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Convex Algebraic Geometry with Applications to Power Systems, Statistics and Optimization.

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Convex Algebraic Geometry with Applications to Power Systems, Statistics and Optimization./
作者:	Lindberg, Julia.
出版者:	Ann Arbor : ProQuest Dissertations & Theses, : 2022,
面頁冊數:	167 p.
附註:	Source: Dissertations Abstracts International, Volume: 83-11, Section: B.
Contained By:	Dissertations Abstracts International83-11B.
標題:	Electrical engineering. -
ISBN:	9798438756828

Convex Algebraic Geometry with Applications to Power Systems, Statistics and Optimization.
Lindberg, Julia.

Convex Algebraic Geometry with Applications to Power Systems, Statistics and Optimization. - Ann Arbor : ProQuest Dissertations & Theses, 2022 - 167 p.

Source: Dissertations Abstracts International, Volume: 83-11, Section: B.

Thesis (Ph.D.)--The University of Wisconsin - Madison, 2022.

This item must not be sold to any third party vendors.

Many important problems in engineering are large scale and nonlinear -- two things that are inherently at odds. As a result, it is desirable to make use of underlying structure to reduce the computational complexity. This work considers problems in power systems, statistics and optimization that have underlying algebraic structure and exploits this structure to answer relevant questions.The first part of this thesis focuses on the power flow equations. Using techniques from numerical algebraic geometry, we design a novel algorithm that dramatically increases the speed with which we can find all solutions to the power flow equations. We study the real solution sets for various families of networks and show that there exist electrical parameter values for which the number of complex solutions is achieved by all real solutions. Finally, we find outer approximations of the convex hull of the real solutions to these equations. The next chapter studies data center geographic load shifting. We propose a metric to allow data center operators to shift load independently of collaboration with independent system operators. We demonstrate the superiority of our metric over other commonly used metrics using a years worth of data.The next part of this thesis focuses on a classical problem in statistics, density estimation for Gaussian mixture models (GMMs). We study the variety stemming from the moment equations of GMMs and provide upper bounds on the number of solutions to these equations. We apply these results to GMMs in high dimensions and design a homotopy algorithm that performs density estimation where the number of paths scales linearly in the dimension. We conclude by considering problems in optimization. We first study maximum likelihood estimation where the statistical model is defined by system of polynomial equations. We give an expression for the number of critical points for this problem and show that the number of critical points is determined by the Newton polytopes of the model. Next, we study the Shor relaxation of quadratic programs. We characterize the geometry of the set of objective functions for which this relaxation is exact and apply these results to the quadratic binary programs.

ISBN: 9798438756828Subjects--Topical Terms:

649834
Electrical engineering.
Subjects--Index Terms:

Algebraic geometry

Convex Algebraic Geometry with Applications to Power Systems, Statistics and Optimization.
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245 1 0 $a Convex Algebraic Geometry with Applications to Power Systems, Statistics and Optimization.
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500 $a Source: Dissertations Abstracts International, Volume: 83-11, Section: B.
500 $a Advisor: Lesieutre, Bernard C.;Rodriguez, Jose I.
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506 $a This item must not be sold to any third party vendors.
520 $a Many important problems in engineering are large scale and nonlinear -- two things that are inherently at odds. As a result, it is desirable to make use of underlying structure to reduce the computational complexity. This work considers problems in power systems, statistics and optimization that have underlying algebraic structure and exploits this structure to answer relevant questions.The first part of this thesis focuses on the power flow equations. Using techniques from numerical algebraic geometry, we design a novel algorithm that dramatically increases the speed with which we can find all solutions to the power flow equations. We study the real solution sets for various families of networks and show that there exist electrical parameter values for which the number of complex solutions is achieved by all real solutions. Finally, we find outer approximations of the convex hull of the real solutions to these equations. The next chapter studies data center geographic load shifting. We propose a metric to allow data center operators to shift load independently of collaboration with independent system operators. We demonstrate the superiority of our metric over other commonly used metrics using a years worth of data.The next part of this thesis focuses on a classical problem in statistics, density estimation for Gaussian mixture models (GMMs). We study the variety stemming from the moment equations of GMMs and provide upper bounds on the number of solutions to these equations. We apply these results to GMMs in high dimensions and design a homotopy algorithm that performs density estimation where the number of paths scales linearly in the dimension. We conclude by considering problems in optimization. We first study maximum likelihood estimation where the statistical model is defined by system of polynomial equations. We give an expression for the number of critical points for this problem and show that the number of critical points is determined by the Newton polytopes of the model. Next, we study the Shor relaxation of quadratic programs. We characterize the geometry of the set of objective functions for which this relaxation is exact and apply these results to the quadratic binary programs.
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