# 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 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,iact,iter,f]=qpsolve(Q,p,C,b,ci,cs,me)
    //Only linear constraints (1 to 4) are active

## See Also

- optim
- qp_solve
- qld

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