2,018 research outputs found
A potential theoretic minimax problem on the torus
We investigate an extension of an equilibrium-type result, conjectured by
Ambrus, Ball and Erd\'elyi, and proved recently by Hardin, Kendall and Saff.
These results were formulated on the torus, hence we also work on the torus,
but one of the main motivations for our extension comes from an analogous setup
on the unit interval, investigated earlier by Fenton.
Basically, the problem is a minimax one, i.e. to minimize the maximum of a
function , defined as the sum of arbitrary translates of certain fixed
"kernel functions", minimization understood with respect to the translates. If
these kernels are assumed to be concave, having certain singularities or cusps
at zero, then translates by will have singularities at (while in
between these nodes the sum function still behaves realtively regularly). So
one can consider the maxima on each subintervals between the nodes ,
and look for the minimization of .
Here also a dual question of maximization of arises. This type
of minimax problems were treated under some additional assumptions on the
kernels. Also the problem is normalized so that .
In particular, Hardin, Kendall and Saff assumed that we have one single
kernel on the torus or circle, and . Fenton considered situations on the interval with
two fixed kernels and , also satisfying additional assumptions, and . Here we consider the situation (on the circle)
when \emph{all the kernel functions can be different}, and . Also an emphasis is put
on relaxing all other technical assumptions and give alternative, rather
minimal variants of the set of conditions on the kernel
Tight Inefficiency Bounds for Perception-Parameterized Affine Congestion Games
Congestion games constitute an important class of non-cooperative games which
was introduced by Rosenthal in 1973. In recent years, several extensions of
these games were proposed to incorporate aspects that are not captured by the
standard model. Examples of such extensions include the incorporation of risk
sensitive players, the modeling of altruistic player behavior and the
imposition of taxes on the resources. These extensions were studied intensively
with the goal to obtain a precise understanding of the inefficiency of
equilibria of these games. In this paper, we introduce a new model of
congestion games that captures these extensions (and additional ones) in a
unifying way. The key idea here is to parameterize both the perceived cost of
each player and the social cost function of the system designer. Intuitively,
each player perceives the load induced by the other players by an extent of
{\rho}, while the system designer estimates that each player perceives the load
of all others by an extent of {\sigma}. The above mentioned extensions reduce
to special cases of our model by choosing the parameters {\rho} and {\sigma}
accordingly. As in most related works, we concentrate on congestion games with
affine latency functions here. Despite the fact that we deal with a more
general class of congestion games, we manage to derive tight bounds on the
price of anarchy and the price of stability for a large range of pa- rameters.
Our bounds provide a complete picture of the inefficiency of equilibria for
these perception-parameterized congestion games. As a result, we obtain tight
bounds on the price of anarchy and the price of stability for the above
mentioned extensions. Our results also reveal how one should "design" the cost
functions of the players in order to reduce the price of anar- chy
On the job rotation problem
The job rotation problem (JRP) is the following: Given an matrix over \Re \cup \{\ -\infty\ \}\ and , find a principal submatrix of whose optimal assignment problem value is maximum. No polynomial algorithm is known for solving this problem if is an input variable. We analyse JRP and present polynomial solution methods for a number of special cases
An inverse dynamics approach to trajectory optimization for an aerospace plane
An inverse dynamics approach for trajectory optimization is proposed. This technique can be useful in many difficult trajectory optimization and control problems. The application of the approach is exemplified by ascent trajectory optimization for an aerospace plane. Both minimum-fuel and minimax types of performance indices are considered. When rocket augmentation is available for ascent, it is shown that accurate orbital insertion can be achieved through the inverse control of the rocket in the presence of disturbances
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