qpsolve
=======

linear quadratic programming solver



Calling Sequence
~~~~~~~~~~~~~~~~


::

    [x [,iact [,iter [,f]]]]=qpsolve(Q,p,C,b,ci,cs,me)




Arguments
~~~~~~~~~

:Q real positive definite symmetric matrix (dimension `n x n` ).
: :p real (column) vector (dimension `n`)
: :C real matrix (dimension `(me + md) x n`). This matrix may be dense
  or sparse.
: :b RHS column vector (dimension `m=(me + md)` )
: :ci column vector of lower-bounds (dimension `n`). If there are no
  lower bound constraints, put `ci = []`. If some components of `x` are
  bounded from below, set the other (unconstrained) values of `ci` to a
  very large negative number (e.g. `ci(j) = -number_properties('huge')`
  .
: :cs column vector of upper-bounds. (Same remarks as above).
: :me number of equality constraints (i.e. `C(1:me,:)*x = b(1:me)` )
: :x optimal solution found.
: :iact vector, indicator of active constraints. The first non zero
  entries give the index of the active constraints
: :iter . 2x1 vector, first component gives the number of "main"
  iterations, the second one says how many constraints were deleted
  after they became active.
:



Description
~~~~~~~~~~~

This function requires `Q` to be symmetric positive definite. If that
hypothesis is not satisfied, one may use the quapro function, which is
provided in the Scilab quapro toolbox.

The qpsolve solver is implemented as a Scilab script, which calls the
compiled qp_solve primitive. It is provided as a facility, in order to
be a direct replacement for the former quapro solver : indeed, the
qpsolve solver has been designed so that it provides the same
interface, that is, the same input/output arguments. But the x0 and
imp input arguments are available in quapro, but not in qpsolve.



Examples
~~~~~~~~


::

    //Find x in R^6 such that:
    //C1*x = b1 (3 equality constraints i.e me=3)
    C1= [1,-1,1,0,3,1;
        -1,0,-3,-4,5,6;
         2,5,3,0,1,0];
    b1=[1;2;3];
    
    //C2*x <= b2 (2 inequality constraints)
    C2=[0,1,0,1,2,-1;
        -1,0,2,1,1,0];
    b2=[-1;2.5];
    
    //with  x between ci and cs:
    ci=[-1000;-10000;0;-1000;-1000;-1000];
    cs=[10000;100;1.5;100;100;1000];
    
    //and minimize 0.5*x'*Q*x + p'*x with
    p=[1;2;3;4;5;6]; Q=`eye`_(6,6);
    
    //No initial point is given;
    C=[C1;C2];
    b=[b1;b2];
    me=3;
    [x,iact,iter,f]=qpsolve(Q,p,C,b,ci,cs,me)
    //Only linear constraints (1 to 4) are active




See Also
~~~~~~~~


+ `optim`_ non-linear optimization routine
+ `qp_solve`_ linear quadratic programming solver builtin
+ `qld`_ linear quadratic programming solver


The contributed toolbox "quapro" may also be of interest, in
particular for singular `Q`.



Memory requirements
~~~~~~~~~~~~~~~~~~~

Let r be


::

    r=`min`_(m,n)


Then the memory required by qpsolve during the computations is


::

    2*n+r*(r+5)/2 + 2*m +1




References
~~~~~~~~~~


+ Goldfarb, D. and Idnani, A. (1982). "Dual and Primal-Dual Methods
  for Solving Strictly Convex Quadratic Programs", in J.P. Hennart
  (ed.), Numerical Analysis, Proceedings, Cocoyoc, Mexico 1981, Vol. 909
  of Lecture Notes in Mathematics, Springer-Verlag, Berlin, pp. 226-239.
+ Goldfarb, D. and Idnani, A. (1983). "A numerically stable dual
  method for solving strictly convex quadratic programs", Mathematical
  Programming 27: 1-33.
+ QuadProg (Quadratic Programming Routines), Berwin A
  Turlach,`http://www.maths.uwa.edu.au/~berwin/software/quadprog.html`_




Used Functions
~~~~~~~~~~~~~~

qpgen1.f (also named QP.solve.f) developed by Berwin A. Turlach
according to the Goldfarb/Idnani algorithm

.. _qp_solve: qp_solve.html
.. _qld: qld.html
.. _optim: optim.html
.. _http://www.maths.uwa.edu.au/~berwin/software/quadprog.html: http://www.maths.uwa.edu.au/~berwin/software/quadprog.html