4,271 research outputs found
Equilibria, Fixed Points, and Complexity Classes
Many models from a variety of areas involve the computation of an equilibrium
or fixed point of some kind. Examples include Nash equilibria in games; market
equilibria; computing optimal strategies and the values of competitive games
(stochastic and other games); stable configurations of neural networks;
analysing basic stochastic models for evolution like branching processes and
for language like stochastic context-free grammars; and models that incorporate
the basic primitives of probability and recursion like recursive Markov chains.
It is not known whether these problems can be solved in polynomial time. There
are certain common computational principles underlying different types of
equilibria, which are captured by the complexity classes PLS, PPAD, and FIXP.
Representative complete problems for these classes are respectively, pure Nash
equilibria in games where they are guaranteed to exist, (mixed) Nash equilibria
in 2-player normal form games, and (mixed) Nash equilibria in normal form games
with 3 (or more) players. This paper reviews the underlying computational
principles and the corresponding classes
Evolutionary algorithm-based analysis of gravitational microlensing lightcurves
A new algorithm developed to perform autonomous fitting of gravitational
microlensing lightcurves is presented. The new algorithm is conceptually
simple, versatile and robust, and parallelises trivially; it combines features
of extant evolutionary algorithms with some novel ones, and fares well on the
problem of fitting binary-lens microlensing lightcurves, as well as on a number
of other difficult optimisation problems. Success rates in excess of 90% are
achieved when fitting synthetic though noisy binary-lens lightcurves, allowing
no more than 20 minutes per fit on a desktop computer; this success rate is
shown to compare very favourably with that of both a conventional (iterated
simplex) algorithm, and a more state-of-the-art, artificial neural
network-based approach. As such, this work provides proof of concept for the
use of an evolutionary algorithm as the basis for real-time, autonomous
modelling of microlensing events. Further work is required to investigate how
the algorithm will fare when faced with more complex and realistic microlensing
modelling problems; it is, however, argued here that the use of parallel
computing platforms, such as inexpensive graphics processing units, should
allow fitting times to be constrained to under an hour, even when dealing with
complicated microlensing models. In any event, it is hoped that this work might
stimulate some interest in evolutionary algorithms, and that the algorithm
described here might prove useful for solving microlensing and/or more general
model-fitting problems.Comment: 14 pages, 3 figures; accepted for publication in MNRA
Convergence of the restricted Nelder-Mead algorithm in two dimensions
The Nelder-Mead algorithm, a longstanding direct search method for
unconstrained optimization published in 1965, is designed to minimize a
scalar-valued function f of n real variables using only function values,
without any derivative information. Each Nelder-Mead iteration is associated
with a nondegenerate simplex defined by n+1 vertices and their function values;
a typical iteration produces a new simplex by replacing the worst vertex by a
new point. Despite the method's widespread use, theoretical results have been
limited: for strictly convex objective functions of one variable with bounded
level sets, the algorithm always converges to the minimizer; for such functions
of two variables, the diameter of the simplex converges to zero, but examples
constructed by McKinnon show that the algorithm may converge to a nonminimizing
point.
This paper considers the restricted Nelder-Mead algorithm, a variant that
does not allow expansion steps. In two dimensions we show that, for any
nondegenerate starting simplex and any twice-continuously differentiable
function with positive definite Hessian and bounded level sets, the algorithm
always converges to the minimizer. The proof is based on treating the method as
a discrete dynamical system, and relies on several techniques that are
non-standard in convergence proofs for unconstrained optimization.Comment: 27 page
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Adaptive grid semidefinite programming for finding optimal designs
We find optimal designs for linear models using anovel algorithm that iteratively combines a semidefinite programming(SDP) approach with adaptive grid techniques.The proposed algorithm is also adapted to find locally optimaldesigns for nonlinear models. The search space is firstdiscretized, and SDP is applied to find the optimal designbased on the initial grid. The points in the next grid set arepoints that maximize the dispersion function of the SDPgeneratedoptimal design using nonlinear programming. Theprocedure is repeated until a user-specified stopping rule isreached. The proposed algorithm is broadly applicable, andwe demonstrate its flexibility using (i) models with one ormore variables and (ii) differentiable design criteria, suchas A-, D-optimality, and non-differentiable criterion like Eoptimality,including the mathematically more challengingcasewhen theminimum eigenvalue of the informationmatrixof the optimal design has geometric multiplicity larger than 1. Our algorithm is computationally efficient because it isbased on mathematical programming tools and so optimalityis assured at each stage; it also exploits the convexity of theproblems whenever possible. Using several linear and nonlinearmodelswith one or more factors, we showthe proposedalgorithm can efficiently find optimal designs
Jointly Optimal Routing and Caching for Arbitrary Network Topologies
We study a problem of fundamental importance to ICNs, namely, minimizing
routing costs by jointly optimizing caching and routing decisions over an
arbitrary network topology. We consider both source routing and hop-by-hop
routing settings. The respective offline problems are NP-hard. Nevertheless, we
show that there exist polynomial time approximation algorithms producing
solutions within a constant approximation from the optimal. We also produce
distributed, adaptive algorithms with the same approximation guarantees. We
simulate our adaptive algorithms over a broad array of different topologies.
Our algorithms reduce routing costs by several orders of magnitude compared to
prior art, including algorithms optimizing caching under fixed routing.Comment: This is the extended version of the paper "Jointly Optimal Routing
and Caching for Arbitrary Network Topologies", appearing in the 4th ACM
Conference on Information-Centric Networking (ICN 2017), Berlin, Sep. 26-28,
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