# Affine scaling

### From Glossary

A variant of Karmarkar's algorithm for
the linear program in standard form:
, where
has full row rank.

Here are the basic steps (of the primal method).

- Given , let .
- Estimate dual: and .
- Move: , where is a parameter in .

Multiplication of and by
is a scaling operation, using current levels as
scale values -i.e. multiplies column
of by , and
multiplies by . Note that the
reduced cost
is in the null space of -i.e. ,
so is a feasible direction.
This *affine scaling direction* is used in its own right in
interior methods. It minimizes the
duality gap subject to
and in an ellipsoid (replacing
).