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Algebraic Aspects of Conditional Independence and Graphical Models
Published in Marloes Maathuis, Mathias Drton, Steffen Lauritzen, Martin Wainwright, Handbook of Graphical Models, 2018
Thomas Kahle, Johannes Rauh, Seth Sullivant
In addition to polynomial equations, in many situations in statistics it is useful to consider solutions to polynomial inequalities as well. This is the subject of the field real algebraic geometry. Inequalities only make sense over an ordered field like R $ \mathbb R $ (but not over C $ \mathbb C $ ). For simplicity, the following definitions and results are formulated with R $ \mathbb R $ . Again, this text only contains the basic definitions. For more details the reader is referred to [4,5].
SOS-based policy iteration for H∞ control of polynomial systems with uncertain parameters
Published in International Journal of Control, 2023
Sajjad Pakkhesal, Saeed Shamaghdari
In many systems, we don’t know the exact values of the physical parameters, and only the bounds of the parameters are known. This kind of uncertainty is called parametric uncertainty (Mueller, 2011). A system with parametric uncertainty can be seen as an infinite number of systems, where each of them corresponds to a certain value of the parameter. For stability analysis of such a system, we should examine the stability for all of these infinite number of systems, which is an NP-hard task (Apkarian et al., 2015) and is known as parametric robust stability. Parametric robust synthesis is to design a controller that guarantees the parametric robustness of the closed-loop system. Using the well-known results of the real algebraic geometry, some parametric robust controllers have been proposed for polynomial systems (Pak Khesal & & Mohammadzaman, 2018; Pozo & Rodellar, 2010; Yu et al., 2014).
Solving SDP completely with an interior point oracle
Published in Optimization Methods and Software, 2021
Bruno F. Lourenço, Masakazu Muramatsu, Takashi Tsuchiya
Nevertheless, the first finite infeasibility certificate was obtained by Ramana in [41] using his extended duality theory. Since then, Sturm mentioned the possibility of obtaining a finite certificate for infeasibility by using the directions produced in his regularization procedure, see page 1243 of [45]. More recently, Liu and Pataki have also obtained finite certificates through elementary reformulations [19]. Interestingly, Klep and Schweighofer [16] also obtained certificates through a completely different approach using tools from real algebraic geometry. As we move from SDPs to conic linear programs over arbitrary cones, facial reduction seems to one of the few approaches that can provide finite certificates of infeasibility see, for example, [18].
SPECTRA – a Maple library for solving linear matrix inequalities in exact arithmetic
Published in Optimization Methods and Software, 2019
Didier Henrion, Simone Naldi, Mohab Safey El Din
Optimization of a linear function on a spectrahedron is called semidefinite programming (SDP), a broad generalization of linear programming with many applications in control engineering, signal processing, combinatorial optimization, mechanical structure design, etc, see [20,22]. The algebra and geometry of spectrahedra is an active area of study in real algebraic geometry, especially in connection with the problem of moments and the decomposition of real multivariate polynomials as sums-of-squares (SOS), see [1,11,17] and references therein. SDP-based methods have recently been developed in the setting of error analysis of roundoff during floating-point computations, see [3,12], or in non-commutative real algebraic geometry, see [4,10].