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quapro(1) Scilab Function quapro(1)
NAME
quapro - linear quadratic programming solver
CALLING SEQUENCE
[x,lagr,f]=quapro(Q,p,C,b [,x0])
[x,lagr,f]=quapro(Q,p,C,b,ci,cs [,x0])
[x,lagr,f]=quapro(Q,p,C,b,ci,cs,mi [,x0])
[x,lagr,f]=quapro(Q,p,C,b,ci,cs,mi,x0 [,imp])
PARAMETERS
Q : real symmetric matrix (dimension n x n).
p : real (column) vector (dimension n)
C : real matrix (dimension (mi + md) x n) (If no constraints are given,
you can set C = [])
b : RHS vector (dimension 1 x (mi + md))
ci : (column) vector of lower-bounds (dimension 1 x 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) = -(% eps)^(-1).
cs : (column) vector of upper-bounds. (Same remarks as above).
mi : number of equality constraints (i.e. C(1:mi,:)*x = b(1:mi))
x0 : either an initial guess for x
or one of the character strings 'v' or 'g'. If x0='v' the calcu-
lated initial feasible point is a vertex. If x0='g' the calculated
initial feasible point is arbitrary.
imp : verbose option (optional parameter) (Try imp=7,8,...).
x : optimal solution found.
f : optimal value of the cost function (i.e. f=p'*x).
lagr : vector of Lagrange multipliers. If lower and upper-bounds ci,cs are
provided, lagr has n + mi + md components and lagr(1:n) is the
Lagrange vector associated with the bound constraints and lagr (n+1 :
n + mi + md) is the Lagrange vector associated with the linear con-
straints. (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 mi + md components.
DESCRIPTION
[x,lagr,f]=quapro(Q,p,C,b [,x0])
Minimize 0.5*x'*Q*x + p'*x
under the constraint
C*x <= b
[x,lagr,f]=quapro(Q,p,C,b,ci,cs [,x0])
Minimize 0.5*x'*Q*x + p'*x
under the constraints
C*x <= b ci <= x <= cs
[x,lagr,f]=quapro(Q,p,C,b,ci,cs,mi [,x0])
Minimize 0.5*x'*Q*x + p'*x
under the constraints
C(j,:) x = b(j), j=1,...,mi
C(j,:) x <= b(j), j=mi+1,...,mi+md
ci <= x <= cs
If no initial point is given the program computes a feasible initial point
which is a vertex of the region of feasible points if x0='v'.
If x0='g', the program computes a feasible initial point which is not
necessarily a vertex. This mode is advisable when the quadratic form is
positive definite and there are few constraints in the problem or when
there are large bounds on the variables that are just security bounds and
very likely not active at the optimal solution.
Note that Q is not necessarily non-negative, i.e. Q may have negative
eigenvalues.
EXAMPLE
//Find x in R^6 such that:
//C1*x = b1 (3 equality constraints i.e mi=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] ; //
mi=3;
[x,lagr,f]=quapro(Q,p,C,b,ci,cs,mi)
//Only linear constraints (1 to 4) are active (lagr(1:6)=0):
[x,lagr,f]=quapro(Q,p,C,b,[],[],mi) //Same result as above
SEE ALSO
linpro
AUTHOR
E. Casas, C. Pola Mendez
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