Linear Programming, the Simplex Algorithm and Simple Polytopes

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Linear Programming, the Simplex Algorithm and Simple Polytopes (1997)

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by Gil Kalai

Venue:	Math. Programming
Citations:	31 - 1 self

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@ARTICLE{Kalai97linearprogramming,,
    author = {Gil Kalai},
    title = {Linear Programming, the Simplex Algorithm and Simple Polytopes},
    journal = {Math. Programming},
    year = {1997},
    volume = {79},
    pages = {217--233}
}

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Abstract

In the first part of the paper we survey some far-reaching applications of the basic facts of linear programming to the combinatorial theory of simple polytopes. In the second part we discuss some recent developments concerning the simplex algorithm. We describe subexponential randomized pivot rules and upper bounds on the diameter of graphs of polytopes. 1 Introduction: A convex polyhedron is the intersection P of a finite number of closed halfspaces in R d . P is a d-dimensional polyhedron (briefly, a d-polyhedron) if the points in P affinely span R d . A convex d-dimensional polytope (briefly, a d-polytope) is a bounded convex d-polyhedron. Alternatively, a convex d-polytope is the convex hull of a finite set of points which affinely span R d . A (nontrivial) face F of a d-polyhedron P is the intersection of P with a supporting hyperplane. F itself is a polyhedron of some lower dimension. If the dimension of F is k we call F a k-face of P . The empty set and P itself are...

Keyphrases

simple polytopes linear programming simplex algorithm second part bounded convex d-polyhedron first part convex polyhedron d-dimensional polyhedron far-reaching application recent development combinatorial theory basic fact finite number empty set finite set upper bound convex d-polytope convex d-dimensional polytope closed halfspaces pivot rule convex hull

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