1,253,059 research outputs found
On the Curvature of the Central Path of Linear Programming Theory
We prove a linear bound on the average total curvature of the central path of
linear programming theory in terms on the number of independent variables of
the primal problem, and independent on the number of constraints.Comment: 24 pages. This is a fully revised version, and the last section of
the paper was rewritten, for clarit
Building a path-integral calculus: a covariant discretization approach
Path integrals are a central tool when it comes to describing quantum or
thermal fluctuations of particles or fields. Their success dates back to
Feynman who showed how to use them within the framework of quantum mechanics.
Since then, path integrals have pervaded all areas of physics where fluctuation
effects, quantum and/or thermal, are of paramount importance. Their appeal is
based on the fact that one converts a problem formulated in terms of operators
into one of sampling classical paths with a given weight. Path integrals are
the mirror image of our conventional Riemann integrals, with functions
replacing the real numbers one usually sums over. However, unlike conventional
integrals, path integration suffers a serious drawback: in general, one cannot
make non-linear changes of variables without committing an error of some sort.
Thus, no path-integral based calculus is possible. Here we identify which are
the deep mathematical reasons causing this important caveat, and we come up
with cures for systems described by one degree of freedom. Our main result is a
construction of path integration free of this longstanding problem, through a
direct time-discretization procedure.Comment: 22 pages, 2 figures, 1 table. Typos correcte
Weak disorder asymptotics in the stochastic mean-field model of distance
In the recent past, there has been a concerted effort to develop mathematical
models for real-world networks and to analyze various dynamics on these models.
One particular problem of significant importance is to understand the effect of
random edge lengths or costs on the geometry and flow transporting properties
of the network. Two different regimes are of great interest, the weak disorder
regime where optimality of a path is determined by the sum of edge weights on
the path and the strong disorder regime where optimality of a path is
determined by the maximal edge weight on the path. In the context of the
stochastic mean-field model of distance, we provide the first mathematically
tractable model of weak disorder and show that no transition occurs at finite
temperature. Indeed, we show that for every finite temperature, the number of
edges on the minimal weight path (i.e., the hopcount) is and
satisfies a central limit theorem with asymptotic means and variances of order
, with limiting constants expressible in terms of the
Malthusian rate of growth and the mean of the stable-age distribution of an
associated continuous-time branching process. More precisely, we take
independent and identically distributed edge weights with distribution
for some parameter , where is an exponential random variable with mean
1. Then the asymptotic mean and variance of the central limit theorem for the
hopcount are and , respectively. We also find limiting
distributional asymptotics for the value of the minimal weight path in terms of
extreme value distributions and martingale limits of branching processes.Comment: Published in at http://dx.doi.org/10.1214/10-AAP753 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Analytic central path, sensitivity analysis and parametric linear programming
In this paper we consider properties of the central path and the analytic center of the optimal face in the context of parametric
linear programming. We first show that if the right-hand side vector of a standard linear program is perturbed, then the analytic center of the optimal face is one-side differentiable with respect to the perturbation parameter. In that case we also show that the whole analytic central path shifts in a uniform fashion. When the
objective vector is perturbed, we show that the last part of the analytic central path
is tangent to a central path defined on
the optimal face of the original problem
Inflation targeting: why it works and how to make it work better
Inflation targeting has worked so well because it leads policymakers to debate, decide on, and communicate the inflation objective. In practice, this process has led the public to believe that the central bank has a long-term inflation objective. Inflation targeting has been successful, then, because the central bank decides on an objective and announces it, not because of a change in its day-to-day behavior in money markets or the way it reacts to news about unemployment or real GDP. By deciding on an inflation rate and announcing it, the central bank is providing information the public needs to concentrate expectations on a common trend. The central bank gains control indirectly by creating information that makes it more likely that people will price things in a way that is consistent with the central bank's goal. The way to improve inflation targeting is to be more explicit about the average inflation rate expected over all relevant horizons. Building a target path for the price level, growing at the desired inflation rate, is the best way to institutionalize a low-inflation environment. In a wide variety of economic models, a price-path target mitigates the zero lower bound problem, eliminates worries about deflation, and improves the central bank's ability to stabilize the real economy.Inflation (Finance) ; Monetary policy
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