.TH linpro 1 "April 1993" "Scilab Group" "Scilab Function"
.so man1/sci.an 
.SH NAME
linpro - linear programming solver
.SH CALLING SEQUENCE
.nf
[x,lagr,f]=linpro(p,C,b [,x0])
[x,lagr,f]=linpro(p,C,b,ci,cs [,x0])
[x,lagr,f]=linpro(p,C,b,ci,cs,mi [,x0])
[x,lagr,f]=linpro(p,C,b,ci,cs,mi,x0 [,imp])
.fi
.SH PARAMETERS
.TP 10
p 
: real (column) vector (dimension \fV n\fR)
.TP
C 
: real matrix (dimension \fV (mi + md) x n\fR)
(If no constraints are given, you can set \fVC = []\fR)
.TP
b 
: RHS vector (dimension \fV1 x (mi + md)\fR)
.TP
ci 
: (column) vector of lower-bounds (dimension \fV n\fR).
If there are no lower bound constraints, put \fVci = []\fR.
If some components of \fVx\fR are bounded from below,
set the other (unconstrained) values of \fVci\fR to a very 
large negative  number (e.g. \fVci(j) = -(% eps)^(-1)\fR.
.TP
cs 
: (column) vector of upper-bounds. (Same remarks as above).
.TP
mi 
: number of equality constraints (i.e. \fVC(1:mi,:)*x = b(1:mi)\fR)
.TP
x0 
: either an initial guess for \fVx\fR 
  or one of the character strings \fV'v'\fR or \fV'g'\fR.
If \fVx0='v'\fR the calculated initial feasible point is a vertex.
If \fVx0='g'\fR the calculated initial feasible point is arbitrary.
.TP
imp 
: verbose option (optional parameter)  (Try \fVimp=7,8,...\fR)
.TP
x 
: optimal solution found.
.TP
f 
: optimal value of the cost function (i.e. \fVf=p'*x\fR).
.TP
lagr 
: vector of Lagrange multipliers. 
If lower and upper-bounds \fVci,cs\fR are provided, \fVlagr\fR has 
\fVn + mi + md\fR components and \fVlagr(1:n)\fR is the Lagrange 
vector associated with the bound constraints and 
\fVlagr (n+1 : n + mi + md)\fR is the Lagrange vector associated 
with the linear constraints.
(If an upper-bound (resp. lower-bound) constraint \fVi\fR is active 
\fVlagr(i)\fR is > 0 (resp. <0).
If no bounds are provided, \fVlagr\fR has only \fVmi + md\fR components.
.SH DESCRIPTION
.Vb [x,lagr,f]=linpro(p,C,b [,x0])
.LP
Minimize \fVp'*x\fR
.PP
under the constraints 
.nf
C*x <= b
.fi
.LP
.Vb [x,lagr,f]=linpro(p,C,b,ci,cs [,x0])
.LP
Minimize \fVp'*x\fR
.PP
under the constraints 
.nf
C*x <= b        ci <= x <= cs
.fi
.Vb [x,lagr,f]=linpro(p,C,b,ci,cs,mi [,x0])
.LP
Minimize \fVp'*x\fR
.PP
under the constraints 
.nf
 C(j,:) x = b(j),  j=1,...,mi
 C(j,:) x <= b(j), j=mi+1,...,mi+md
 ci <= x <= cs
.fi
.LP 
If no initial point is given the
program computes a feasible initial point
which is a vertex of the region of feasible points if
\fVx0='v'\fR.
.LP
If \fVx0='g'\fR, 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 a few constraints in
the problem or when there are large bounds
on the variables that are security bounds and
very likely not active at the optimal solution.
.SH EXAMPLE
.nf
//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 p'*x with
p=[1;2;3;4;5;6]
//No initial point is given: x0='v';
C=[C1;C2]; b=[b1;b2] ; mi=3; x0='v';
[x,lagr,f]=linpro(p,C,b,ci,cs,mi,x0)
// Lower bound constraints 3 and 4 are active and upper bound
// constraint 5 is active --> lagr(3:4) < 0 and lagr(5) > 0.
// Linear (equality) constraints 1 to 3 are active --> lagr(7:9) <> 0
.fi
.SH SEE ALSO
quapro
.SH AUTHOR
E. Casas, C. Pola Mendez