Klee–Minty cube

The Klee–Minty cube or Klee–Minty polytope (named after

visit all 8 corners in the worst case.

In particular, many optimization

Description

The Klee–Minty cube was originally specified with a parameterized system of linear inequalities, with the dimension as the parameter. The cube in two-dimensional space is a squashed square, and the "cube" in three-dimensional space is a squashed cube. Illustrations of the "cube" have appeared besides algebraic descriptions.[3] The Klee–Minty polytope is given by:^[4]

This has ${\displaystyle D}$ variables, ${\displaystyle D}$ constraints other than the ${\displaystyle D}$ non-negativity constraints, and ${\displaystyle 2^{D}}$ vertices, just as a ${\displaystyle D}$-dimensional hypercube does. If the objective function to be maximized is

and if the initial vertex for the simplex algorithm is the origin, then the algorithm as formulated by Dantzig visits all ${\displaystyle 2^{D}}$ vertices, finally reaching the optimal vertex ${\displaystyle (0,0,\dots ,5^{D})}$.

Computational complexity

The Klee–Minty cube has been used to analyze the performance of many algorithms, both in the worst case and on average. The

${\displaystyle D^{3}}$ operations, and so it is said to have polynomial time-complexity because its complexity is bounded by a ). Because exponential functions eventually grow much faster than polynomial functions, an exponential complexity implies that an algorithm has slow performance on large problems.

Worst case

In mathematical optimization, the Klee–Minty cube is an example that shows the

${\displaystyle D}$. Klee and Minty showed that Dantzig's simplex algorithm visits all corners of a (perturbed) cube in dimension ${\displaystyle D}$ in the worst case.^[5]

Modifications of the Klee–Minty construction showed similar exponential time complexity for other pivoting rules of simplex type, which maintain primal feasibility, such as Bland's rule.^[6] Another modification showed that the criss-cross algorithm, which does not maintain primal feasibility, also visits all the corners of a modified Klee–Minty cube.^[7] Like the simplex algorithm, the criss-cross algorithm visits all 8 corners of the three-dimensional cube in the worst case.

Path-following algorithms

Further modifications of the Klee–Minty cube have shown the poor performance of central-path–following algorithms for linear optimization, in that the central path comes arbitrarily close to each of the corners of a cube. This "vertex-stalking" performance is surprising because such path-following algorithms have polynomial-time complexity for linear optimization.^[8]

Average case

The Klee–Minty cube has also inspired research on

${\displaystyle O(D^{2})}$.^[9] Standard variants of the simplex algorithm take on average ${\displaystyle D}$ steps for a cube.^[a] However according to Fukuda & Namiki (1994), when it is initialized at a random corner of the cube, the criss-cross algorithm visits only ${\displaystyle D}$ additional corners.^[11] Both the simplex algorithm and the criss-cross algorithm visit exactly 3 additional corners of the three-dimensional cube on average.

See also

• Projective algorithm of Karmarkar
• Ellipsoidal algorithm of Khachiyan

References

Notes

Citations

Bibliography

External links

Algebraic description with illustration

The first two links have both an algebraic construction and a picture of a three-dimensional Klee–Minty cube:

• Deza, Antoine; Nematollahi, Eissa; Terlaky, Tamás (May 2008). "How good are interior point methods? Klee–Minty cubes tighten iteration-complexity bounds" (PDF). Mathematical Programming. 113 (1): 1–14.
  S2CID 156325
  .
• Gartner, B.;
  S2CID 8090478
  . IEEE mis-spells the name "Gärnter" as "Gartner" (sic.).

Pictures with no linear system

• Articles of E. Nematollahi, which discuss the Klee-Minty cube with illustrations.
• A picture of a Klee-Minty cube showing a simplex-algorithm path (automatic translation of German) by
  Günter Ziegler
  . The picture in the second half of the page.