# qpsolve

linear quadratic programming solver

### Syntax

[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 iprint 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 — 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

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