# The Average Number of Pivot Steps

• Karl Heinz Borgwardt
Part of the Algorithms and Combinatorics book series (AC, volume 1)

## Abstract

Let us define what we mean by “average number of pivot steps” precisely. For this purpose consider the matrix of the input data
$${\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\frown}}}{A}}: = \left[ \begin{gathered} a_1^T \hfill \\ . \hfill \\ a_m^T \hfill \\ {v^T} \hfill \\ \end{gathered} \right] \in {R^{{\left( {m + 1} \right)n}}}$$
(2.1.1)
of a problem
$$\begin{gathered} Maximize\quad {v^T}x \hfill \\ subject\,to\quad a_1^Tx \leqslant 1,...,a_m^Tx \leqslant 1 \hfill \\ where\quad v,x,{a_1},...,{a_m} \in {R^n},m \geqslant n \hfill \\ \end{gathered}$$
(2.1.2)
We regard this matrix Â as a random variable in a probability space
$$\left( {{R^{{\left( {m + 1} \right)n}}},A,P} \right)$$
(2.1.3)
, Where A is the σ-algebra of the Lebesgue-measurable sets of R(m+1)n, and where P is a probability measure defined on A.

## Keywords

Random Vector Probability Space Intersection Plane Integral Formula Rotational Symmetry
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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