# qld

linear quadratic programming solver

## Calling Sequence

                [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 i.e md=2)
    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
- optim

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`