321,803 research outputs found
Canonical Duality-Triality Theory: Bridge Between Nonconvex Analysis/Mechanics and Global Optimization in Complex Systems
Canonical duality-triality is a breakthrough methodological theory, which can
be used not only for modeling complex systems within a unified framework, but
also for solving a wide class of challenging problems from real-world
applications. This paper presents a brief review on this theory, its
philosophical origin, physics foundation, and mathematical statements in both
finite and infinite dimensional spaces, with emphasizing on its role for
bridging the gap between nonconvex analysis/mechanics and global optimization.
Special attentions are paid on unified understanding the fundamental
difficulties in large deformation mechanics, bifurcation/chaos in nonlinear
science, and the NP-hard problems in global optimization, as well as the
theorems, methods, and algorithms for solving these challenging problems.
Misunderstandings and confusions on some basic concepts, such as objectivity,
nonlinearity, Lagrangian, and generalized convexities are discussed and
classified. Breakthrough from recent challenges and conceptual mistakes by M.
Voisei, C. Zalinescu and his co-worker are addressed. Some open problems and
future works in global optimization and nonconvex mechanics are proposed.Comment: 43 pages, 4 figures. appears in Mathematics and Mechanics of Solids,
201
Particle Swarm Optimization: A survey of historical and recent developments with hybridization perspectives
Particle Swarm Optimization (PSO) is a metaheuristic global optimization
paradigm that has gained prominence in the last two decades due to its ease of
application in unsupervised, complex multidimensional problems which cannot be
solved using traditional deterministic algorithms. The canonical particle swarm
optimizer is based on the flocking behavior and social co-operation of birds
and fish schools and draws heavily from the evolutionary behavior of these
organisms. This paper serves to provide a thorough survey of the PSO algorithm
with special emphasis on the development, deployment and improvements of its
most basic as well as some of the state-of-the-art implementations. Concepts
and directions on choosing the inertia weight, constriction factor, cognition
and social weights and perspectives on convergence, parallelization, elitism,
niching and discrete optimization as well as neighborhood topologies are
outlined. Hybridization attempts with other evolutionary and swarm paradigms in
selected applications are covered and an up-to-date review is put forward for
the interested reader.Comment: 34 pages, 7 table
Convergence analysis of beetle antennae search algorithm and its applications
The beetle antennae search algorithm was recently proposed and investigated
for solving global optimization problems. Although the performance of the
algorithm and its variants were shown to be better than some existing
meta-heuristic algorithms, there is still a lack of convergence analysis. In
this paper, we provide theoretical analysis on the convergence of the beetle
antennae search algorithm. We test the performance of the BAS algorithm via
some representative benchmark functions. Meanwhile, some applications of the
BAS algorithm are also presented.Comment: n
A Computation Offloading Incentive Mechanism with Delay and Cost Constraints under 5G Satellite-ground IoV architecture
The 5G Internet of Vehicles has become a new paradigm alongside the growing
popularity and variety of computation-intensive applications with high
requirements for computational resources and analysis capabilities. Existing
network architectures and resource management mechanisms may not sufficiently
guarantee satisfactory Quality of Experience and network efficiency, mainly
suffering from coverage limitation of Road Side Units, insufficient resources,
and unsatisfactory computational capabilities of onboard equipment, frequently
changing network topology, and ineffective resource management schemes. To meet
the demands of such applications, in this article, we first propose a novel
architecture by integrating the satellite network with 5G cloud-enabled
Internet of Vehicles to efficiently support seamless coverage and global
resource management. A incentive mechanism based joint optimization problem of
opportunistic computation offloading under delay and cost constraints is
established under the aforementioned framework, in which a vehicular user can
either significantly reduce the application completion time by offloading
workloads to several nearby vehicles through opportunistic vehicle-to-vehicle
channels while effectively controlling the cost or protect its own profit by
providing compensated computing service. As the optimization problem is
non-convex and NP-hard, simulated annealing based on the Markov Chain Monte
Carlo as well as the metropolis algorithm is applied to solve the optimization
problem, which can efficaciously obtain both high-quality and cost-effective
approximations of global optimal solutions. The effectiveness of the proposed
mechanism is corroborated through simulation results
Generalized Eigenvalue Problems with Specified Eigenvalues
We consider the distance from a (square or rectangular) matrix pencil to the
nearest matrix pencil in 2-norm that has a set of specified eigenvalues. We
derive a singular value optimization characterization for this problem and
illustrate its usefulness for two applications. First, the characterization
yields a singular value formula for determining the nearest pencil whose
eigenvalues lie in a specified region in the complex plane. For instance, this
enables the numerical computation of the nearest stable descriptor system in
control theory. Second, the characterization partially solves the problem posed
in [Boutry et al. 2005] regarding the distance from a general rectangular
pencil to the nearest pencil with a complete set of eigenvalues. The involved
singular value optimization problems are solved by means of BFGS and
Lipschitz-based global optimization algorithms.Comment: 23 pages with 3 pdf figure
Filled function method for nonlinear equations
AbstractSystems of nonlinear equations are ubiquitous in engineering, physics and mechanics, and have myriad applications. Generally, they are very difficult to solve. In this paper, we will present a filled function method to solve nonlinear systems. We will first convert the nonlinear systems into equivalent global optimization problems with the property: x∗ is a global minimizer if and only if its function value is zero. A filled function method is proposed to solve the converted global optimization problem. Numerical examples are presented to illustrate our new techniques
Porcupine Neural Networks: (Almost) All Local Optima are Global
Neural networks have been used prominently in several machine learning and
statistics applications. In general, the underlying optimization of neural
networks is non-convex which makes their performance analysis challenging. In
this paper, we take a novel approach to this problem by asking whether one can
constrain neural network weights to make its optimization landscape have good
theoretical properties while at the same time, be a good approximation for the
unconstrained one. For two-layer neural networks, we provide affirmative
answers to these questions by introducing Porcupine Neural Networks (PNNs)
whose weight vectors are constrained to lie over a finite set of lines. We show
that most local optima of PNN optimizations are global while we have a
characterization of regions where bad local optimizers may exist. Moreover, our
theoretical and empirical results suggest that an unconstrained neural network
can be approximated using a polynomially-large PNN
A Distributed Hierarchical SGD Algorithm with Sparse Global Reduction
Reducing communication in training large-scale machine learning applications
on distributed platform is still a big challenge. To address this issue, we
propose a distributed hierarchical averaging stochastic gradient descent
(Hier-AVG) algorithm with infrequent global reduction by introducing local
reduction. As a general type of parallel SGD, Hier-AVG can reproduce several
popular synchronous parallel SGD variants by adjusting its parameters. We show
that Hier-AVG with infrequent global reduction can still achieve standard
convergence rate for non-convex optimization problems. In addition, we show
that more frequent local averaging with more participants involved can lead to
faster training convergence. By comparing Hier-AVG with another popular
distributed training algorithm K-AVG, we show that through deploying local
averaging with fewer number of global averaging, Hier-AVG can still achieve
comparable training speed while frequently get better test accuracy. This
indicates that local averaging can serve as an alternative remedy to
effectively reduce communication overhead when the number of learners is large.
Experimental results of Hier-AVG with several state-of-the-art deep neural nets
on CIFAR-10 and IMAGENET-1K are presented to validate our analysis and show its
superiority.Comment: 38 page
A Practical Approach to Quasi-convex Optimization
A new and simple method for quasi-convex optimization is introduced from
which its various applications can be derived. Especially, a global optimum
under constrains can be approximated for all continuous functions.Comment: 8 pages, 5 figure
Rational Optimization using Sum-of-Squares Techniques
Motivated by many control applications, this paper
deals with the global solutions of unconstrained optimization problems. First, a simple SOS method is presented to find the infimum of a polynomial, which can be handled efficiently using the relevant software tools. The main idea of this method is to introduce a perturbation variable whose approaching to zero results in a solution with any arbitrary precision. The proposed technique is then extended to the case of rational functions. The primary advantages of this approach over the existing ones are its simplicity and capability of treating problems for which the existing methods are not efficient, as demonstrated in three numerical examples
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