Stochastic Programming - Index of

262 STOCHASTIC PROGRAMMING
procedure framebylp(W :(m × n) matrix);
begin
n1 :=n;
q := 0;
for i := n1 downto 1 do begin
LP(W \ W i ,W i ,q,y,feasible);
if feasible then begin
W i := W n ;
n := n − 1;
end;
end;
end;
Figure 1
Finding a frame.
5.1.1 Finding a Frame
Let us repeat the definition of pos W :
pos W = {t | t = Wy, y ≥ 0}.
In words, pos W is the set of all positive (nonnegative) linear combinations
of columns of the matrix W . A subset of the columns, determining a matrix
W ′ , is called a frame if pos W =posW ′ , and equality is not preserved if
any one column is removed from W ′ . So, by finding the frame of a given
matrix, we remove all columns that are not needed to describe the pointed
cone pos W . As an example, if we use a two-phase simplex method to solve a
linear programming problem, only the columns of W ′ are needed in phase 1.
If W is a matrix, and j is an index, let W \ W j be the matrix W with
column j removed.
A simple approach for finding a frame is outlined in Figure 1. To do that, we
need a procedure that solves LPs. It can be found in Figure 7 in Chapter 3. The
matrix W in procedure framebylp in Figure 1 is both input and output. On
entry, it contains the matrix for which we seek the frame; on exit, it contains
those columns that were in the frame. The number of columns, n, is changed
accordingly.
To summarize, the effect of the frame algorithm is that as many columns
as possible are removed from a matrix W without changing the pointed cone
spanned by the columns. We have earlier discussed generators of cones. In this
case we may say that the columns in W , after the application of procedure
framebylp, are generators of pos W . Let us now turn to the use of this
algorithm.