Scilab Function
Last update : 24/06/2005

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])`

### Parameters

- `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 précision.
- `x`: optimal solution found.
- `lagr`: vector of Lagrange multipliers. If lower and upper-bounds `ci,cs` are provided, `lagr` has `n + me + md` components and `lagr(1:n)` is the Lagrange vector associated with the bound constraints and `lagr (n+1 : n + me + md)` is the Lagrange vector associated with the linear 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.
- `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

Minimize `0.5*x'*Q*x + p'*x`

under the constraints

     C(j,:) x = b(j),  j=1,...,me
     C(j,:) x <= b(j), j=me+1,...,me+md
     ci <= x <= cs

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

### 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

`quapro`, `optim`,

### Authors

  K.Schittkowski, University of Bayreuth, Germany
  A.L. Tits and J.L. Zhou , University of Maryland

### Used Function

ql0001.f in routines/optim directory