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# qld

linear quadratic programming solver

### Syntax

[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
Floating 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 satisfy 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

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`

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