3,098 research outputs found
Some Applications of Polynomial Optimization in Operations Research and Real-Time Decision Making
We demonstrate applications of algebraic techniques that optimize and certify
polynomial inequalities to problems of interest in the operations research and
transportation engineering communities. Three problems are considered: (i)
wireless coverage of targeted geographical regions with guaranteed signal
quality and minimum transmission power, (ii) computing real-time certificates
of collision avoidance for a simple model of an unmanned vehicle (UV)
navigating through a cluttered environment, and (iii) designing a nonlinear
hovering controller for a quadrotor UV, which has recently been used for load
transportation. On our smaller-scale applications, we apply the sum of squares
(SOS) relaxation and solve the underlying problems with semidefinite
programming. On the larger-scale or real-time applications, we use our recently
introduced "SDSOS Optimization" techniques which result in second order cone
programs. To the best of our knowledge, this is the first study of real-time
applications of sum of squares techniques in optimization and control. No
knowledge in dynamics and control is assumed from the reader
Lower Bounds on Complexity of Lyapunov Functions for Switched Linear Systems
We show that for any positive integer , there are families of switched
linear systems---in fixed dimension and defined by two matrices only---that are
stable under arbitrary switching but do not admit (i) a polynomial Lyapunov
function of degree , or (ii) a polytopic Lyapunov function with facets, or (iii) a piecewise quadratic Lyapunov function with
pieces. This implies that there cannot be an upper bound on the size of the
linear and semidefinite programs that search for such stability certificates.
Several constructive and non-constructive arguments are presented which connect
our problem to known (and rather classical) results in the literature regarding
the finiteness conjecture, undecidability, and non-algebraicity of the joint
spectral radius. In particular, we show that existence of an extremal piecewise
algebraic Lyapunov function implies the finiteness property of the optimal
product, generalizing a result of Lagarias and Wang. As a corollary, we prove
that the finiteness property holds for sets of matrices with an extremal
Lyapunov function belonging to some of the most popular function classes in
controls
Robust-to-Dynamics Optimization
A robust-to-dynamics optimization (RDO) problem is an optimization problem
specified by two pieces of input: (i) a mathematical program (an objective
function and a feasible set
), and (ii) a dynamical system (a map
). Its goal is to minimize over the
set of initial conditions that forever remain in
under . The focus of this paper is on the case where the
mathematical program is a linear program and the dynamical system is either a
known linear map, or an uncertain linear map that can change over time. In both
cases, we study a converging sequence of polyhedral outer approximations and
(lifted) spectrahedral inner approximations to . Our inner
approximations are optimized with respect to the objective function and
their semidefinite characterization---which has a semidefinite constraint of
fixed size---is obtained by applying polar duality to convex sets that are
invariant under (multiple) linear maps. We characterize three barriers that can
stop convergence of the outer approximations from being finite. We prove that
once these barriers are removed, our inner and outer approximating procedures
find an optimal solution and a certificate of optimality for the RDO problem in
a finite number of steps. Moreover, in the case where the dynamics are linear,
we show that this phenomenon occurs in a number of steps that can be computed
in time polynomial in the bit size of the input data. Our analysis also leads
to a polynomial-time algorithm for RDO instances where the spectral radius of
the linear map is bounded above by any constant less than one. Finally, in our
concluding section, we propose a broader research agenda for studying
optimization problems with dynamical systems constraints, of which RDO is a
special case
An architecture for a focused trend parallel web crawler with the application of clickstream analysis
The tremendous growth of the Web poses many challenges for all-purpose single-process crawlers including the presence of some irrelevant answers among search results and the coverage and scaling issues regarding the enormous dimension of the World Wide Web. Hence, more enhanced and convincing algorithms are on demand to yield more precise and relevant search results in an appropriate amount of time. Since employing link based Web page importance metrics within a multi-processes crawler bears a considerable communication overhead on the overall system and cannot produce the precise answer set, employing these metrics in search engines is not an absolute solution to identify the best search answer set by the overall search system. Thus considering the employment of a link independent Web page importance metric is required to govern the priority rule within the queue of fetched URLs. The aim of this paper is to propose a modest weighted architecture for a focused structured parallel Web crawler which employs a link independent clickstream based Web page importance metric. The experiments of this metric over the restricted boundary Web zone of our crowded UTM University Web site shows the efficiency of the proposed metric
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