.. _c-modeling:

********
Modeling
********

The module :mod:`cvxopt.modeling`  can be used to specify and solve 
optimization problems  with convex piecewise-linear objective and 
constraint functions.  Using this modeling tool, one can specify an 
optimization problem by first defining the optimization variables (see the 
section :ref:`s-variables`), and then specifying the objective and 
constraint functions using linear operations (vector addition and 
subtraction, matrix-vector multiplication, indexing and slicing)
and nested evaluations of :func:`max <cvxopt.modeling.max>`, 
:func:`min <cvxopt.modeling.min>`, 
:func:`abs <cvxopt.modeling.abs>` and 
:func:`sum <cvxopt.modeling.sum>` (see the section :ref:`s-functions`).

A more general Python convex modeling package is 
`CVXPY <http://cvxpy.org>`_.

.. _s-variables:

Variables 
=========

Optimization variables are represented by :class:`variable` objects.

.. function:: cvxopt.modeling.variable([size[, name]])

    A vector variable.  The first argument is the dimension of the vector
    (a positive integer with default value 1).  The second argument is a 
    string with a name for the variable.  The name is optional and has 
    default value :const:`""`. It is only used when displaying variables 
    (or objects that depend on variables, such as functions or constraints) 
    using :func:`print` statements, when calling the built-in functions
    :func:`repr`  or :func:`str`, or when writing linear programs to MPS 
    files.

The function :func:`!len` returns the length of a :class:`variable`.  
A :class:`variable` ``x`` has two attributes.

.. attribute:: variable.name 

    The name of the variable.  


.. attribute:: variable.value

    Either :const:`None` or a dense :const:`'d'` matrix of size 
    ``len(x)`` by 1.

    The attribute ``x.value`` is set to :const:`None` when the variable
    ``x`` is created.   It can be given a numerical value later, typically 
    by solving an LP that has ``x`` as one of its variables.  One can also 
    make an explicit assignment ``x.value = y``.  The assigned value 
    ``y`` must be an integer or float, or a dense :const:`'d'` matrix of 
    size ``(len(x), 1)``.  If ``y`` is an integer or float, all the 
    elements of ``x.value`` are set to the value of ``y``.



>>> from cvxopt import matrix
>>> from cvxopt.modeling import variable
>>> x = variable(3,'a')
>>> len(x)
3
>>> print(x.name)
a
>>> print(x.value)
None
>>> x.value = matrix([1.,2.,3.])
>>> print(x.value)
[ 1.00e+00]
[ 2.00e+00]
[ 3.00e+00]
>>> x.value = 1
>>> print(x.value)
[ 1.00e+00]
[ 1.00e+00]
[ 1.00e+00]


.. _s-functions:

Functions 
=========

Objective and constraint functions can be defined via overloaded operations
on variables and other functions.  A function ``f`` is interpreted as a 
column vector, with length ``len(f)`` and with a value that depends on 
the values of its variables.  Functions have two public attributes.  

.. attribute:: variables

    Returns a copy of the list of variables of the function.


.. attribute:: value

    The function value.  If any of the variables of ``f`` has value 
    :const:`None`, then ``f.value()`` returns :const:`None`.  Otherwise,
    it returns a dense :const:`'d'` matrix of size ``(len(f),1)`` with 
    the function value computed from the :attr:`value` attributes of the 
    variables of ``f``.  

Three types of functions are supported: affine, convex piecewise-linear, 
and concave piecewise-linear.

**Affine functions** represent vector valued functions of the form

.. math::

    f(x_1,\ldots,x_n) = A_1 x_1 + \cdots + A_n x_n + b.

The coefficients can be scalars or dense or sparse matrices. The 
constant term is a scalar or a column vector.

Affine functions result from the following operations.

**Unary operations** 
    For a variable ``x``, the unary operation ``+x`` results in an 
    affine function with ``x`` as variable, coefficient 1.0, and constant 
    term 0.0.  The unary operation ``-x`` returns an affine function 
    with ``x`` as variable, coefficient -1.0, and constant term 0.0.  For 
    an affine function ``f``, ``+f`` is a copy of ``f``, and  
    ``-f`` is a copy of ``f`` with the signs of its coefficients and 
    constant term reversed.

**Addition and subtraction**
    Sums and differences of affine functions, variables and constants result
    in new affine functions.  The constant terms in the sum can be of type 
    integer or float, or dense or sparse :const:`'d'` matrices with one 
    column. 

    The rules for addition and subtraction follow the conventions for 
    matrix addition and subtraction in the section :ref:`s-arithmetic`, 
    with variables and affine functions interpreted as dense :const:`'d'` 
    matrices with one column.  In particular, a scalar term (integer, float,
    1 by 1 dense :const:`'d'` matrix, variable of length 1, or affine 
    function of length 1) can be added to an affine function or variable of
    length greater than 1.

**Multiplication**
    Suppose ``v`` is an affine function or a variable, and ``a`` is an 
    integer, float, sparse or dense :const:`'d'` matrix.  The products 
    ``a * v`` and  ``v * a`` are valid affine functions whenever 
    the product is allowed under the rules for matrix and scalar 
    multiplication of the section :ref:`s-arithmetic`, with ``v`` 
    interpreted
    as a :const:`'d'` matrix with one column.  In particular, the product 
    ``a * v`` is defined if ``a`` is a scalar (integer, float, or 
    1 by 1 dense :const:`'d'` matrix), or a matrix (dense or sparse) with 
    ``a.size[1]`` equal to ``len(v)``.   The operation ``v * a``
    is defined if ``a`` is scalar, or if ``len(v)`` is 1 and ``a`` is a
    matrix with one column.

**Inner products**
    The following two functions return scalar affine functions defined
    as inner products of a constant vector with  a variable or affine
    function.

    .. function:: cvxopt.modeling.sum(v)

        The argument is an affine function or a variable.  The result is an
        affine function of length 1, with the sum of the components of the
        argument ``v``.  

    .. function:: cvxopt.modeling.dot(u, v)

        If ``v`` is a variable or affine function and ``u`` is a 
        :const:`'d'` matrix of size ``(len(v), 1)``, then 
        ``dot(u, v)`` and ``dot(v, u)`` are equivalent to 
        ``u.trans() * v``.

        If ``u`` and ``v`` are dense matrices, then :func:`dot` 
        is equivalent to the function :func:`blas.dot <cvxopt.blas.dot>`,
        i.e., it returns the inner product of the two matrices.


In the following example, the variable ``x`` has length 1 and ``y`` has 
length 2.  The functions ``f`` and ``g`` are given by

.. math::

    f(x,y) &= \left[ \begin{array}{c} 2 \\ 2 \end{array}\right] x 
        + y + \left[ \begin{array}{c} 3 \\ 3 \end{array}\right], \\
    g(x,y) &= 
        \left[ \begin{array}{cc} 1 & 3 \\ 2 & 4 \end{array}\right] f(x,y) +
        \left[ \begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array} \right] y + 
        \left[ \begin{array}{c} 1 \\ -1 \end{array} \right] \\
           &= \left[ \begin{array}{c} 8 \\ 12 \end{array}\right] x + 
       \left[ \begin{array}{cc} 2 & 4 \\ 3 & 5 \end{array}\right] y + 
       \left[ \begin{array}{c} 13 \\ 17\end{array}\right].


>>> from cvxopt.modeling import variable
>>> x = variable(1,'x')
>>> y = variable(2,'y')
>>> f = 2*x + y + 3  
>>> A = matrix([[1., 2.], [3.,4.]])
>>> b = matrix([1.,-1.])
>>> g = A*f + sum(y) + b 
>>> print(g)
affine function of length 2
constant term:
[ 1.30e+01]
[ 1.70e+01]
linear term: linear function of length 2
coefficient of variable(2,'y'):
[ 2.00e+00  4.00e+00]
[ 3.00e+00  5.00e+00]
coefficient of variable(1,'x'):
[ 8.00e+00]
[ 1.20e+01]


**In-place operations** 
    For an affine function ``f`` the operations ``f += u`` and 
    ``f -= u``, with ``u`` a constant, a variable or an affine function,
    are allowed if they do not change the length of ``f``, i.e., if ``u`` 
    has length ``len(f)`` or length 1.  In-place multiplication 
    ``f *= u`` and division ``f /= u`` are allowed if ``u`` is an 
    integer, float, or 1 by 1 matrix.


**Indexing and slicing** 
    Variables and affine functions admit single-argument indexing of the 
    four types described in the section :ref:`s-indexing`.  The result of 
    an indexing or slicing operation is an affine function.  


>>> x = variable(4,'x')
>>> f = x[::2]
>>> print(f)
linear function of length 2
linear term: linear function of length 2
coefficient of variable(4,'x'):
[ 1.00e+00     0         0         0    ]
[    0         0      1.00e+00     0    ]
>>> y = variable(3,'x')
>>> g = matrix(range(12),(3,4),'d')*x - 3*y + 1
>>> print(g[0] + g[2])
affine function of length 1
constant term:
[ 2.00e+00]
linear term: linear function of length 1
coefficient of variable(4,'x'):
[ 2.00e+00  8.00e+00  1.40e+01  2.00e+01]
coefficient of variable(3,'x'):
[-3.00e+00     0     -3.00e+00]


The general expression of a **convex piecewise-linear** function is

.. math::

    f(x_1,\ldots,x_n) = b + A_1 x_1 + \cdots + A_n x_n + 
        \sum_{k=1}^K \max (y_1, y_2, \ldots, y_{m_k}).

The maximum in this expression is a componentwise maximum of its vector 
arguments, which can be constant vectors, variables, affine functions or 
convex piecewise-linear functions.  The general expression for a 
**concave piecewise-linear** function is

.. math::

    f(x_1,\ldots,x_n) = b + A_1 x_1 + \cdots + A_n x_n + 
        \sum_{k=1}^K \min (y_1, y_2, \ldots, y_{m_k}).

Here the arguments of the :func:`!min` 
can be constants, variables, affine 
functions or concave piecewise-linear functions.

Piecewise-linear functions can be created using the following 
operations.

**Maximum**  
    If the arguments in ``f = max(y1, y2, ...)`` do not include any 
    variables or functions, then the Python built-in :func:`!max` is 
    evaluated.  

    If one or more of the arguments are variables or functions, 
    :func:`!max` 
    returns a piecewise-linear function defined as the elementwise maximum 
    of its arguments.  In other words, 
    ``f[k] = max(y1[k], y2[k], ...)`` for ``k`` = 0, ...,  
    ``len(f) - 1``.  The length of ``f`` is equal to the maximum of the
    lengths of the arguments.  Each argument must have length equal to 
    ``len(f)`` or length one.  Arguments with length one are interpreted
    as vectors of length ``len(f)`` with identical entries.

    The arguments can be scalars of type integer or float, dense 
    :const:`'d'` matrices with one column, variables, affine functions or 
    convex piecewise-linear functions.
     
    With one argument, ``f = max(u)`` is interpreted as
    ``f = max(u[0], u[1], ..., u[len(u)-1])``.  

**Minimum** 
    Similar to :func:`!max` but returns a concave piecewise-linear 
    function.
    The arguments can be scalars of type integer or float, dense 
    :const:`'d'` matrices with one column, variables, affine functions or 
    concave piecewise-linear functions.

**Absolute value** 
    If ``u`` is a variable or affine function then ``f = abs(u)`` 
    returns the convex piecewise-linear function ``max(u, -u)``.

**Unary plus and minus** 
    ``+f`` creates a copy of ``f``.  ``-f`` is a concave 
    piecewise-linear function if ``f`` is convex and a convex 
    piecewise-linear function if ``f`` is concave.

**Addition and subtraction**  
    Sums and differences involving piecewise-linear functions are allowed 
    if they result in convex or concave functions.  For example, one can add
    two convex or two concave functions, but not a convex and a concave 
    function.  The command ``sum(f)`` is equivalent to 
    ``f[0] + f[1] + ... + f[len(f) - 1]``.

**Multiplication** 
    Scalar multiplication ``a * f`` of a piecewise-linear function ``f``
    is defined if ``a`` is an integer, float, 1 by 1 :const:`'d'` matrix. 
    Matrix-matrix multiplications ``a * f`` or ``f * a`` are only 
    defined if ``a`` is a dense or sparse 1 by 1 matrix.

**Indexing and slicing** 
    Piecewise-linear functions admit single-argument indexing of the four 
    types described in the section :ref:`s-indexing`.  The result of an 
    indexing or slicing operation is a new piecewise-linear function.


In the following example, ``f`` is the 1-norm of a vector variable ``x`` of 
length 10, ``g`` is its infinity-norm, and ``h`` is the function

.. math::

    h(x) = \sum_k \phi(x[k]), \qquad
    \phi(u) = \left\{\begin{array}{ll}
        0       & |u| \leq 1 \\
        |u|-1   & 1 \leq |u| \leq 2 \\
        2|u|-3  & |u| \geq 2. 
    \end{array}\right.


>>> from cvxopt.modeling import variable, max
>>> x = variable(10, 'x')
>>> f = sum(abs(x))    
>>> g = max(abs(x))   
>>> h = sum(max(0, abs(x)-1, 2*abs(x)-3))  


**In-place operations**
    If ``f`` is piecewise-linear then the in-place operations  
    ``f += u``, ``f -= u``, ``f *= u``, ``f /= u`` are 
    defined if the corresponding expanded operations ``f = f + u``, 
    ``f = f - u``, ``f = f * u``, and ``f = f/u`` are defined 
    and if they do not change the length of ``f``.


.. _s-constraints:

Constraints
===========

Linear equality and inequality constraints of the form

.. math::

    f(x_1,\ldots,x_n) = 0, \qquad f(x_1,\ldots,x_n) \preceq  0, 

where :math:`f` is a convex function, are represented by :class:`constraint`
objects.  Equality constraints are created by expressions of the form 

::

    f1 == f2 

Here ``f1`` and ``f2`` can be any objects for which the difference 
``f1 - f2`` yields an affine function.  Inequality constraints are 
created by expressions of the form 

::

    f1 <= f2 
    f2 >= f1

where ``f1`` and ``f2`` can be any objects for which the difference 
``f1 - f2`` yields a convex piecewise-linear function.  The comparison 
operators first convert the expressions to ``f1 - f2 == 0``, resp., 
``f1 - f2 <= 0``, and then return a new :class:`constraint` object with
constraint function ``f1 - f2``.

In the following example we create three constraints

.. math::

    \newcommand{\ones}{{\bf 1}}
    0 \preceq x \preceq \ones, \qquad \ones^T x = 2,

for a variable of length 5.

>>> x = variable(5,'x')
>>> c1 = (x <= 1)
>>> c2 = (x >= 0)
>>> c3 = (sum(x) == 2)


The built-in function :func:`!len` returns the dimension of the 
constraint function.

Constraints have four public attributes.

.. attribute:: constraint.type

    Returns :const:`'='` if the constraint is an equality constraint, and 
    **'<'** if the constraint is an inequality constraint.


.. attribute:: constraint.value 

    Returns the value of the constraint function.  


.. attribute:: constraint.multiplier

    For a constraint ``c``, ``c.multiplier`` is a :class:`variable` 
    object of dimension ``len(c)``.  It is used to represent the 
    Lagrange multiplier or dual variable associated with the constraint.
    Its value is initialized as :const:`None`, and can be modified by making
    an assignment to ``c.multiplier.value``.


.. attribute:: constraint.name

    The name of the constraint.  Changing the name of a constraint also 
    changes the name of the multiplier of ``c``.  For example, the command  
    ``c.name = 'newname'`` also changes
    ``c.multiplier.name`` to ``'newname_mul'``.



.. _s-lp:

Optimization Problems 
=====================

Optimization problems are be constructed by calling the following
function.

.. function:: cvxopt.modeling.op([objective[, constraints[, name]]])

    The first argument specifies the objective function to be minimized.
    It can be an affine or convex piecewise-linear function with length 1, 
    a :class:`variable` with length 1, or a scalar constant (integer, float,
    or 1 by 1 dense :const:`'d'` matrix).  The default value is 0.0.

    The second argument is a single :class:`constraint`, or a list of 
    :class:`constraint` objects.  The default value is an empty list.

    The third argument is a string with a name for the problem.
    The default value is the empty string.

The following attributes and methods are useful for examining
and modifying optimization problems.

.. attribute:: op.objective

    The objective or cost function.  One can write to this attribute to 
    change the objective of an existing problem.  


.. method:: op.variables

    Returns a list of the variables of the problem.


.. method:: op.constraints

    Returns a list of the constraints.


.. method:: op.inequalities

    Returns a list of the inequality constraints.


.. method:: op.equalities

    Returns a list of the equality constraints.


.. method:: op.delconstraint(c)

    Deletes constraint ``c`` from the problem.


.. method:: op.addconstraint(c)

    Adds constraint ``c`` to the problem.


An optimization problem with convex piecewise-linear objective and
constraints can be solved by calling the method :func:`solve`.

.. method:: op.solve([format[, solver]]) 

    This function converts the optimization problem to a linear program in 
    matrix form and then solves it using the solver described in 
    the section :ref:`s-lpsolver`.

    The first argument is either :const:`'dense'` or :const:`'sparse'`, and 
    denotes the matrix types used in the matrix representation of the LP.
    The default value is :const:`'dense'`.

    The second argument is either :const:`None`, :const:`'glpk'`, or 
    :const:`'mosek'`, and selects one of three available LP solvers: the 
    default solver written in Python, the GLPK solver (if installed) or the
    MOSEK LP solver (if installed); see the section :ref:`s-lpsolver`.  The 
    default value is :const:`None`.

    The solver reports the outcome of optimization by setting the attribute 
    :attr:`self.status` and by modifying the :attr:`value` attributes of 
    the variables and the constraint multipliers of the problem.


    * If the problem is solved to optimality, :attr:`self.status` is set to
      :const:`'optimal'`.  The :attr:`value` attributes of the variables in
      the problem  are set to their computed solutions, and the 
      :attr:`value` attributes of the multipliers of the constraints of the
      problem are set to the computed dual optimal solution.

    * If it is determined that the problem is infeasible, 
      :attr:`self.status` is set to :const:`'primal infeasible'`.  
      The :attr:`value` attributes of the variables are set to 
      :const:`None`.  The :attr:`value` attributes of the multipliers of 
      the constraints of the problem are set to a certificate of primal 
      infeasibility.  With the :const:`'glpk'` option, :func:`solve` does 
      not provide certificates of infeasibility.

    * If it is determined that the problem is dual infeasible, 
      :attr:`self.status` is set to :const:`'dual infeasible'`.  
      The :attr:`value` attributes of the multipliers of the constraints of 
      the problem are set to :const:`None`.  The :attr:`value` attributes 
      of the variables are set to a certificate of dual infeasibility. 
      With the :const:`'glpk'` option, :func:`solve` does not provide 
      certificates of infeasibility.

    * If the problem was not solved successfully, :attr:`self.status` is set
      to :const:`'unknown'`.  The :attr:`value` attributes of the variables
      and the constraint multipliers are set to :const:`None`.

We refer to the section :ref:`s-lpsolver` for details on the algorithms and
the different solver options.

As an example we solve the LP

.. math::
     \begin{array}{ll}
     \mbox{minimize}   & -4x - 5y \\
     \mbox{subject to} &  2x +y \leq 3 \\
                       &  x +2y \leq 3 \\
                       & x \geq 0, \quad y \geq 0.
     \end{array}


>>> from cvxopt.modeling import op
>>> x = variable()
>>> y = variable()
>>> c1 = ( 2*x+y <= 3 ) 
>>> c2 = ( x+2*y <= 3 )
>>> c3 = ( x >= 0 )
>>> c4 = ( y >= 0 ) 
>>> lp1 = op(-4*x-5*y, [c1,c2,c3,c4]) 
>>> lp1.solve()
>>> lp1.status
'optimal'
>>> print(lp1.objective.value())
[-9.00e+00]
>>> print(x.value)
[ 1.00e+00]
>>> print(y.value)
[ 1.00e+00]
>>> print(c1.multiplier.value)
[ 1.00e+00]
>>> print(c2.multiplier.value)
[ 2.00e+00]
>>> print(c3.multiplier.value)
[ 2.87e-08]
>>> print(c4.multiplier.value)
[ 2.80e-08]


We can solve the same LP in  matrix form as follows.

>>> from cvxopt.modeling import op, dot
>>> x = variable(2)
>>> A = matrix([[2.,1.,-1.,0.], [1.,2.,0.,-1.]])
>>> b = matrix([3.,3.,0.,0.])
>>> c = matrix([-4.,-5.])
>>> ineq = ( A*x <= b )
>>> lp2 = op(dot(c,x), ineq)
>>> lp2.solve()
>>> print(lp2.objective.value())
[-9.00e+00]
>>> print(x.value)
[ 1.00e+00]
[ 1.00e+00]
>>> print(ineq.multiplier.value)
[1.00e+00]
[2.00e+00]
[2.87e-08]
[2.80e-08]


The :class:`op` class also includes two methods for writing and reading
files in 
`MPS format <http://lpsolve.sourceforge.net/5.5/mps-format.htm>`_.

.. method:: tofile(filename) :noindex:

    If the problem is an LP, writes it to the file `filename` using the 
    MPS format.  Row and column labels are assigned based on the variable 
    and constraint names in the LP.  


.. method:: fromfile(filename) :noindex:

    Reads the LP from the file `filename`.  The file must be a fixed-format
    MPS file.  Some features of the MPS format are not supported: comments 
    beginning with dollar signs, the row types 'DE', 'DL', 'DG', and 'DN', 
    and the capability of reading multiple righthand side, bound or range 
    vectors.


Examples
========


**Norm and Penalty Approximation**

    In the first example we solve the norm approximation problems

    .. math::

        \begin{array}{ll} 
        \mbox{minimize} & \|Ax - b\|_\infty,
        \end{array} 
        \qquad
        \begin{array}{ll} 
        \mbox{minimize} & \|Ax - b\|_1
        \end{array},

    and the penalty approximation problem

    .. math::

        \begin{array}{ll} 
        \mbox{minimize} & \sum_k \phi((Ax-b)_k), 
        \end{array} \qquad
        \phi(u) = \left\{\begin{array}{ll}
            0        & |u| \leq 3/4 \\
            |u|-3/4  & 3/4 \leq |u| \leq 3/2 \\
            2|u|-9/4 & |u| \geq 3/2.
        \end{array}\right.

    We use randomly generated data.

    The code uses the `Matplotlib <http://matplotlib.sourceforge.net>`_
    package for plotting the histograms of the residual vectors for the
    two solutions.  It generates the figure shown below.

    :: 

        from cvxopt import normal
        from cvxopt.modeling import variable, op, max, sum
        import pylab

        m, n = 500, 100
        A = normal(m,n)
        b = normal(m)

        x1 = variable(n)
        op(max(abs(A*x1-b))).solve()

        x2 = variable(n)
        op(sum(abs(A*x2-b))).solve()

        x3 = variable(n)
        op(sum(max(0, abs(A*x3-b)-0.75, 2*abs(A*x3-b)-2.25))).solve()

        pylab.subplot(311)
        pylab.hist(A*x1.value-b, m/5)
        pylab.subplot(312)
        pylab.hist(A*x2.value-b, m/5)
        pylab.subplot(313)
        pylab.hist(A*x3.value-b, m/5)
        pylab.show()


    .. image:: normappr.png
       :width: 600px


    Equivalently, we can formulate and solve the problems as LPs.
    
    ::

        t = variable()
        x1 = variable(n)
        op(t, [-t <= A*x1-b, A*x1-b<=t]).solve()

        u = variable(m)
        x2 = variable(n)
        op(sum(u), [-u <= A*x2+b, A*x2+b <= u]).solve()

        v = variable(m)
        x3 = variable(n)
        op(sum(v), [v >= 0, v >= A*x3+b-0.75, v >= -(A*x3+b)-0.75, v >= 2*(A*x3-b)-2.25, v >= -2*(A*x3-b)-2.25]).solve()



**Robust Linear Programming**

    The robust LP

    .. math::

        \begin{array}{ll}
        \mbox{minimize}   & c^T x \\
        \mbox{subject to} & \sup_{\|v\|_\infty \leq 1} 
                            (a_i+v)^T x \leq b_i, \qquad i=1,\ldots,m
        \end{array}

    is equivalent to the problem

    .. math::

        \begin{array}{ll}
        \mbox{minimize} & c^Tx \\
        \mbox{subject to} & a_i^Tx + \|x\|_1 \leq b_i, \qquad i=1,\ldots,m.
        \end{array}

    The following code computes the solution and the solution of the 
    equivalent LP

    .. math::

        \newcommand{\ones}{{\bf 1}}
        \begin{array}{ll}
        \mbox{minimize}   & c^Tx \\
        \mbox{subject to} & a_i^Tx + \ones^Ty \leq b_i, 
                            \qquad i=1,\ldots,m \\
                          & -y \preceq x \preceq y
        \end{array}

    for randomly generated data.

    :: 

        from cvxopt import normal, uniform
        from cvxopt.modeling import variable, dot, op, sum 

        m, n = 500, 100
        A = normal(m,n)
        b = uniform(m)
        c = normal(n)

        x = variable(n)
        op(dot(c,x), A*x+sum(abs(x)) <= b).solve()

        x2 = variable(n)
        y = variable(n)
        op(dot(c,x2), [A*x2+sum(y) <= b, -y <= x2, x2 <= y]).solve()



**1-Norm Support Vector Classifier**

    The following problem arises in classification:

    .. math::

        \newcommand{\ones}{{\bf 1}}
        \begin{array}{ll}
        \mbox{minimize}   & \|x\|_1 + \ones^Tu \\
        \mbox{subject to} & Ax \succeq \ones -u \\
                          & u \succeq 0.
        \end{array}


    It can be solved as follows.

    ::

        x = variable(A.size[1],'x')
        u = variable(A.size[0],'u')
        op(sum(abs(x)) + sum(u), [A*x >= 1-u, u >= 0]).solve()

    An equivalent unconstrained formulation is

    :: 

        x = variable(A.size[1],'x')
        op(sum(abs(x)) + sum(max(0,1-A*x))).solve()

