11 research outputs found
A Semidefinite Hierarchy for Containment of Spectrahedra
A spectrahedron is the positivity region of a linear matrix pencil and thus
the feasible set of a semidefinite program. We propose and study a hierarchy of
sufficient semidefinite conditions to certify the containment of a
spectrahedron in another one. This approach comes from applying a moment
relaxation to a suitable polynomial optimization formulation. The hierarchical
criterion is stronger than a solitary semidefinite criterion discussed earlier
by Helton, Klep, and McCullough as well as by the authors. Moreover, several
exactness results for the solitary criterion can be brought forward to the
hierarchical approach. The hierarchy also applies to the (equivalent) question
of checking whether a map between matrix (sub-)spaces is positive. In this
context, the solitary criterion checks whether the map is completely positive,
and thus our results provide a hierarchy between positivity and complete
positivity.Comment: 24 pages, 2 figures; minor corrections; to appear in SIAM J. Opti
Classifying Linear Matrix Inequalities via Abstract Operator Systems
We systematically study how properties of abstract operator systems help
classifying linear matrix inequality definitions of sets. Our main focus is on
polyhedral cones, the 3-dimensional Lorentz cone, where we can completely
describe all defining linear matrix inequalities, and on the cone of positive
semidefinite matrices. Here we use results on isometries between matrix
algebras to describe linear matrix inequality definitions of relatively small
size. We conversely use the theory of operator systems to characterize special
such isometries
Calculus of unbounded spectrahedral shadows and their polyhedral approximation
The present thesis deals with the polyhedral approximation and calculus of spectrahedral shadows that are not necessarily bounded. These sets are the images of the feasible regions of semidefinite programs under linear transformations. Spectrahedral shadows contain polyhedral sets as a proper subclass. Therefore, the method of polyhedral approximation is a useful device to approximately describe them using members of the same class with a simpler structure. In the first part we develop a calculus for spectrahedral shadows. Besides showing their closedness under numerous set operations, we derive explicit descriptions of the resulting sets as spectrahedral shadows. Special attention is paid to operations that result in unbounded sets, such as the polar cone, conical hull and recession cone. The second part is dedicated to the approximation of compact spectrahedral shadows with respect to the Hausdorff distance. We present two algorithms for the computation of polyhedral approximations of such sets. Convergence as well as correctness of both algorithms are proved. As a supplementary tool we also present an algorithm that generates points from the relative interior of a spectrahedral shadow and computes its affine hull. Finally, we investigate the limits of polyhedral approximation in the Hausdorff distance in general and, extending known results, characterize the sets that admit such approximations. In the last part we develop concepts and tools for the approximation of spectrahedral shadows that are compatible with unboundedness. We present two notions of polyhedral approximation and show that sequences of approximations converge to the true set if the approximation errors diminish. In combination with algorithms for their computation we develop an algorithm for the polyhedral approximation of recession cones of spectrahedral shadows. Finiteness and correctness of all algorithms are proved and properties of the approximation concepts are investigated
Geometric Ramifications of the Lovász Theta Function and Their Interplay with Duality
The Lovasz theta function and the associated convex sets known as theta bodies are fundamental objects in combinatorial and
semidefinite optimization. They are accompanied by a rich duality theory and
deep connections to the geometric concept of orthonormal representations of graphs. In this thesis, we investigate several ramifications of the theory underlying these objects, including those arising from the illuminating viewpoint of duality. We study some optimization problems over unit-distance representations of graphs, which are intimately related to the Lovasz theta function and orthonormal representations. We also strengthen some known results about dual descriptions of theta bodies and their variants. Our main goal throughout the thesis is to lay some of the foundations for using semidefinite optimization and convex analysis in a way analogous to how polyhedral combinatorics has been using linear optimization to prove min-max theorems.
A unit-distance representation of a graph maps its nodes to some Euclidean space so that adjacent nodes are sent to pairs of points at distance one. The hypersphere number of , denoted by , is the (square of the) minimum radius of a hypersphere that contains a unit-distance representation of . Lovasz proved a min-max relation describing as a function of , the theta number of the complement of . This relation provides a dictionary between unit-distance representations in hyperspheres and orthonormal representations, which we exploit in a number of ways: we develop a weighted generalization of , parallel to the weighted version of ; we prove that is equal to the (square of the) minimum radius of an Euclidean ball that contains a unit-distance representation of ; we abstract some properties of that yield the famous Sandwich Theorem and use them to define another weighted generalization of , called ellipsoidal number of , where the unit-distance representation of is required to be in an ellipsoid of a given shape with minimum volume. We determine an analytic formula for the ellipsoidal number of the complete graph on nodes whenever there exists a Hadamard matrix of order .
We then study several duality aspects of the description of the theta body . For a graph , the convex corner is known to be the projection of a certain convex set, denoted by , which lies in a much higher-dimensional matrix space. We prove that the vertices of are precisely the symmetric tensors of incidence vectors of stable sets in , thus broadly generalizing previous results about vertices of the elliptope due to Laurent and Poljak from 1995. Along the way, we also identify all the vertices of several variants of and of the elliptope. Next we introduce an axiomatic framework for studying generalized theta bodies, based on the concept of diagonally scaling invariant cones, which allows us to prove in a unified way several characterizations of and the variants and , introduced independently by Schrijver, and by McEliece, Rodemich, and Rumsey in the late 1970's, and by Szegedy in 1994. The beautiful duality equation which states that the antiblocker of is is extended to this setting. The framework allows us to treat the stable set polytope and its classical polyhedral relaxations as generalized theta bodies, using the completely positive cone and its dual, and it allows us to derive a (weighted generalization of a) copositive formulation for the fractional chromatic number due to Dukanovic and Rendl in 2010 from a completely positive formulation for the stability number due to de Klerk and Pasechnik in 2002. Finally, we study a non-convex constraint for semidefinite programs (SDPs) that may be regarded as analogous to the usual integrality constraint for linear programs. When applied to certain classical SDPs, it specializes to the standard rank-one constraint. More importantly, the non-convex constraint also applies to the dual SDP, and for a certain SDP formulation of , the modified dual yields precisely the clique covering number. This opens the way to study some exactness properties of SDP relaxations for combinatorial optimization problems akin to the corresponding classical notions from polyhedral combinatorics, as well as approximation algorithms based on SDP relaxations
Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization
Many real-life decision problems are discrete in nature. To solve such problems as mathematical optimization problems, integrality constraints are commonly incorporated in the model to reflect the choice of finitely many alternatives. At the same time, it is known that semidefinite programming is very suitable for obtaining strong relaxations of combinatorial optimization problems. In this dissertation, we study the interplay between semidefinite programming and integrality, where a special focus is put on the use of cutting-plane methods. Although the notions of integrality and cutting planes are well-studied in linear programming, integer semidefinite programs (ISDPs) are considered only recently. We show that manycombinatorial optimization problems can be modeled as ISDPs. Several theoretical concepts, such as the Chvátal-Gomory closure, total dual integrality and integer Lagrangian duality, are studied for the case of integer semidefinite programming. On the practical side, we introduce an improved branch-and-cut approach for ISDPs and a cutting-plane augmented Lagrangian method for solving semidefinite programs with a large number of cutting planes. Throughout the thesis, we apply our results to a wide range of combinatorial optimization problems, among which the quadratic cycle cover problem, the quadratic traveling salesman problem and the graph partition problem. Our approaches lead to novel, strong and efficient solution strategies for these problems, with the potential to be extended to other problem classes