AMSC 607 / CMSC 764 Fall 2006
HMWK 3: Due November 2
Show all work. All work must be your own (i.e., no group efforts are allowed).
If you use a reference book, cite it, or you will lose credit!
In this homework you will program a feasible direction method and an
interior point method for solving linear programming problems
min
xcTx
Ax =b
x≥0
where x∈ Rnand b∈ Rmwith m < n. Assume a constraint qualification.
The feasible direction method. For this method, see the feasible direction
notes, pp.6-8. This is the simplex method except that we allow iterates that
are not vertices of the feasible set.
Write a Matlab function xopt = lpfeasdir(A,b,c,x). The parameters
to your feasible direction algorithm are A,b,c, and an initial feasible point
x.
•Use qrupdate (instead of the Band Nmethod in the notes) to update
a factorization of ˆ
AT, where the rows of the basis matrix ˆ
Aare the rows
of Aand the rows of the identity matrix corresponding to the currently
active inequality constraints x≥0.
•At each iteration, ˆ
ATgains one column, and it may also lose one: if
there is no feasible downhill direction, remove the column correspond-
ing to the most negative (estimated) Lagrange multiplier.
•The next point is x+αp, where pis determined from solving the system
involving a column of the identity matrix, and αdefines the longest step
that is possible without violating any of the constraints x≥0. The
constraint that we hit becomes the added one.
•Stop when there is no feasible downhill direction.
The IPM. For this method, see the LP-IPM notes, pp. 9-10. Write a Matlab
function xopt = lpipm(A,b,c,x,y,z,eta). The parameters to your IPM
are A,b,c, a convergence tolerance η, and an initial feasible point x, y, z.
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