qld
===

linear quadratic programming solver



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


::

    [x, lagr] = qld(Q, p, C, b, ci, cs, me [,tol])
    [x, lagr, info] = qld(Q, p, C, b, ci, cs, me [,tol])




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` )
: :b RHS column vector (dimension `(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)` )
: :tol Floatting point number, required precision.
: :x optimal solution found.
: :lagr vector of Lagrange multipliers. If lower and upper-bounds `ci,
  cs` are provided, `lagr` has `me + md + 2* n` components. The
  components `lagr(1:me + md)` are associated with the linear
  constraints and `lagr (me + md + 1 : 2 * n)` are associated with the
  lower and upper bounds constraints. If an upper-bound (resp. lower-
  bound) constraint `i` is active `lagr(i)` is > 0 (resp. <0). If no
  bounds are provided, `lagr` has only `me + md` components. On
  successful termination, all values of `lagr` with respect to
  inequalities and bounds should be greater or equal to zero.
: :info integer, return the execution status instead of sending
  errors. info==1 : Too many iterations needed info==2 : Accuracy
  insufficient to statisfy convergence criterion info==5 : Length of
  working array is too short info==10: The constraints are inconsistent
:



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

This function requires `Q` to be positive definite, if it is not the
case, one may use the contributed toolbox " **quapro**".



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,lagr]=qld(Q,p,C,b,ci,cs,me)
    //Only linear constraints (1 to 4) are active (lagr(1:6)=0):




See Also
~~~~~~~~


+ `qpsolve`_ linear quadratic programming solver
+ `optim`_ non-linear optimization routine


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



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

`ql0001.f` in `modules/optimization/src/fortran/ql0001.f`

.. _optim: optim.html
.. _qpsolve: qpsolve.html