163 research outputs found
Jordan symmetry reduction for conic optimization over the doubly nonnegative cone: theory and software
A common computational approach for polynomial optimization problems (POPs)
is to use (hierarchies of) semidefinite programming (SDP) relaxations. When the
variables in the POP are required to be nonnegative, these SDP problems
typically involve nonnegative matrices, i.e. they are conic optimization
problems over the doubly nonnegative cone. The Jordan reduction, a symmetry
reduction method for conic optimization, was recently introduced for symmetric
cones by Parrilo and Permenter [Mathematical Programming 181(1), 2020]. We
extend this method to the doubly nonnegative cone, and investigate its
application to known relaxations of the quadratic assignment and maximum stable
set problems. We also introduce new Julia software where the symmetry reduction
is implemented.Comment: 19 pages, titled change from earlier version. arXiv admin note: text
overlap with arXiv:1908.0087
Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization
The affine rank minimization problem consists of finding a matrix of minimum
rank that satisfies a given system of linear equality constraints. Such
problems have appeared in the literature of a diverse set of fields including
system identification and control, Euclidean embedding, and collaborative
filtering. Although specific instances can often be solved with specialized
algorithms, the general affine rank minimization problem is NP-hard. In this
paper, we show that if a certain restricted isometry property holds for the
linear transformation defining the constraints, the minimum rank solution can
be recovered by solving a convex optimization problem, namely the minimization
of the nuclear norm over the given affine space. We present several random
ensembles of equations where the restricted isometry property holds with
overwhelming probability. The techniques used in our analysis have strong
parallels in the compressed sensing framework. We discuss how affine rank
minimization generalizes this pre-existing concept and outline a dictionary
relating concepts from cardinality minimization to those of rank minimization
Symmetry reduction in convex optimization with applications in combinatorics
This dissertation explores different approaches to and applications of symmetry reduction in convex optimization. Using tools from semidefinite programming, representation theory and algebraic combinatorics, hard combinatorial problems are solved or bounded. The first chapters consider the Jordan reduction method, extend the method to optimization over the doubly nonnegative cone, and apply it to quadratic assignment problems and energy minimization on a discrete torus. The following chapter uses symmetry reduction as a proving tool, to approach a problem from queuing theory with redundancy scheduling. The final chapters propose generalizations and reductions of flag algebras, a powerful tool for problems coming from extremal combinatorics
Positive trace polynomials and the universal Procesi-Schacher conjecture
Positivstellensatz is a fundamental result in real algebraic geometry
providing algebraic certificates for positivity of polynomials on semialgebraic
sets. In this article Positivstellens\"atze for trace polynomials positive on
semialgebraic sets of matrices are provided. A Krivine-Stengle-type
Positivstellensatz is proved characterizing trace polynomials nonnegative on a
general semialgebraic set using weighted sums of hermitian squares with
denominators. The weights in these certificates are obtained from generators of
and traces of hermitian squares. For compact semialgebraic sets
Schm\"udgen- and Putinar-type Positivstellens\"atze are obtained: every trace
polynomial positive on has a sum of hermitian squares decomposition with
weights and without denominators. The methods employed are inspired by
invariant theory, classical real algebraic geometry and functional analysis.
Procesi and Schacher in 1976 developed a theory of orderings and positivity
on central simple algebras with involution and posed a Hilbert's 17th problem
for a universal central simple algebra of degree : is every totally positive
element a sum of hermitian squares? They gave an affirmative answer for .
In this paper a negative answer for is presented. Consequently, including
traces of hermitian squares as weights in the Positivstellens\"atze is
indispensable
Automorphisms of rank-one generated hyperbolicity cones and their derivative relaxations
A hyperbolicity cone is said to be rank-one generated (ROG) if all its
extreme rays have rank one, where the rank is computed with respect to the
underlying hyperbolic polynomial. This is a natural class of hyperbolicity
cones which are strictly more general than the ROG spectrahedral cones. In this
work, we present a study of the automorphisms of ROG hyperbolicity cones and
their derivative relaxations. One of our main results states that the
automorphisms of the derivative relaxations are exactly the automorphisms of
the original cone fixing a certain direction. As an application, we completely
determine the automorphisms of the derivative relaxations of the nonnegative
orthant and of the cone of positive semidefinite matrices. More generally, we
also prove relations between the automorphisms of a spectral cone and the
underlying permutation-invariant set, which might be of independent interest.Comment: 25 pages. Some minor fixes and changes. To appear at the SIAM Journal
on Applied Algebra and Geometr
Decomposed Structured Subsets for Semidefinite and Sum-of-Squares Optimization
Semidefinite programs (SDPs) are standard convex problems that are frequently
found in control and optimization applications. Interior-point methods can
solve SDPs in polynomial time up to arbitrary accuracy, but scale poorly as the
size of matrix variables and the number of constraints increases. To improve
scalability, SDPs can be approximated with lower and upper bounds through the
use of structured subsets (e.g., diagonally-dominant and scaled-diagonally
dominant matrices). Meanwhile, any underlying sparsity or symmetry structure
may be leveraged to form an equivalent SDP with smaller positive semidefinite
constraints. In this paper, we present a notion of decomposed structured
subsets}to approximate an SDP with structured subsets after an equivalent
conversion. The lower/upper bounds found by approximation after conversion
become tighter than the bounds obtained by approximating the original SDP
directly. We apply decomposed structured subsets to semidefinite and
sum-of-squares optimization problems with examples of H-infinity norm
estimation and constrained polynomial optimization. An existing basis pursuit
method is adapted into this framework to iteratively refine bounds.Comment: 23 pages, 10 figures, 9 table
Jordan Algebras of Symmetric Matrices
We study linear spaces of symmetric matrices whose reciprocal is also a
linear space. These are Jordan algebras. We classify such algebras in low
dimensions, and we study the associated Jordan loci in the Grassmannian.Comment: 16 page
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