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On the power of symmetric linear programs
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On the power of symmetric linear programs
We consider families of symmetric linear programs (LPs) that decide a
property of graphs (or other relational structures) in the sense that, for each
size of graph, there is an LP defining a polyhedral lift that separates the
integer points corresponding to graphs with the property from those
corresponding to graphs without the property. We show that this is equivalent,
with at most polynomial blow-up in size, to families of symmetric Boolean
circuits with threshold gates. In particular, when we consider polynomial-size
LPs, the model is equivalent to definability in a non-uniform version of
fixed-point logic with counting (FPC). Known upper and lower bounds for FPC
apply to the non-uniform version. In particular, this implies that the class of
graphs with perfect matchings has polynomial-size symmetric LPs while we obtain
an exponential lower bound for symmetric LPs for the class of Hamiltonian
graphs. We compare and contrast this with previous results (Yannakakis 1991)
showing that any symmetric LPs for the matching and TSP polytopes have
exponential size. As an application, we establish that for random, uniformly
distributed graphs, polynomial-size symmetric LPs are as powerful as general
Boolean circuits. We illustrate the effect of this on the well-studied
planted-clique problem
Steady-State Voltage Security Assessment Using Symmetric Eigenvalue Analysis for Weak Area Identification in Large Power Transmission Network
The central focus of this thesis is on long-term static voltage stability analysis of large power transmission grid. This thesis work is a product of an attempt to comprehend the numerous researches that has been done over the years on voltage security assessment. Voltage stability is one of the essential components influencing the reliability of a power network. There are several Transmission planning and operation compliance standards pertaining to voltage criterion from NERC and Independent System Operators (ISO) directed toward the utilities to operate their grid within tight voltage limits. This requires the utility to perform comprehensive planning studies of the power system frequently for different load profiles like summer and winter - peak load and light load conditions taking into account several contingency scenarios. The humongous number of nodes and branches in a typical preset-day power network has increased the complexity of conventional voltage stability analysis methods like PV / QV curves.
Initially, this study discusses various linear algebraic techniques used in steady-state power system analysis and presents the results on the simulations of IEEE test systems - 14 bus, 30 bus and 118 bus system. Later, it introduces an idea of performing a spectral (Symmetric Eigenvalue) analysis of the power system Jacobian and a rigorous testing of the same IEEE bus test systems was performed. Finally, it concludes by presenting a comparative result against other eigenvalue-based methods. The entire analysis has been performed by a combination of custom-written MATLAB programs, Python scripts and Siemens PTI PSS/E software for its one-line diagram capabilities
-permutability and linear Datalog implies symmetric Datalog
We show that if is a core relational structure such that
CSP() can be solved by a linear Datalog program, and is
-permutable for some , then CSP() can be solved by a symmetric
Datalog program (and thus CSP() lies in deterministic logspace). At
the moment, it is not known for which structures will CSP() be solvable by a linear Datalog program. However, once somebody obtains a
characterization of linear Datalog, our result immediately gives a
characterization of symmetric Datalog
Tools for active control system design
Efficient control law analysis and design tools which properly account for the interaction of flexible structures, unsteady aerodynamics and active controls are developed. Development, application, validation and documentation of efficient multidisciplinary computer programs for analysis and design of active control laws are also discussed
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
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