791 research outputs found
Compact Routing on Internet-Like Graphs
The Thorup-Zwick (TZ) routing scheme is the first generic stretch-3 routing
scheme delivering a nearly optimal local memory upper bound. Using both direct
analysis and simulation, we calculate the stretch distribution of this routing
scheme on random graphs with power-law node degree distributions, . We find that the average stretch is very low and virtually
independent of . In particular, for the Internet interdomain graph,
, the average stretch is around 1.1, with up to 70% of paths
being shortest. As the network grows, the average stretch slowly decreases. The
routing table is very small, too. It is well below its upper bounds, and its
size is around 50 records for -node networks. Furthermore, we find that
both the average shortest path length (i.e. distance) and width of
the distance distribution observed in the real Internet inter-AS graph
have values that are very close to the minimums of the average stretch in the
- and -directions. This leads us to the discovery of a unique
critical quasi-stationary point of the average TZ stretch as a function of
and . The Internet distance distribution is located in a
close neighborhood of this point. This observation suggests the analytical
structure of the average stretch function may be an indirect indicator of some
hidden optimization criteria influencing the Internet's interdomain topology
evolution.Comment: 29 pages, 16 figure
Computing local minimizers in polynomial optimization under genericity conditions
In this paper, we aim at computing all local minimizers of a polynomial
optimization problem under genericity conditions. By using a technique in
computer algebra, we provide a univariate representation for the set of local
minimizers. In particular, for an unconstrained problem, the coordinates of all
local minimizers can be represented by several univariate polynomial equalities
and one univariate polynomial matrix inequality. We also develop the technique
for constrained problems having equality constraints. Based on the above
technique, we design algorithms to enumerate the local minimizers.
At the end of the paper, we provide some experimental examples.Comment: 24 pages, submitte
Efficient approaches for escaping higher order saddle points in non-convex optimization
Local search heuristics for non-convex optimizations are popular in applied
machine learning. However, in general it is hard to guarantee that such
algorithms even converge to a local minimum, due to the existence of
complicated saddle point structures in high dimensions. Many functions have
degenerate saddle points such that the first and second order derivatives
cannot distinguish them with local optima. In this paper we use higher order
derivatives to escape these saddle points: we design the first efficient
algorithm guaranteed to converge to a third order local optimum (while existing
techniques are at most second order). We also show that it is NP-hard to extend
this further to finding fourth order local optima
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