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Lifting for Simplicity: Concise Descriptions of Convex Sets
TLDR
The connection between the existence of lifts of a convex set and certain structured factorizations of its associated slack operator is explained, and a uniform approach to the construction of spectrahedral lifts of convex sets is described.Abstract:
This paper presents a selected tour through the theory and applications of lifts of convex sets. A lift of a convex set is a higher-dimensional convex set that projects onto the original set. Many convex sets have lifts that are dramatically simpler to describe than the original set. Finding such simple lifts has significant algorithmic implications, particularly for optimization problems. We consider both the classical case of polyhedral lifts, described by linear inequalities, as well as spectrahedral lifts, defined by linear matrix inequalities, with a focus on recent developments related to spectrahedral lifts.
Given a convex set, ideally we would either like to find a (low-complexity) polyhedral or spectrahedral lift, or find an obstruction proving that no such lift is possible. To this end, we explain the connection between the existence of lifts of a convex set and certain structured factorizations of its associated slack operator. Based on this characterization, we describe a uniform approach, via sums of squares, to the construction of spectrahedral lifts of convex sets and illustrate the method on several families of examples. Finally, we discuss two flavors of obstruction to the existence of lifts: one related to facial structure, and the other related to algebraic properties of the set in question.
Rather than being exhaustive, our aim is to illustrate the richness of the area. We touch on a range of different topics related to the existence of lifts, and present many examples of lifts from different areas of mathematics and its applications.read more
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On Approximations of the PSD Cone by a Polynomial Number of Smaller-sized PSD Cones
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Proceedings ArticleDOI
Lyapunov Differential Equation Hierarchy and Polynomial Lyapunov Functions for Switched Implicit Systems
TL;DR: In this article, the authors investigated stability analysis for implicit, switched linear systems using homogeneous Lyapunov functions (HLF) and presented linear matrix inequalities sufficient conditions for asymptotic stability of these systems based on HLFs.
References
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Journal ArticleDOI
Graph-Based Algorithms for Boolean Function Manipulation
TL;DR: In this paper, the authors present a data structure for representing Boolean functions and an associated set of manipulation algorithms, which have time complexity proportional to the sizes of the graphs being operated on, and hence are quite efficient as long as the graphs do not grow too large.
Journal ArticleDOI
Semidefinite programming
Lieven Vandenberghe,Stephen Boyd +1 more
TL;DR: A survey of the theory and applications of semidefinite programs and an introduction to primaldual interior-point methods for their solution are given.
Journal ArticleDOI
Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming
TL;DR: This algorithm gives the first substantial progress in approximating MAX CUT in nearly twenty years, and represents the first use of semidefinite programming in the design of approximation algorithms.
Combinatorial optimization. Polyhedra and efficiency.
TL;DR: This book shows the combinatorial optimization polyhedra and efficiency as your friend in spending the time in reading a book.