python-cvxopt 1.1.4-1.4 / usr / lib / python2.7 / dist-packages / cvxopt / msk.py

"""
CVXOPT interface for MOSEK 6.0
"""

# Copyright 2010-2011 L. Vandenberghe.
# Copyright 2004-2009 J. Dahl and L. Vandenberghe.
# 
# This file is part of CVXOPT version 1.1.4.
#
# CVXOPT is free software; you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation; either version 3 of the License, or
# (at your option) any later version.
#
# CVXOPT is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
# GNU General Public License for more details.
# 
# You should have received a copy of the GNU General Public License
# along with this program.  If not, see <http://www.gnu.org/licenses/>.


import mosek
from cvxopt import matrix, spmatrix, sparse
from mosek.array import array, zeros
env = mosek.Env()
env.init()
def streamprinter(text): print(text)
env.set_Stream (mosek.streamtype.log, streamprinter)

inf = 0.0

options = {}

def lp(c, G, h, A=None, b=None):
    """
    Solves a pair of primal and dual LPs 

        minimize    c'*x             maximize    -h'*z - b'*y 
        subject to  G*x + s = h      subject to  G'*z + A'*y + c = 0
                    A*x = b                      z >= 0.
                    s >= 0
                    
    using MOSEK 6.0.

    (solsta, x, z, y) = lp(c, G, h, A=None, b=None).

    Input arguments 

        G is m x n, h is m x 1, A is p x n, b is p x 1.  G and A must be 
        dense or sparse 'd' matrices.   h and b are dense 'd' matrices 
        with one column.  The default values for A and b are empty 
        matrices with zero rows.


    Return values

        solsta is a MOSEK solution status key.

            If solsta is mosek.solsta.optimal, then (x, y, z) contains the 
                primal-dual solution.
            If solsta is mosek.solsta.prim_infeas_cer, then (x, y, z) is a 
                certificate of primal infeasibility.
            If solsta is mosek.solsta.dual_infeas_cer, then (x, y, z) is a 
                certificate of dual infeasibility.
            If solsta is mosek.solsta.unknown, then (x, y, z) are all None.

            Other return values for solsta include:  
                mosek.solsta.dual_feas  
                mosek.solsta.near_dual_feas
                mosek.solsta.near_optimal
                mosek.solsta.near_prim_and_dual_feas
                mosek.solsta.near_prim_feas
                mosek.solsta.prim_and_dual_feas
                mosek.solsta.prim_feas
             in which case the (x,y,z) value may not be well-defined,
             c.f., section 17.48 of the MOSEK Python API manual.
        
        x, y, z  the primal-dual solution.                    

    Options are passed to MOSEK solvers via the msk.options dictionary. 
    For example, the following turns off output from the MOSEK solvers
    
        >>> msk.options = {mosek.iparam.log: 0} 
    
    see chapter 15 of the MOSEK Python API manual.                    
    """

    if type(c) is not matrix or c.typecode != 'd' or c.size[1] != 1: 
        raise TypeError("'c' must be a dense column matrix")
    n = c.size[0]
    if n < 1: raise ValueError("number of variables must be at least 1")

    if (type(G) is not matrix and type(G) is not spmatrix) or \
        G.typecode != 'd' or G.size[1] != n:
        raise TypeError("'G' must be a dense or sparse 'd' matrix "\
            "with %d columns" %n)
    m = G.size[0]
    if m is 0: raise ValueError("m cannot be 0")

    if type(h) is not matrix or h.typecode != 'd' or h.size != (m,1):
        raise TypeError("'h' must be a 'd' matrix of size (%d,1)" %m)

    if A is None:  A = spmatrix([], [], [], (0,n), 'd')
    if (type(A) is not matrix and type(A) is not spmatrix) or \
        A.typecode != 'd' or A.size[1] != n:
        raise TypeError("'A' must be a dense or sparse 'd' matrix "\
            "with %d columns" %n)
    p = A.size[0]
    if b is None: b = matrix(0.0, (0,1))
    if type(b) is not matrix or b.typecode != 'd' or b.size != (p,1): 
        raise TypeError("'b' must be a dense matrix of size (%d,1)" %p)
 
    bkc = m*[ mosek.boundkey.up ] + p*[ mosek.boundkey.fx ]
    blc = m*[ -inf ] + [ bi for bi in b ]
    buc = matrix([h, b])

    bkx = n*[mosek.boundkey.fr] 
    blx = n*[ -inf ] 
    bux = n*[ +inf ]

    colptr, asub, acof = sparse([G,A]).CCS
    aptrb, aptre = colptr[:-1], colptr[1:]

    task = env.Task(0,0) 
    task.set_Stream (mosek.streamtype.log, streamprinter) 

    # set MOSEK options 
    for (param, val) in options.items():
        if str(param)[:6] == "iparam":
            task.putintparam(param, val)
        elif str(param)[:6] == "dparam":
            task.putdouparam(param, val)
        elif str(param)[:6] == "sparam":
            task.putstrparam(param, val)
        else:
            raise ValueError("invalid MOSEK parameter: " + str(param))

    task.inputdata (m+p, # number of constraints
                    n,   # number of variables
                    array(c), # linear objective coefficients  
                    0.0, # objective fixed value  
                    array(aptrb), 
                    array(aptre), 
                    array(asub),
                    array(acof), 
                    bkc,
                    blc,
                    buc, 
                    bkx,
                    blx,
                    bux) 

    task.putobjsense(mosek.objsense.minimize)

    task.optimize()

    task.solutionsummary (mosek.streamtype.msg); 

    prosta, solsta = task.getsolutionstatus(mosek.soltype.bas)

    x, z = zeros(n, float), zeros(m, float)
    task.getsolutionslice(mosek.soltype.bas, mosek.solitem.xx, 0, n, x) 
    task.getsolutionslice(mosek.soltype.bas, mosek.solitem.suc, 0, m, z) 
    x, z = matrix(x), matrix(z)
    
    if p is not 0:
        yu, yl = zeros(p, float), zeros(p, float)
        task.getsolutionslice(mosek.soltype.bas, mosek.solitem.suc, m, 
            m+p, yu) 
        task.getsolutionslice(mosek.soltype.bas, mosek.solitem.slc, m, 
            m+p, yl) 
        y = matrix(yu) - matrix(yl)
    else:
        y = matrix(0.0, (0,1))

    if (solsta is mosek.solsta.unknown):
        return (solsta, None, None, None)
    else:
        return (solsta, x, z, y)


def conelp(c, G, h, dims = None):
    """
    Solves a pair of primal and dual SOCPs

        minimize    c'*x
        subject to  G*x + s = h
                    s >= 0

        maximize    -h'*z 
        subject to  G'*z + c = 0
                    z >= 0 

    using MOSEK 6.0.   

    The inequalities are with respect to a cone C defined as the Cartesian
    product of N + 1 cones:
    
        C = C_0 x C_1 x .... x C_N x C_{N+1}.

    The first cone C_0 is the nonnegative orthant of dimension ml.
    The other cones are second order cones of dimension mq[0], ..., 
    mq[N-1].  The second order cone of dimension m is defined as
    
        { (u0, u1) in R x R^{m-1} | u0 >= ||u1||_2 }.

    The formats of G and h are identical to that used in solvers.conelp(), 
    except that only componentwise and second order cone inequalities are 
    (dims['s'] must be zero, if defined).

    Input arguments.
   
        c is a dense 'd' matrix of size (n,1).

        dims is a dictionary with the dimensions of the components of C.  
        It has three fields.
        - dims['l'] = ml, the dimension of the nonnegative orthant C_0.
          (ml >= 0.)
        - dims['q'] = mq = [ mq[0], mq[1], ..., mq[N-1] ], a list of N 
          integers with the dimensions of the second order cones C_1, ..., 
          C_N.  (N >= 0 and mq[k] >= 1.)
        The default value of dims is {'l': G.size[0], 'q': []}.

        G is a dense or sparse 'd' matrix of size (K,n), where

            K = ml + mq[0] + ... + mq[N-1].

        Each column of G describes a vector 

            v = ( v_0, v_1, ..., v_N, vec(v_{N+1}) )

        in V = R^ml x R^mq[0] x ... x R^mq[N-1] stored as a column vector.

        h is a dense 'd' matrix of size (K,1), representing a vector in V,
        in the same format as the columns of G.
    

 
    Return values

        solsta is a MOSEK solution status key.

            If solsta is mosek.solsta.optimal,
                then (x, zl, zq) contains the primal-dual solution.
            If solsta is moseksolsta.prim_infeas_cer,
                then (x, zl, zq) is a certificate of dual infeasibility.
            If solsta is moseksolsta.dual_infeas_cer,
                then (x, zl, zq) is a certificate of primal infeasibility.
            If solsta is mosek.solsta.unknown,
                then (x, zl, zq) are all None

            Other return values for solsta include:  
                mosek.solsta.dual_feas  
                mosek.solsta.near_dual_feas
                mosek.solsta.near_optimal
                mosek.solsta.near_prim_and_dual_feas
                mosek.solsta.near_prim_feas
                mosek.solsta.prim_and_dual_feas
                mosek.solsta.prim_feas
            in which case the (x,y,z) value may not be well-defined,
            c.f., section 17.48 of the MOSEK Python API manual.
        
        x, z the primal-dual solution.


    Options are passed to MOSEK solvers via the msk.options dictionary, 
    e.g., the following turns off output from the MOSEK solvers
    
        >>> msk.options = {mosek.iparam.log:0} 
    
    see chapter 15 of the MOSEK Python API manual.                    
    """

    if dims is None: 
        (solsta, x, y, z) = lp(c, G, h)
        return (solsta, x, z, None)

    try:
        if len(dims['s']) > 0: raise ValueError("dims['s'] must be zero")
    except:
        pass

    N, n = G.size
    ml, mq = dims['l'], dims['q']
    cdim = ml + sum(mq)
    if cdim is 0: raise ValueError("ml+mq cannot be 0")

    # Data for kth 'q' constraint are found in rows indq[k]:indq[k+1] of G.
    indq = [ dims['l'] ]  
    for k in dims['q']:  indq = indq + [ indq[-1] + k ] 

    if type(h) is not matrix or h.typecode != 'd' or h.size[1] != 1:
        raise TypeError("'h' must be a 'd' matrix with 1 column")
    if type(G) is matrix or type(G) is spmatrix:
        if G.typecode != 'd' or G.size[0] != cdim:
            raise TypeError("'G' must be a 'd' matrix with %d rows " %cdim)
        if h.size[0] != cdim:
            raise TypeError("'h' must have %d rows" %cdim)
    else: 
        raise TypeError("'G' must be a matrix")

    if min(dims['q'])<1: raise TypeError(
        "dimensions of quadratic cones must be positive")

    bkc = n*[ mosek.boundkey.fx ] 
    blc = array(-c)
    buc = array(-c)

    bkx = ml*[ mosek.boundkey.lo ] + sum(mq)*[ mosek.boundkey.fr ]
    blx = ml*[ 0.0 ] + sum(mq)*[ -inf ]
    bux = N*[ +inf ] 

    c   = array(-h)       
    
    colptr, asub, acof = sparse([G.T]).CCS
    aptrb, aptre = colptr[:-1], colptr[1:]

    task = env.Task(0,0) 
    task.set_Stream (mosek.streamtype.log, streamprinter) 

    # set MOSEK options 
    for (param, val) in options.items():
        if str(param)[:6] == "iparam":
            task.putintparam(param, val)
        elif str(param)[:6] == "dparam":
            task.putdouparam(param, val)
        elif str(param)[:6] == "sparam":
            task.putstrparam(param, val)
        else:
            raise ValueError("invalid MOSEK parameter: "+str(param))

    task.inputdata (n,   # number of constraints
                    N,   # number of variables
                    c,   # linear objective coefficients  
                    0.0, # objective fixed value  
                    array(aptrb), 
                    array(aptre), 
                    array(asub),
                    array(acof), 
                    bkc,
                    blc,
                    buc, 
                    bkx,
                    blx,
                    bux) 

    task.putobjsense(mosek.objsense.maximize)

    for k in range(len(mq)):
        task.appendcone(mosek.conetype.quad, 0.0, 
                        array(range(ml+sum(mq[:k]),ml+sum(mq[:k+1]))))
    task.optimize()

    task.solutionsummary (mosek.streamtype.msg); 

    prosta, solsta = task.getsolutionstatus(mosek.soltype.itr)

    xu, xl, zq = zeros(n, float), zeros(n, float), zeros(sum(mq), float)
    task.getsolutionslice(mosek.soltype.itr, mosek.solitem.slc, 0, n, xl) 
    task.getsolutionslice(mosek.soltype.itr, mosek.solitem.suc, 0, n, xu) 
    task.getsolutionslice(mosek.soltype.itr, mosek.solitem.xx, ml, N, zq) 
    x = matrix(xu-xl)
    zq = matrix(zq)

    if ml:
        zl = zeros(ml, float)
        task.getsolutionslice(mosek.soltype.itr, mosek.solitem.xx, 0, ml, 
            zl) 
        zl = matrix(zl)
    else:
        zl = matrix(0.0, (0,1))

    if (solsta is mosek.solsta.unknown):
        return (solsta, None, None)
    else:
        return (solsta, x, matrix([zl, zq]))



def socp(c, Gl = None, hl = None, Gq = None, hq = None):
    """
    Solves a pair of primal and dual SOCPs

        minimize    c'*x             
        subject to  Gl*x + sl = hl      
                    Gq[k]*x + sq[k] = hq[k],  k = 0, ..., N-1
                    sl >= 0,  
                    sq[k] >= 0, k = 0, ..., N-1

        maximize    -hl'*zl - sum_k hq[k]'*zq[k] 
        subject to  Gl'*zl + sum_k Gq[k]'*zq[k] + c = 0
                    zl >= 0,  zq[k] >= 0, k = 0, ..., N-1.
                    
    using MOSEK 6.0.

    solsta, x, zl, zq = socp(c, Gl = None, hl = None, Gq = None, hq = None)

    Return values

        solsta is a MOSEK solution status key.
            If solsta is mosek.solsta.optimal,
                then (x, zl, zq) contains the primal-dual solution.
            If solsta is mosek.solsta.prim_infeas_cer,
                then (x, zl, zq) is a certificate of dual infeasibility.
            If solsta is mosek.solsta.dual_infeas_cer,
                then (x, zl, zq) is a certificate of primal infeasibility.
            If solsta is mosek.solsta.unknown,
                then (x, zl, zq) are all None

            Other return values for solsta include:  
                mosek.solsta.dual_feas  
                mosek.solsta.near_dual_feas
                mosek.solsta.near_optimal
                mosek.solsta.near_prim_and_dual_feas
                mosek.solsta.near_prim_feas
                mosek.solsta.prim_and_dual_feas
                mosek.solsta.prim_feas
             in which case the (x,y,z) value may not be well-defined,
             c.f., section 17.48 of the MOSEK Python API manual.
        
        x, zl, zq  the primal-dual solution.


    Options are passed to MOSEK solvers via the msk.options dictionary, 
    e.g., the following turns off output from the MOSEK solvers
    
        >>> msk.options = {mosek.iparam.log: 0} 
    
    see chapter 15 of the MOSEK Python API manual.                    
    """

    if type(c) is not matrix or c.typecode != 'd' or c.size[1] != 1: 
        raise TypeError("'c' must be a dense column matrix")
    n = c.size[0]
    if n < 1: raise ValueError("number of variables must be at least 1")

    if Gl is None:  Gl = spmatrix([], [], [], (0,n), tc='d')
    if (type(Gl) is not matrix and type(Gl) is not spmatrix) or \
        Gl.typecode != 'd' or Gl.size[1] != n:
        raise TypeError("'Gl' must be a dense or sparse 'd' matrix "\
            "with %d columns" %n)
    ml = Gl.size[0]
    if hl is None: hl = matrix(0.0, (0,1))
    if type(hl) is not matrix or hl.typecode != 'd' or \
        hl.size != (ml,1):
        raise TypeError("'hl' must be a dense 'd' matrix of " \
            "size (%d,1)" %ml)

    if Gq is None: Gq = []
    if type(Gq) is not list or [ G for G in Gq if (type(G) is not matrix 
        and type(G) is not spmatrix) or G.typecode != 'd' or 
        G.size[1] != n ]:
        raise TypeError("'Gq' must be a list of sparse or dense 'd' "\
            "matrices with %d columns" %n)
    mq = [ G.size[0] for G in Gq ]
    a = [ k for k in range(len(mq)) if mq[k] == 0 ] 
    if a: raise TypeError("the number of rows of Gq[%d] is zero" %a[0])
    if hq is None: hq = []
    if type(hq) is not list or len(hq) != len(mq) or [ h for h in hq if
        (type(h) is not matrix and type(h) is not spmatrix) or 
        h.typecode != 'd' ]: 
            raise TypeError("'hq' must be a list of %d dense or sparse "\
                "'d' matrices" %len(mq))
    a = [ k for k in range(len(mq)) if hq[k].size != (mq[k], 1) ]
    if a:
        k = a[0]
        raise TypeError("'hq[%d]' has size (%d,%d).  Expected size "\
            "is (%d,1)." %(k, hq[k].size[0], hq[k].size[1], mq[k]))

    N = ml + sum(mq)
    h = matrix(0.0, (N,1))
    if type(Gl) is matrix or [ Gk for Gk in Gq if type(Gk) is matrix ]:
        G = matrix(0.0, (N, n))
    else:
        G = spmatrix([], [], [], (N, n), 'd')
    h[:ml] = hl
    G[:ml,:] = Gl
    ind = ml
    for k in range(len(mq)):
        h[ind : ind + mq[k]] = hq[k]
        G[ind : ind + mq[k], :] = Gq[k]
        ind += mq[k]

    bkc = n*[ mosek.boundkey.fx ] 
    blc = array(-c)
    buc = array(-c)

    bkx = ml*[ mosek.boundkey.lo ] + sum(mq)*[ mosek.boundkey.fr ]
    blx = ml*[ 0.0 ] + sum(mq)*[ -inf ]
    bux = N*[ +inf ] 

    c   = -h        
    
    colptr, asub, acof = sparse([G.T]).CCS
    aptrb, aptre = colptr[:-1], colptr[1:]

    task = env.Task(0,0) 
    task.set_Stream (mosek.streamtype.log, streamprinter) 

    # set MOSEK options 
    for (param, val) in options.items():
        if str(param)[:6] == "iparam":
            task.putintparam(param, val)
        elif str(param)[:6] == "dparam":
            task.putdouparam(param, val)
        elif str(param)[:6] == "sparam":
            task.putstrparam(param, val)
        else:
            raise ValueError("invalid MOSEK parameter: "+str(param))

    task.inputdata (n,   # number of constraints
                    N,   # number of variables
                    array(c), # linear objective coefficients  
                    0.0, # objective fixed value  
                    array(aptrb), 
                    array(aptre), 
                    array(asub),
                    array(acof), 
                    bkc,
                    blc,
                    buc, 
                    bkx,
                    blx,
                    bux) 

    task.putobjsense(mosek.objsense.maximize)

    for k in range(len(mq)):
        task.appendcone(mosek.conetype.quad, 0.0, 
                        array(range(ml+sum(mq[:k]),ml+sum(mq[:k+1]))))
    task.optimize()

    task.solutionsummary (mosek.streamtype.msg); 

    prosta, solsta = task.getsolutionstatus(mosek.soltype.itr)

    xu, xl, zq = zeros(n, float), zeros(n, float), zeros(sum(mq), float)
    task.getsolutionslice(mosek.soltype.itr, mosek.solitem.slc, 0, n, xl) 
    task.getsolutionslice(mosek.soltype.itr, mosek.solitem.suc, 0, n, xu) 
    task.getsolutionslice(mosek.soltype.itr, mosek.solitem.xx, ml, N, zq) 
    x = matrix(xu-xl)

    zq = [ matrix(zq[sum(mq[:k]):sum(mq[:k+1])]) for k in range(len(mq)) ]
    
    if ml:
        zl = zeros(ml, float)
        task.getsolutionslice(mosek.soltype.itr, mosek.solitem.xx, 0, ml, 
            zl) 
        zl = matrix(zl)
    else:
        zl = matrix(0.0, (0,1))

    if (solsta is mosek.solsta.unknown):
        return (solsta, None, None, None)
    else:
        return (solsta, x, zl, zq)


def qp(P, q, G=None, h=None, A=None, b=None):
    """
    Solves a quadratic program

        minimize    (1/2)*x'*P*x + q'*x 
        subject to  G*x <= h      
                    A*x = b.                    
                    
    using MOSEK 6.0.

    solsta, x, z, y = qp(P, q, G=None, h=None, A=None, b=None)

    Return values

        solsta is a MOSEK solution status key.

            If solsta is mosek.solsta.optimal,
                then (x, y, z) contains the primal-dual solution.
            If solsta is mosek.solsta.prim_infeas_cer,
                then (x, y, z) is a certificate of primal infeasibility.
            If solsta is mosek.solsta.dual_infeas_cer,
                then (x, y, z) is a certificate of dual infeasibility.
            If solsta is mosek.solsta.unknown, then (x, y, z) are all None.

            Other return values for solsta include:  
                mosek.solsta.dual_feas  
                mosek.solsta.near_dual_feas
                mosek.solsta.near_optimal
                mosek.solsta.near_prim_and_dual_feas
                mosek.solsta.near_prim_feas
                mosek.solsta.prim_and_dual_feas
                mosek.solsta.prim_feas
            in which case the (x,y,z) value may not be well-defined,
            c.f., section 17.48 of the MOSEK Python API manual.
        
        x, z, y  the primal-dual solution.                    

    Options are passed to MOSEK solvers via the msk.options dictionary, 
    e.g., the following turns off output from the MOSEK solvers
    
        >>> msk.options = {mosek.iparam.log: 0} 
    
    see chapter 15 of the MOSEK Python API manual.                    
    """

    if (type(P) is not matrix and type(P) is not spmatrix) or \
        P.typecode != 'd' or P.size[0] != P.size[1]:
        raise TypeError("'P' must be a square dense or sparse 'd' matrix ")
    n = P.size[0]

    if n < 1: raise ValueError("number of variables must be at least 1")

    if type(q) is not matrix or q.typecode != 'd' or q.size != (n,1):
        raise TypeError("'q' must be a 'd' matrix of size (%d,1)" %n)

    if G is None: G = spmatrix([], [], [], (0,n), 'd')
    if (type(G) is not matrix and type(G) is not spmatrix) or \
        G.typecode != 'd' or G.size[1] != n:
        raise TypeError("'G' must be a dense or sparse 'd' matrix "\
            "with %d columns" %n)

    m = G.size[0]
    if h is None: h = matrix(0.0, (0,1))
    if type(h) is not matrix or h.typecode != 'd' or h.size != (m,1):
        raise TypeError("'h' must be a 'd' matrix of size (%d,1)" %m)

    if A is None:  A = spmatrix([], [], [], (0,n), 'd')
    if (type(A) is not matrix and type(A) is not spmatrix) or \
        A.typecode != 'd' or A.size[1] != n:
        raise TypeError("'A' must be a dense or sparse 'd' matrix "\
            "with %d columns" %n)
    p = A.size[0]
    if b is None: b = matrix(0.0, (0,1))
    if type(b) is not matrix or b.typecode != 'd' or b.size != (p,1): 
        raise TypeError("'b' must be a dense matrix of size (%d,1)" %p)
 
    if m+p is 0: raise ValueError("m + p must be greater than 0")

    c = array(q)        

    bkc = m*[ mosek.boundkey.up ] + p*[ mosek.boundkey.fx ]
    blc = m*[ -inf ] + [ bi for bi in b ]
    buc = matrix([h, b])

    bkx = n*[mosek.boundkey.fr] 
    blx = n*[ -inf ] 
    bux = n*[ +inf ]

    colptr, asub, acof = sparse([G,A]).CCS
    aptrb, aptre = colptr[:-1], colptr[1:]

    task = env.Task(0,0) 
    task.set_Stream (mosek.streamtype.log, streamprinter) 

    # set MOSEK options 
    for (param, val) in options.items():
        if str(param)[:6] == "iparam":
            task.putintparam(param, val)
        elif str(param)[:6] == "dparam":
            task.putdouparam(param, val)
        elif str(param)[:6] == "sparam":
            task.putstrparam(param, val)
        else:
            raise ValueError("invalid MOSEK parameter: "+str(param))

    task.inputdata (m+p, # number of constraints
                    n,   # number of variables
                    array(c), # linear objective coefficients  
                    0.0, # objective fixed value  
                    array(aptrb), 
                    array(aptre), 
                    array(asub),
                    array(acof), 
                    bkc,
                    blc,
                    buc, 
                    bkx,
                    blx,
                    bux) 

    Ps = sparse(P)
    I, J = Ps.I, Ps.J
    tril = [ k for k in range(len(I)) if I[k] >= J[k] ]
    task.putqobj(array(I[tril]), array(J[tril]), array(Ps.V[tril]))
    
    task.putobjsense(mosek.objsense.minimize)

    task.optimize()

    task.solutionsummary (mosek.streamtype.msg); 

    prosta, solsta = task.getsolutionstatus(mosek.soltype.itr)

    x = zeros(n, float)
    task.getsolutionslice(mosek.soltype.itr, mosek.solitem.xx, 0, n, x) 
    x = matrix(x)

    if m is not 0:
        z = zeros(m, float)
        task.getsolutionslice(mosek.soltype.itr, mosek.solitem.suc, 0, m, 
            z) 
        z = matrix(z)
    else:
        z = matrix(0.0, (0,1))

    if p is not 0:
        yu, yl = zeros(p, float), zeros(p, float)
        task.getsolutionslice(mosek.soltype.itr, mosek.solitem.suc, m, m+p,
            yu) 
        task.getsolutionslice(mosek.soltype.itr, mosek.solitem.slc, m, m+p,
            yl) 
        y = matrix(yu) - matrix(yl)
    else:
        y = matrix(0.0, (0,1))

    if (solsta is mosek.solsta.unknown):
        return (solsta, None, None, None)
    else:
        return (solsta, x, z, y)


def ilp(c, G, h, A=None, b=None, I=None):
    """
    Solves the mixed integer LP

        minimize    c'*x       
        subject to  G*x + s = h
                    A*x = b    
                    s >= 0
                    xi integer, forall i in I
                    
    using MOSEK 6.0.

    solsta, x = ilp(c, G, h, A=None, b=None, I=None).

    Input arguments 

        G is m x n, h is m x 1, A is p x n, b is p x 1.  G and A must be 
        dense or sparse 'd' matrices.   h and b are dense 'd' matrices 
        with one column.  The default values for A and b are empty 
        matrices with zero rows.

        I is a Python set with indices of integer elements of x.  By 
        default all elements in x are constrained to be integer, i.e.,
        the default value of I is I = set(range(n))

        Dual variables are not returned for MOSEK.


    Return values

        solsta is a MOSEK solution status key.
            
            If solsta is mosek.solsta.integer_optimal, then x contains 
                the solution.
            If solsta is mosek.solsta.unknown, then x is None.

            Other return values for solsta include:  
                mosek.solsta.near_integer_optimal
            in which case the x value may not be well-defined,
            c.f., section 17.48 of the MOSEK Python API manual.
        
        x is the solution

    Options are passed to MOSEK solvers via the msk.options dictionary, 
    e.g., the following turns off output from the MOSEK solvers
    
    >>> msk.options = {mosek.iparam.log: 0} 
    
    see chapter 15 of the MOSEK Python API manual.                    
    """

    if type(c) is not matrix or c.typecode != 'd' or c.size[1] != 1: 
        raise TypeError("'c' must be a dense column matrix")
    n = c.size[0]
    if n < 1: raise ValueError("number of variables must be at least 1")

    if (type(G) is not matrix and type(G) is not spmatrix) or \
        G.typecode != 'd' or G.size[1] != n:
        raise TypeError("'G' must be a dense or sparse 'd' matrix "\
            "with %d columns" %n)
    m = G.size[0]
    if m is 0: raise ValueError("m cannot be 0")

    if type(h) is not matrix or h.typecode != 'd' or h.size != (m,1):
        raise TypeError("'h' must be a 'd' matrix of size (%d,1)" %m)

    if A is None:  A = spmatrix([], [], [], (0,n), 'd')
    if (type(A) is not matrix and type(A) is not spmatrix) or \
        A.typecode != 'd' or A.size[1] != n:
        raise TypeError("'A' must be a dense or sparse 'd' matrix "\
            "with %d columns" %n)
    p = A.size[0]
    if b is None: b = matrix(0.0, (0,1))
    if type(b) is not matrix or b.typecode != 'd' or b.size != (p,1): 
        raise TypeError("'b' must be a dense matrix of size (%d,1)" %p)
 
    c = array(c)        

    if I is None: I = set(range(n))

    if type(I) is not set: 
        raise TypeError("invalid argument for integer index set")

    for i in I:
        if type(i) is not int: 
            raise TypeError("invalid integer index set I")

    if len(I) > 0 and min(I) < 0: raise IndexError(
            "negative element in integer index set I")
    if len(I) > 0 and max(I) > n-1: raise IndexError(
            "maximum element in in integer index set I is larger than n-1")

    bkc = m*[ mosek.boundkey.up ] + p*[ mosek.boundkey.fx ]
    blc = m*[ -inf ] + [ bi for bi in b ]
    buc = matrix([h, b])

    bkx = n*[mosek.boundkey.fr] 
    blx = n*[ -inf ] 
    bux = n*[ +inf ]

    colptr, asub, acof = sparse([G,A]).CCS
    aptrb, aptre = colptr[:-1], colptr[1:]

    task = env.Task(0,0) 
    task.set_Stream (mosek.streamtype.log, streamprinter) 

    # set MOSEK options 
    for (param, val) in options.items():
        if str(param)[:6] == "iparam":
            task.putintparam(param, val)
        elif str(param)[:6] == "dparam":
            task.putdouparam(param, val)
        elif str(param)[:6] == "sparam":
            task.putstrparam(param, val)
        else:
            raise ValueError("invalid MOSEK parameter: "+str(param))
    
    task.inputdata (m+p, # number of constraints
                    n,   # number of variables
                    array(c), # linear objective coefficients  
                    0.0, # objective fixed value  
                    array(aptrb), 
                    array(aptre), 
                    array(asub),
                    array(acof), 
                    bkc,
                    blc,
                    buc, 
                    bkx,
                    blx,
                    bux) 

    task.putobjsense(mosek.objsense.minimize)

    # Define integer variables 
    if len(I) > 0:
        task.putvartypelist(list(I), len(I)*[ mosek.variabletype.type_int ])

    task.putintparam (mosek.iparam.mio_mode, mosek.miomode.satisfied) 

    task.optimize()

    task.solutionsummary (mosek.streamtype.msg); 

    if len(I) > 0:
        prosta, solsta = task.getsolutionstatus(mosek.soltype.itg)
    else:
        prosta, solsta = task.getsolutionstatus(mosek.soltype.bas)
        
    x = zeros(n, float)
    if len(I) > 0:
        task.getsolutionslice(mosek.soltype.itg, mosek.solitem.xx, 0, n, x) 
    else:
        task.getsolutionslice(mosek.soltype.bas, mosek.solitem.xx, 0, n, x) 
    x = matrix(x)

    if (solsta is mosek.solsta.unknown):
        return (solsta, None)
    else:
        return (solsta, x)