156,055 research outputs found
An Optimal Control Theory for the Traveling Salesman Problem and Its Variants
We show that the traveling salesman problem (TSP) and its many variants may
be modeled as functional optimization problems over a graph. In this
formulation, all vertices and arcs of the graph are functionals; i.e., a
mapping from a space of measurable functions to the field of real numbers. Many
variants of the TSP, such as those with neighborhoods, with forbidden
neighborhoods, with time-windows and with profits, can all be framed under this
construct. In sharp contrast to their discrete-optimization counterparts, the
modeling constructs presented in this paper represent a fundamentally new
domain of analysis and computation for TSPs and their variants. Beyond its
apparent mathematical unification of a class of problems in graph theory, the
main advantage of the new approach is that it facilitates the modeling of
certain application-specific problems in their home space of measurable
functions. Consequently, certain elements of economic system theory such as
dynamical models and continuous-time cost/profit functionals can be directly
incorporated in the new optimization problem formulation. Furthermore, subtour
elimination constraints, prevalent in discrete optimization formulations, are
naturally enforced through continuity requirements. The price for the new
modeling framework is nonsmooth functionals. Although a number of theoretical
issues remain open in the proposed mathematical framework, we demonstrate the
computational viability of the new modeling constructs over a sample set of
problems to illustrate the rapid production of end-to-end TSP solutions to
extensively-constrained practical problems.Comment: 24 pages, 8 figure
Quadratically-Regularized Optimal Transport on Graphs
Optimal transportation provides a means of lifting distances between points
on a geometric domain to distances between signals over the domain, expressed
as probability distributions. On a graph, transportation problems can be used
to express challenging tasks involving matching supply to demand with minimal
shipment expense; in discrete language, these become minimum-cost network flow
problems. Regularization typically is needed to ensure uniqueness for the
linear ground distance case and to improve optimization convergence;
state-of-the-art techniques employ entropic regularization on the
transportation matrix. In this paper, we explore a quadratic alternative to
entropic regularization for transport over a graph. We theoretically analyze
the behavior of quadratically-regularized graph transport, characterizing how
regularization affects the structure of flows in the regime of small but
nonzero regularization. We further exploit elegant second-order structure in
the dual of this problem to derive an easily-implemented Newton-type
optimization algorithm.Comment: 27 page
Towards Optimal Discrete Online Hashing with Balanced Similarity
When facing large-scale image datasets, online hashing serves as a promising
solution for online retrieval and prediction tasks. It encodes the online
streaming data into compact binary codes, and simultaneously updates the hash
functions to renew codes of the existing dataset. To this end, the existing
methods update hash functions solely based on the new data batch, without
investigating the correlation between such new data and the existing dataset.
In addition, existing works update the hash functions using a relaxation
process in its corresponding approximated continuous space. And it remains as
an open problem to directly apply discrete optimizations in online hashing. In
this paper, we propose a novel supervised online hashing method, termed
Balanced Similarity for Online Discrete Hashing (BSODH), to solve the above
problems in a unified framework. BSODH employs a well-designed hashing
algorithm to preserve the similarity between the streaming data and the
existing dataset via an asymmetric graph regularization. We further identify
the "data-imbalance" problem brought by the constructed asymmetric graph, which
restricts the application of discrete optimization in our problem. Therefore, a
novel balanced similarity is further proposed, which uses two equilibrium
factors to balance the similar and dissimilar weights and eventually enables
the usage of discrete optimizations. Extensive experiments conducted on three
widely-used benchmarks demonstrate the advantages of the proposed method over
the state-of-the-art methods.Comment: 8 pages, 11 figures, conferenc
Distributed convex optimization via continuous-time coordination algorithms with discrete-time communication
This paper proposes a novel class of distributed continuous-time coordination
algorithms to solve network optimization problems whose cost function is a sum
of local cost functions associated to the individual agents. We establish the
exponential convergence of the proposed algorithm under (i) strongly connected
and weight-balanced digraph topologies when the local costs are strongly convex
with globally Lipschitz gradients, and (ii) connected graph topologies when the
local costs are strongly convex with locally Lipschitz gradients. When the
local cost functions are convex and the global cost function is strictly
convex, we establish asymptotic convergence under connected graph topologies.
We also characterize the algorithm's correctness under time-varying interaction
topologies and study its privacy preservation properties. Motivated by
practical considerations, we analyze the algorithm implementation with
discrete-time communication. We provide an upper bound on the stepsize that
guarantees exponential convergence over connected graphs for implementations
with periodic communication. Building on this result, we design a
provably-correct centralized event-triggered communication scheme that is free
of Zeno behavior. Finally, we develop a distributed, asynchronous
event-triggered communication scheme that is also free of Zeno with asymptotic
convergence guarantees. Several simulations illustrate our results.Comment: 12 page
Graph Theoretic Algorithms Adaptable to Quantum Computing
Computational methods are rapidly emerging as an essential tool for understanding and solving complex engineering problems, which complement the traditional tools of experimentation and theory. When considered in a discrete computational setting, many engineering problems can be reduced to a graph coloring problem. Examples range from systems design, airline scheduling, image segmentation to pattern recognition, where energy cost functions with discrete variables are extremized. However, using discrete variables over continuous variables introduces some complications when defining differential quantities, such as gradients and Hessians involved in scientific computations within solid and fluid mechanics. Consequently, graph techniques are under-utilized in this important domain. However, we have recently witnessed great developments in quantum computing where physical devices can solve discrete optimization problems faster than most well-known classical algorithms. This warrants further investigation into the re-formulation of scientific computation problems into graph-theoretic problems, thus enabling rapid engineering simulations in a soon-to-be quantum computing world.
The computational techniques developed in this thesis allow the representation of surface scalars, such as perimeter and area, using discrete variables in a graph. Results from integral geometry, specifically Cauchy-Crofton relations, are used to estimate these scalars via submodular functions. With this framework, several quantities important to engineering applications can be represented in graph-based algorithms. These include the surface energy of cracks for fracture prediction, grain boundary energy to model microstructure evolution, and surface area estimates (of grains and fibers) for generating conformal meshes. Combinatorial optimization problems for these applications are presented first.
The last two chapters describe two new graph coloring algorithms implemented on a physical quantum computing device: the D-wave quantum annealer. The first algorithm describes a functional minimization approach to solve differential equations. The second algorithm describes a realization of the Boltzmann machine learning algorithm on a quantum annealer. The latter allows generative and discriminative learning of data, which has vast applications in many fields. Theoretical aspects and the implementation of these problems are outlined with a focus on engineering applications.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/168116/1/sidsriva_1.pd
Discrete and Continuous Optimization Methods for Self-Organization in Small Cell Networks - Models and Algorithms
Self-organization is discussed in terms of distributed computational methods and algorithms for resource allocation in cellular networks. In order to develop algorithms for different self-organization problems pertinent to small cell networks (SCN), a number of concepts from discrete and continuous optimization theory are employed. Self-organized resource allocation problems such as physical cell identifier (PCI) assignment and primary component carrier selection are formulated as discrete optimization problems. Distributed graph coloring and constraint satisfaction algorithms are used to solve these problems. The PCI assignment is also discussed for multi-operator heterogeneous networks. Furthermore, different variants of simulated annealing are proposed for solving a graph coloring formulation of the orthogonal resource allocation problem.
In the continuous optimization domain, a network utility maximization approach is considered for solving different resource allocation problems. Network synchronization is addressed using greedy and gradient search algorithms. Primal and dual decomposition are discussed for transmit power and scheduling weight optimizations, under a network-wide power constraint. Joint optimization over transmit powers and multi-user scheduling weights is considered in a multi-carrier SCN, for both maximum rate and proportional-fair rate utilities. This formulation is extended for multiple-input multiple-output (MIMO) SCNs, where apart from transmit powers and multi-user scheduling weights, the transmit precoders are also optimized, for a generic alpha-fair utility function. Optimization of network resources over multiple degrees of freedom is particularly effective in reducing mutual interference, leading to significant gains in network utility. Finally, an alternate formulation of transmit power allocation is considered, in which the network transmit power is minimized subject to the data rate constraints of users. Thus, network resource allocation algorithms inspired by optimization theory constitute an effective approach for self-organization in contemporary as well as future cellular networks
Reflection methods for user-friendly submodular optimization
Recently, it has become evident that submodularity naturally captures widely
occurring concepts in machine learning, signal processing and computer vision.
Consequently, there is need for efficient optimization procedures for
submodular functions, especially for minimization problems. While general
submodular minimization is challenging, we propose a new method that exploits
existing decomposability of submodular functions. In contrast to previous
approaches, our method is neither approximate, nor impractical, nor does it
need any cumbersome parameter tuning. Moreover, it is easy to implement and
parallelize. A key component of our method is a formulation of the discrete
submodular minimization problem as a continuous best approximation problem that
is solved through a sequence of reflections, and its solution can be easily
thresholded to obtain an optimal discrete solution. This method solves both the
continuous and discrete formulations of the problem, and therefore has
applications in learning, inference, and reconstruction. In our experiments, we
illustrate the benefits of our method on two image segmentation tasks.Comment: Neural Information Processing Systems (NIPS), \'Etats-Unis (2013
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