501 research outputs found
The simplex algorithm with the pivot rule of maximizing criterion improvement
AbstractWe extend a result of Klee and Minty by showing that the Simplex Algorithm with the pivot rule of maximizing criterion improvement is not a good algorithm in the sense of Edmonds. The method of proof extends to other similar pivot rules
A unified worst case for classical simplex and policy iteration pivot rules
We construct a family of Markov decision processes for which the policy
iteration algorithm needs an exponential number of improving switches with
Dantzig's rule, with Bland's rule, and with the Largest Increase pivot rule.
This immediately translates to a family of linear programs for which the
simplex algorithm needs an exponential number of pivot steps with the same
three pivot rules. Our results yield a unified construction that simultaneously
reproduces well-known lower bounds for these classical pivot rules, and we are
able to infer that any (deterministic or randomized) combination of them cannot
avoid an exponential worst-case behavior. Regarding the policy iteration
algorithm, pivot rules typically switch multiple edges simultaneously and our
lower bound for Dantzig's rule and the Largest Increase rule, which perform
only single switches, seem novel. Regarding the simplex algorithm, the
individual lower bounds were previously obtained separately via deformed
hypercube constructions. In contrast to previous bounds for the simplex
algorithm via Markov decision processes, our rigorous analysis is reasonably
concise
Tropicalizing the simplex algorithm
We develop a tropical analog of the simplex algorithm for linear programming.
In particular, we obtain a combinatorial algorithm to perform one tropical
pivoting step, including the computation of reduced costs, in O(n(m+n)) time,
where m is the number of constraints and n is the dimension.Comment: v1: 35 pages, 7 figures, 4 algorithms; v2: improved presentation, 39
pages, 9 figures, 4 algorithm
The Complexity of the Simplex Method
The simplex method is a well-studied and widely-used pivoting method for
solving linear programs. When Dantzig originally formulated the simplex method,
he gave a natural pivot rule that pivots into the basis a variable with the
most violated reduced cost. In their seminal work, Klee and Minty showed that
this pivot rule takes exponential time in the worst case. We prove two main
results on the simplex method. Firstly, we show that it is PSPACE-complete to
find the solution that is computed by the simplex method using Dantzig's pivot
rule. Secondly, we prove that deciding whether Dantzig's rule ever chooses a
specific variable to enter the basis is PSPACE-complete. We use the known
connection between Markov decision processes (MDPs) and linear programming, and
an equivalence between Dantzig's pivot rule and a natural variant of policy
iteration for average-reward MDPs. We construct MDPs and show
PSPACE-completeness results for single-switch policy iteration, which in turn
imply our main results for the simplex method
Learning to Pivot as a Smart Expert
Linear programming has been practically solved mainly by simplex and interior
point methods. Compared with the weakly polynomial complexity obtained by the
interior point methods, the existence of strongly polynomial bounds for the
length of the pivot path generated by the simplex methods remains a mystery. In
this paper, we propose two novel pivot experts that leverage both global and
local information of the linear programming instances for the primal simplex
method and show their excellent performance numerically. The experts can be
regarded as a benchmark to evaluate the performance of classical pivot rules,
although they are hard to directly implement. To tackle this challenge, we
employ a graph convolutional neural network model, trained via imitation
learning, to mimic the behavior of the pivot expert. Our pivot rule, learned
empirically, displays a significant advantage over conventional methods in
various linear programming problems, as demonstrated through a series of
rigorous experiments
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