12,884 research outputs found
Algorithm 950: Ncpol2sdpa---Sparse Semidefinite Programming Relaxations for Polynomial Optimization Problems of Noncommuting Variables
A hierarchy of semidefinite programming (SDP) relaxations approximates the
global optimum of polynomial optimization problems of noncommuting variables.
Generating the relaxation, however, is a computationally demanding task, and
only problems of commuting variables have efficient generators. We develop an
implementation for problems of noncommuting problems that creates the
relaxation to be solved by SDPA -- a high-performance solver that runs in a
distributed environment. We further exploit the inherent sparsity of
optimization problems in quantum physics to reduce the complexity of the
resulting relaxations. Constrained problems with a relaxation of order two may
contain up to a hundred variables. The implementation is available in Python.
The tool helps solve problems such as finding the ground state energy or
testing quantum correlations.Comment: 17 pages, 3 figures, 1 table, 2 algorithms, the algorithm is
available at http://peterwittek.github.io/ncpol2sdpa
New Dependencies of Hierarchies in Polynomial Optimization
We compare four key hierarchies for solving Constrained Polynomial
Optimization Problems (CPOP): Sum of Squares (SOS), Sum of Diagonally Dominant
Polynomials (SDSOS), Sum of Nonnegative Circuits (SONC), and the Sherali Adams
(SA) hierarchies. We prove a collection of dependencies among these hierarchies
both for general CPOPs and for optimization problems on the Boolean hypercube.
Key results include for the general case that the SONC and SOS hierarchy are
polynomially incomparable, while SDSOS is contained in SONC. A direct
consequence is the non-existence of a Putinar-like Positivstellensatz for
SDSOS. On the Boolean hypercube, we show as a main result that Schm\"udgen-like
versions of the hierarchies SDSOS*, SONC*, and SA* are polynomially equivalent.
Moreover, we show that SA* is contained in any Schm\"udgen-like hierarchy that
provides a O(n) degree bound.Comment: 26 pages, 4 figure
Applications of Finite Model Theory: Optimisation Problems, Hybrid Modal Logics and Games.
There exists an interesting relationships between two seemingly distinct fields: logic from the field of Model Theory, which deals with the truth of statements about discrete structures; and Computational Complexity, which deals with the classification of problems by how much of a particular computer resource is required in order to compute a solution. This relationship is known as Descriptive Complexity and it is the primary application of the tools from Model Theory when they are restricted to the finite; this restriction is commonly called Finite Model Theory.
In this thesis, we investigate the extension of the results of Descriptive Complexity from classes of decision problems to classes of optimisation problems. When dealing with decision problems the natural mapping from true and false in logic to yes and no instances of a problem is used but when dealing with optimisation problems, other features of a logic need to be used. We investigate what these features are and provide results in the form of logical frameworks that can be used for describing optimisation problems in particular classes, building on the existing research into this area.
Another application of Finite Model Theory that this thesis investigates is the relative expressiveness of various fragments of an extension of modal logic called hybrid modal logic. This is achieved through taking the Ehrenfeucht-Fraïssé game from Model Theory and modifying it so that it can be applied to hybrid modal logic. Then, by developing winning strategies for the players in the game, results are obtained that show strict hierarchies of expressiveness for fragments of hybrid modal logic that are generated by varying the quantifier depth and the number of proposition and nominal symbols available
Certified Roundoff Error Bounds using Bernstein Expansions and Sparse Krivine-Stengle Representations
Floating point error is an inevitable drawback of embedded systems
implementation. Computing rigorous upper bounds of roundoff errors is
absolutely necessary to the validation of critical software. This problem is
even more challenging when addressing non-linear programs. In this paper, we
propose and compare two new methods based on Bernstein expansions and sparse
Krivine-Stengle representations, adapted from the field of the global
optimization to compute upper bounds of roundoff errors for programs
implementing polynomial functions. We release two related software package
FPBern and FPKiSten, and compare them with state of the art tools. We show that
these two methods achieve competitive performance, while computing accurate
upper bounds by comparison with other tools.Comment: 20 pages, 2 table
Fuchsian bispectral operators
The aim of this paper is to classify the bispectral operators of any rank
with regular singular points (the infinite point is the most important one). We
characterise them in several ways. Probably the most important result is that
they are all Darboux transformations of powers of generalised Bessel operators
(in the terminology of q-alg/9602011). For this reason they can be effectively
parametrised by the points of a certain (infinite) family of algebraic
manifolds as pointed out in q-alg/9602011.Comment: 32 pages, late
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