # z[m:] := d2[m:] .* (-P*x[:n] - x[n:] - bz[m:]) z[:m] = mul(di[:m], u - x[n:] - z[:m]) z[m:] = mul(di[m:], -u - x[n:] - z[m:]) return f sol = solvers.conelp(c, G, h, dims, kktsolver = F) return sol['x'][:n], sol['z'][m:] - sol['z'][:m]
Example: SDP with diagonal linear term
The SDP
can be solved efficiently by exploiting properties of the diag operator.
from cvxopt import blas, lapack, solvers, matrix def mcsdp(w): """ Returns solution x, z to (primal) minimize sum(x) subject to w + diag(x) >= 0 (dual) maximize -tr(w*z) subject to diag(z) = 1 z >= 0. """ n = w.size[0] c = matrix(1.0, (n,1)) def G(x, y, alpha = 1.0, beta = 0.0, trans = 'N'): """ y := alpha*(-diag(x)) + beta*y. """ if trans=='N': # x is a vector; y is a symmetric matrix in column major order. y *= beta y[::n+1] -= alpha * x else: # x is a symmetric matrix in column major order; y is a vector. y *= beta y -= alpha * x[::n+1] def cngrnc(r, x, alpha = 1.0): """ Congruence transformation x := alpha * r'*x*r. r and x are square matrices. """ # Scale diagonal of x by 1/2. x[::n+1] *= 0.5 # a := tril(x)*r a = +r tx = matrix(x, (n,n)) blas.trmm(tx, a, side = 'L') # x := alpha*(a*r' + r*a') blas.syr2k(r, a, tx, trans = 'T', alpha = alpha) x[:] = tx[:] dims = {'l': 0, 'q': [], 's': [n]} def F(W): """ Returns a function f(x, y, z) that solves -diag(z) = bx -diag(x) - r*r'*z*r*r' = bz where r = W['r'][0] = W['rti'][0]^{-T}. """ rti = W['rti'][0] # t = rti*rti' as a nonsymmetric matrix. t = matrix(0.0, (n,n)) blas.gemm(rti, rti, t, transB = 'T') # Cholesky factorization of tsq = t.*t. tsq = t**2 lapack.potrf(tsq) def f(x, y, z): """ On entry, x contains bx, y is empty, and z contains bz stored in column major order. On exit, they contain the solution, with z scaled (vec(r'*z*r) is returned instead of z). We first solve ((rti*rti') .* (rti*rti')) * x = bx - diag(t*bz*t) and take z = - rti' * (diag(x) + bz) * rti. """ # tbst := t * bz * t tbst = +z cngrnc(t, tbst) # x := x - diag(tbst) = bx - diag(rti*rti' * bz * rti*rti') x -= tbst[::n+1] # x := (t.*t)^{-1} * x = (t.*t)^{-1} * (bx - diag(t*bz*t)) lapack.potrs(tsq, x) # z := z + diag(x) = bz + diag(x) z[::n+1] += x # z := -vec(rti' * z * rti) # = -vec(rti' * (diag(x) + bz) * rti cngrnc(rti, z, alpha = -1.0) return f sol = solvers.conelp(c, G, w[:], dims, kktsolver = F) return sol['x'], sol['z']
In the second example, we use a similar trick to solve the problem
The code below is efficient, if we assume that the number of rows in is greater than or equal to the number of columns.
def qcl1(A, b):
"""
Returns the solution u, z of
(primal) minimize || u ||_1
subject to || A * u - b ||_2 <= 1
(dual) maximize b^T z - ||z||_2
subject to || A'*z ||_inf <= 1.
Exploits structure, assuming A is m by n with m >= n.
"""
m, n = A.size
# Solve equivalent cone LP with variables x = [u; v].
#
# minimize [0; 1]' * x
# subject to [ I -I ] * x <= [ 0 ] (componentwise)
# [-I -I ] * x <= [ 0 ] (componentwise)
# [ 0 0 ] * x <= [ 1 ] (SOC)
# [-A 0 ] [ -b ]
#
# maximize -t + b' * w
# subject to z1 - z2 = A'*w
# z1 + z2 = 1
# z1 >= 0, z2 >=0, ||w||_2 <= t.
c = matrix(n*[0.0] + n*[1.0])
h = matrix( 0.0, (2*n + m + 1, 1))
h[2*n] = 1.0
h[2*n+1:] = -b
def G(x, y, alpha = 1.0, beta = 0.0, trans = 'N'):
y *= beta
if trans=='N':
# y += alpha * G * x
y[:n] += alpha * (x[:n] - x[n:2*n])
y[n:2*n] += alpha * (-x[:n] - x[n:2*n])
y[2*n+1:] -= alpha * A*x[:n]
else:
# y += alpha * G'*x
y[:n] += alpha * (x[:n] - x[n:2*n] - A.T * x[-m:])
y[n:] -= alpha * (x[:n] + x[n:2*n])
def Fkkt(W):
"""
Returns a function f(x, y, z) that solves
[ 0 G' ] [ x ] = [ bx ]
[ G -W'*W ] [ z ] [ bz ].
"""
# First factor
#
# S = G' * W**-1 * W**-T * G
# = [0; -A]' * W3^-2 * [0; -A] + 4 * (W1**2 + W2**2)**-1
#
# where
#
# W1 = diag(d1) with d1 = W['d'][:n] = 1 ./ W['di'][:n]
# W2 = diag(d2) with d2 = W['d'][n:] = 1 ./ W['di'][n:]
# W3 = beta * (2*v*v' - J), W3^-1 = 1/beta * (2*J*v*v'*J - J)
# with beta = W['beta'][0], v = W['v'][0], J = [1, 0; 0, -I].
# As = W3^-1 * [ 0 ; -A ] = 1/beta * ( 2*J*v * v' - I ) * [0; A]
beta, v = W['beta'][0], W['v'][0]
As = 2 * v * (v[1:].T * A)
As[1:,:] *= -1.0
As[1:,:] -= A
As /= beta
# S = As'*As + 4 * (W1**2 + W2**2)**-1
S = As.T * As
d1, d2 = W['d'][:n], W['d'][n:]
d = 4.0 * (d1**2 + d2**2)**-1
S[::n+1] += d
lapack.potrf(S)
def f(x, y, z):
# z := - W**-T * z
z[:n] = -div( z[:n], d1 )
z[n:2*n] = -div( z[n:2*n], d2 )
z[2*n:] -= 2.0*v*( v[0]*z[2*n] - blas.dot(v[1:], z[2*n+1:]) )
z[2*n+1:] *= -1.0
z[2*n:] /= beta
# x := x - G' * W**-1 * z
x[:n] -= div(z[:n], d1) - div(z[n:2*n], d2) + As.T * z[-(m+1):]
x[n:] += div(z[:n], d1) + div(z[n:2*n], d2)
# Solve for x[:n]:
#
# S*x[:n] = x[:n] - (W1**2 - W2**2)(W1**2 + W2**2)^-1 * x[n:]
x[:n] -= mul( div(d1**2 - d2**2, d1**2 + d2**2), x[n:])
lapack.potrs(S, x)
# Solve for x[n:]:
#
# (d1**-2 + d2**-2) * x[n:] = x[n:] + (d1**-2 - d2**-2)*x[:n]
x[n:] += mul( d1**-2 - d2**-2, x[:n])
x[n:] = div( x[n:], d1**-2 + d2**-2)
# z := z + W^-T * G*x
z[:n] += div( x[:n] - x[n:2*n], d1)
z[n:2*n] += div( -x[:n] - x[n:2*n], d2)
z[2*n:] += As*x[:n]
return f
dims = {'l': 2*n, 'q': [m+1], 's': []}
sol = solvers.conelp(c, G, h, dims, kktsolver = Fkkt)
if sol['status'] == 'optimal':
return sol['x'][:n], sol['z'][-m:]
else:
return None, None
As an example that illustrates how structure can be exploited in coneqp, we consider the 1-norm regularized least-squares problem
with variable . The problem is equivalent to the quadratic program
with variables and . The implementation below is efficient when has many more columns than rows.
from cvxopt import matrix, spdiag, mul, div, blas, lapack, solvers, sqrt
import math
def l1regls(A, y):
"""
Returns the solution of l1-norm regularized least-squares problem
minimize || A*x - y ||_2^2 + || x ||_1.
"""
m, n = A.size
q = matrix(1.0, (2*n,1))
q[:n] = -2.0 * A.T * y
def P(u, v, alpha = 1.0, beta = 0.0 ):
"""
v := alpha * 2.0 * [ A'*A, 0; 0, 0 ] * u + beta * v
"""
v *= beta
v[:n] += alpha * 2.0 * A.T * (A * u[:n])
def G(u, v, alpha=1.0, beta=0.0, trans='N'):
"""
v := alpha*[I, -I; -I, -I] * u + beta * v (trans = 'N' or 'T')
"""
v *= beta
v[:n] += alpha*(u[:n] - u[n:])
v[n:] += alpha*(-u[:n] - u[n:])
h = matrix(0.0, (2*n,1))
# Customized solver for the KKT system
#
# [ 2.0*A'*A 0 I -I ] [x[:n] ] [bx[:n] ]
# [ 0 0 -I -I ] [x[n:] ] = [bx[n:] ].
# [ I -I -D1^-1 0 ] [zl[:n]] [bzl[:n]]
# [ -I -I 0 -D2^-1 ] [zl[n:]] [bzl[n:]]
#
# where D1 = W['di'][:n]**2, D2 = W['di'][n:]**2.
#
# We first eliminate zl and x[n:]:
#
# ( 2*A'*A + 4*D1*D2*(D1+D2)^-1 ) * x[:n] =
# bx[:n] - (D2-D1)*(D1+D2)^-1 * bx[n:] +
# D1 * ( I + (D2-D1)*(D1+D2)^-1 ) * bzl[:n] -
# D2 * ( I - (D2-D1)*(D1+D2)^-1 ) * bzl[n:]
#
# x[n:] = (D1+D2)^-1 * ( bx[n:] - D1*bzl[:n] - D2*bzl[n:] )
# - (D2-D1)*(D1+D2)^-1 * x[:n]
#
# zl[:n] = D1 * ( x[:n] - x[n:] - bzl[:n] )
# zl[n:] = D2 * (-x[:n] - x[n:] - bzl[n:] ).
#
# The first equation has the form
#
# (A'*A + D)*x[:n] = rhs
#
# and is equivalent to
#
# [ D A' ] [ x:n] ] = [ rhs ]
# [ A -I ] [ v ] [ 0 ].
#
# It can be solved as
#
# ( A*D^-1*A' + I ) * v = A * D^-1 * rhs
# x[:n] = D^-1 * ( rhs - A'*v ).
S = matrix(0.0, (m,m))
Asc = matrix(0.0, (m,n))
v = matrix(0.0, (m,1))
def Fkkt(W):
# Factor
#
# S = A*D^-1*A' + I
#
# where D = 2*D1*D2*(D1+D2)^-1, D1 = d[:n]**-2, D2 = d[n:]**-2.
d1, d2 = W['di'][:n]**2, W['di'][n:]**2
# ds is square root of diagonal of D
ds = math.sqrt(2.0) * div( mul( W['di'][:n], W['di'][n:]), sqrt(d1+d2) )
d3 = div(d2 - d1, d1 + d2)
# Asc = A*diag(d)^-1/2
Asc = A * spdiag(ds**-1)
# S = I + A * D^-1 * A'
blas.syrk(Asc, S)
S[::m+1] += 1.0
lapack.potrf(S)
def g(x, y, z):
x[:n] = 0.5 * ( x[:n] - mul(d3, x[n:]) +
mul(d1, z[:n] + mul(d3, z[:n])) - mul(d2, z[n:] -
mul(d3, z[n:])) )
x[:n] = div( x[:n], ds)
# Solve
#
# S * v = 0.5 * A * D^-1 * ( bx[:n] -
# (D2-D1)*(D1+D2)^-1 * bx[n:] +
# D1 * ( I + (D2-D1)*(D1+D2)^-1 ) * bzl[:n] -
# D2 * ( I - (D2-D1)*(D1+D2)^-1 ) * bzl[n:] )
blas.gemv(Asc, x, v)
lapack.potrs(S, v)
# x[:n] = D^-1 * ( rhs - A'*v ).
blas.gemv(Asc, v, x, alpha=-1.0, beta=1.0, trans='T')
x[:n] = div(x[:n], ds)
# x[n:] = (D1+D2)^-1 * ( bx[n:] - D1*bzl[:n] - D2*bzl[n:] )
# - (D2-D1)*(D1+D2)^-1 * x[:n]
x[n:] = div( x[n:] - mul(d1, z[:n]) - mul(d2, z[n:]), d1+d2 )\
- mul( d3, x[:n] )
# zl[:n] = D1^1/2 * ( x[:n] - x[n:] - bzl[:n] )
# zl[n:] = D2^1/2 * ( -x[:n] - x[n:] - bzl[n:] ).
z[:n] = mul( W['di'][:n], x[:n] - x[n:] - z[:n] )
z[n:] = mul( W['di'][n:], -x[:n] - x[n:] - z[n:] )
return g
return solvers.coneqp(P, q, G, h, kktsolver = Fkkt)['x'][:n]
CVXOPT includes optional interfaces to several other optimization libraries.
GLPK, MOSEK and DSDP are not included in the CVXOPT distribution and need to be installed separately.
In this section we list some algorithm control parameters that can be modified without editing the source code. These control parameters are accessible via the dictionary solvers.options. By default the dictionary is empty and the default values of the parameters are used.
One can change the parameters in the default solvers by adding entries with the following key values.
For example the command
>>> from cvxopt import solvers
>>> solvers.options['show_progress'] = False
turns off the screen output during calls to the solvers.
The tolerances 'abstol', 'reltol' and 'feastol' have the following meaning. conelp terminates with status 'optimal' if
and
and
It returns with status 'primal infeasible' if
It returns with status 'dual infeasible' if
The functions lp <cvxopt.solvers.lp, socp and sdp call conelp and hence use the same stopping criteria.
The function coneqp terminates with status 'optimal' if
and
and at least one of the following three conditions is satisfied:
or
or
Here
The function qp calls coneqp and hence uses the same stopping criteria.
The control parameters listed in the GLPK documentation are set to their default values and can be customized by making an entry in solvers.options. The keys in the dictionary are strings with the name of the GLPK parameter. For example, the command
>>> from cvxopt import solvers
>>> solvers.options['LPX_K_MSGLEV'] = 0
turns off the screen output subsequent calls lp with the 'glpk' option.
The MOSEK interior-point algorithm parameters are set to their default values. They can be modified by adding an entry solvers.options['MOSEK']. This entry is a dictionary with MOSEK parameter/value pairs, with the parameter names imported from mosek. For details see Section 15 of the MOSEK Python API Manual.
For example the commands
>>> from cvxopt import solvers
>>> import mosek
>>> solvers.options['MOSEK'] = {mosek.iparam.log: 0}
turn off the screen output during calls of lp or socp with the 'mosek' option.
The following control parameters in solvers.options affect the execution of the DSDP algorithm: