6 research outputs found
A note on the Minimum Norm Point algorithm
We present a provably more efficient implementation of the Minimum Norm Point
Algorithm conceived by Fujishige than the one presented in \cite{FUJI06}. The
algorithm solves the minimization problem for a class of functions known as
submodular. Many important functions, such as minimum cut in the graph, have
the so called submodular property \cite{FUJI82}. It is known that the problem
can also be efficiently solved in strongly polynomial time \cite{IWAT01},
however known theoretical bounds are far from being practical. We present an
improved implementation of the algorithm, for which unfortunately no worst case
bounds are know, but which performs very well in practice. With the
modifications presented, the algorithm performs an order of magnitude faster
for certain submodular functions
Conditions of discreteness of the spectrum for Schr\"odinger operator and some optimization problems for capacity and measures
For the the Schr\"odinger operator , acting in the
space L_2(\R^d)\,(d\ge 3), with V(x)\ge 0 and V(\cdot)\in L_{1,loc}(\R^d), we
obtain some constructive conditions for discreteness of its spectrum. Basing on
the Mazya-Shubin criterion for discreteness of the spectrum of and using
the isocapacity inequality and the concept of base polyhedron for the harmonic
capacity, we have estimated from below the cost functional of an optimization
problem, involved in this criterion, replacing a submodular constrain (in terms
of the harmonic capacity) by a weaker but additive constrain (in terms of a
measure). By this way we obtain an optimization problem, which can be
considered as an infinite-dimensional analogue of the optimal covering problem.
We have solved this problem for the case of a non-atomic measure. This approach
enables us to obtain for the operator H some sufficient conditions for
discreteness of its spectrum in terms of non-increasing rearrangements, with
respect to measures from the base polyhedron, for some functions connected with
the potential V(x). We construct some counterexamples, which permit to compare
our results between themselves and with results of other authors.Comment: 27 page