1,275 research outputs found
Approximation Hierarchies for Copositive Cone over Symmetric Cone and Their Comparison
We first provide an inner-approximation hierarchy described by a
sum-of-squares (SOS) constraint for the copositive (COP) cone over a general
symmetric cone. The hierarchy is a generalization of that proposed by Parrilo
(2000) for the usual COP cone (over a nonnegative orthant). We also discuss its
dual. Second, we characterize the COP cone over a symmetric cone using the
usual COP cone. By replacing the usual COP cone appearing in this
characterization with the inner- or outer-approximation hierarchy provided by
de Klerk and Pasechnik (2002) or Yildirim (2012), we obtain an inner- or
outer-approximation hierarchy described by semidefinite but not by SOS
constraints for the COP matrix cone over the direct product of a nonnegative
orthant and a second-order cone. We then compare them with the existing
hierarchies provided by Zuluaga et al. (2006) and Lasserre (2014). Theoretical
and numerical examinations imply that we can numerically increase a depth
parameter, which determines an approximation accuracy, in the approximation
hierarchies derived from de Klerk and Pasechnik (2002) and Yildirim (2012),
particularly when the nonnegative orthant is small. In such a case, the
approximation hierarchy derived from Yildirim (2012) can yield nearly optimal
values numerically. Combining the proposed approximation hierarchies with
existing ones, we can evaluate the optimal value of COP programming problems
more accurately and efficiently
Conic Programming Approaches for Polynomial Optimization: Theory and Applications
Historically, polynomials are among the most popular class of functions used for empirical modeling in science and engineering. Polynomials are easy to evaluate, appear naturally in many physical (real-world) systems, and can be used to accurately approximate any smooth function. It is not surprising then, that the task of solving polynomial optimization problems; that is, problems where both the objective function and constraints are multivariate polynomials, is ubiquitous and of enormous interest in these fields. Clearly, polynomial op- timization problems encompass a very general class of non-convex optimization problems, including key combinatorial optimization problems.The focus of the first three chapters of this document is to address the solution of polynomial optimization problems in theory and in practice, using a conic optimization approach. Convex optimization has been well studied to solve quadratic constrained quadratic problems. In the first part, convex relaxations for general polynomial optimization problems are discussed. Instead of using the matrix space to study quadratic programs, we study the convex relaxations for POPs through a lifted tensor space, more specifically, using the completely positive tensor cone and the completely positive semidefinite tensor cone. We show that tensor relaxations theoretically yield no-worse global bounds for a class of polynomial optimization problems than relaxation for a QCQP reformulation of the POPs. We also propose an approximation strategy for tensor cones and show empirically the advantage of the tensor relaxation.In the second part, we propose an alternative SDP and SOCP hierarchy to obtain global bounds for general polynomial optimization problems. Comparing with other existing SDP and SOCP hierarchies that uses higher degree sum of square (SOS) polynomials and scaled diagonally sum of square polynomials (SDSOS) when the hierarchy level increases, these proposed hierarchies, using fixed degree SOS and SDSOS polynomials but more of these polynomials, perform numerically better. Numerical results show that the hierarchies we proposed have better performance in terms of tightness of the bound and solution time compared with other hierarchies in the literature.The third chapter deals with Alternating Current Optimal Power Flow problem via a polynomial optimization approach. The Alternating Current Optimal Power Flow (ACOPF) problem is a challenging non-convex optimization problem in power systems. Prior research mainly focuses on using SDP relaxations and SDP-based hierarchies to address the solution of ACOPF problem. In this Chapter, we apply existing SOCP hierarchies to this problem and explore the structure of the network to propose simplified hierarchies for ACOPF problems. Compared with SDP approaches, SOCP approaches are easier to solve and can be used to approximate large scale ACOPF problems.The last chapter also relates to the use of conic optimization techniques, but in this case to pricing in markets with non-convexities. Indeed, it is an application of conic optimization approach to solve a pricing problem in energy systems. Prior research in energy market pricing mainly focus on linear costs in the objective function. Due to the penetration of renewable energies into the current electricity grid, it is important to consider quadratic costs in the objective function, which reflects the ramping costs for traditional generators. This study address the issue how to find the market clearing prices when considering quadratic costs in the objective function
On the closure of the completely positive semidefinite cone and linear approximations to quantum colorings
We investigate structural properties of the completely positive semidefinite
cone , consisting of all the symmetric matrices
that admit a Gram representation by positive semidefinite matrices of any size.
This cone has been introduced to model quantum graph parameters as conic
optimization problems. Recently it has also been used to characterize the set
of bipartite quantum correlations, as projection of an affine
section of it. We have two main results concerning the structure of the
completely positive semidefinite cone, namely about its interior and about its
closure. On the one hand we construct a hierarchy of polyhedral cones which
covers the interior of , which we use for computing some
variants of the quantum chromatic number by way of a linear program. On the
other hand we give an explicit description of the closure of the completely
positive semidefinite cone, by showing that it consists of all matrices
admitting a Gram representation in the tracial ultraproduct of matrix algebras.Comment: 20 page
Positive semidefinite approximations to the cone of copositive kernels
It has been shown that the maximum stable set problem in some infinite
graphs, and the kissing number problem in particular, reduces to a minimization
problem over the cone of copositive kernels. Optimizing over this infinite
dimensional cone is not tractable, and approximations of this cone have been
hardly considered in literature. We propose two convergent hierarchies of
subsets of copositive kernels, in terms of non-negative and positive definite
kernels. We use these hierarchies and representation theorems for invariant
positive definite kernels on the sphere to construct new SDP-based bounds on
the kissing number. This results in fast-to-compute upper bounds on the kissing
number that lie between the currently existing LP and SDP bounds.Comment: 29 pages, 2 tables, 1 figur
Lower bounds on matrix factorization ranks via noncommutative polynomial optimization
We use techniques from (tracial noncommutative) polynomial optimization to formulate hierarchies of semidefinite programming lower bounds on matrix factorization ranks. In particular, we consider the nonnegative rank, the completely positive rank, and their symmetric analogues: the positive semidefinite rank and the completely positive semidefinite rank. We study the convergence properties of our hierarchies, compare them extensively to known lower bounds, and provide some (numerical) examples
Light-cone Superstring Field Theory, pp-wave background and integrability properties
We show that the three strings vertex coefficients in light--cone open string
field theory satisfy the Hirota equations for the dispersionless Toda lattice
hierarchy. We show that Hirota equations allow us to calculate the correlators
of an associated quantum system where the Neumann coefficients represent the
two--point functions. We consider next the three strings vertex coefficients of
the light--cone string field theory on a maximally supersymmetric pp--wave
background. Using the previous results we are able to show that these Neumann
coefficients satisfy the Hirota equations for the full Toda lattice hierarchy
at least up to second order in the 'string mass' .Comment: 23 pages, 3 figures, footnote and references adde
Rounding Sum-of-Squares Relaxations
We present a general approach to rounding semidefinite programming
relaxations obtained by the Sum-of-Squares method (Lasserre hierarchy). Our
approach is based on using the connection between these relaxations and the
Sum-of-Squares proof system to transform a *combining algorithm* -- an
algorithm that maps a distribution over solutions into a (possibly weaker)
solution -- into a *rounding algorithm* that maps a solution of the relaxation
to a solution of the original problem.
Using this approach, we obtain algorithms that yield improved results for
natural variants of three well-known problems:
1) We give a quasipolynomial-time algorithm that approximates the maximum of
a low degree multivariate polynomial with non-negative coefficients over the
Euclidean unit sphere. Beyond being of interest in its own right, this is
related to an open question in quantum information theory, and our techniques
have already led to improved results in this area (Brand\~{a}o and Harrow, STOC
'13).
2) We give a polynomial-time algorithm that, given a d dimensional subspace
of R^n that (almost) contains the characteristic function of a set of size n/k,
finds a vector in the subspace satisfying ,
where . Aside from being a natural relaxation, this
is also motivated by a connection to the Small Set Expansion problem shown by
Barak et al. (STOC 2012) and our results yield a certain improvement for that
problem.
3) We use this notion of L_4 vs. L_2 sparsity to obtain a polynomial-time
algorithm with substantially improved guarantees for recovering a planted
-sparse vector v in a random d-dimensional subspace of R^n. If v has mu n
nonzero coordinates, we can recover it with high probability whenever , improving for prior methods which
intrinsically required
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