Gruber P. Convex and Discrete Geometry

336 Convex Polytopes
economics, von Neumann [768], Koopmans [608] and, in particular, Dantzig [237],
who specified the simplex algorithm. While the Klee-Minty [597] cube shows that
the common version of the simplex algorithm is not polynomial, Borgwardt [151]
proved that, on the average, it is polynomial. See also a more recent result of
Spielman and Teng [950]. In practice the simplex algorithm works very effectively.
Khachiyan [580] indicated a proof that the ellipsoid algorithm of Shor [933] and
Yudin and Nemirovskiĭ [1033] is polynomial. The ellipsoid algorithm did not replace
the simplex algorithm in practice and was never stably implemented. There is no
running code available, not even for small test problems. A different polynomial
algorithm is that of Karmarkar [566]. While it was not put to practical use, it started
the development of a very efficient, huge class of interior-point methods for linear
programming, some of which are both polynomial and efficient in practice. For many
types of problems the interior point methods are better than the simplex algorithm
and are widely used in practice.
Integer linear optimization (or integer linear programming) is linear optimization
with the variables restricted to the integers. A standard problem is the following,
sup � c · x : x ∈ Z d , Ax ≤ b � ,
where A is a rational m × d matrix and b ∈ E m , c ∈ E d are rational vectors. An
integer linear optimization problem may be interpreted as the search of optimum
points of the integer lattice Z d contained in the convex polyhedron {x : Ax ≤ b}.
Integer linear optimization is essentially different from linear optimization: there is
no duality, no polynomial algorithm is known and, presumably, does not exist.
In this section we first consider a classical duality result, then describe the simplex
algorithm in geometric terms and explain how to find feasible solutions with the
ellipsoid algorithm. In integer optimization we consider so-called totally dual integral
systems for which integer optimization is easier than in the general case. Their
relations to lattice polyhedra are touched and Hilbert bases are used to characterize
totally dual integral systems. We have borrowed freely from Schrijver’s book [915].
To simplify the presentation, we often consider row vectors as being contained
in E d or E m . Using the matrix product we write cx instead of c T · x where c is a row
vector and x a column vector in E d .
There exists a rich literature on linear and integer optimization, including
the classic of Dantzig [238] and the monographs of Schrijver [915], Borgwardt [152],
Grötschel, Lovász and Schrijver [409], Berkovitz [99], Dantzig and Thapa [239] and
Schrijver [916]. See also the surveys of Shamir [928], Burkard [179], Gritzmann and
Klee [396] and Bartels [75]. For information on the history of optimization, see [508]
and [915].
20.1 Preliminaries and Duality
As remarked before, any of the standard linear optimization problems can be reduced
in polynomial time to any of the others. It turns out that certain pairs of linear optimization
problems, one a maximization, the other one a minimization problem, are