1,351 research outputs found
Tight Polyhedral Representations of Discrete Sets Using Projections, Simplices, and Base-2 Expansions
This research effort focuses on the acquisition of polyhedral outer-approximations to the convex hull of feasible solutions for mixed-integer linear and mixed-integer nonlinear programs. The goal is to produce desirable formulations that have superior size and/or relaxation strength. These two qualities often have great influence on the success of underlying solution strategies, and so it is with these qualities in mind that the work of this dissertation presents three distinct contributions. The first studies a family of relatively unknown polytopes that enable the linearization of polynomial expressions involving two discrete variables. Projections of higher-dimensional convex hulls are employed to reduce the dimensionality of the requisite linearizing polyhedra. For certain lower dimensions, a complete characterization of the convex hull is obtained; for others, a family of facets is acquired. Furthermore, a novel linearization for the product of a bounded continuous variable and a general discrete variable is obtained. The second contribution investigates the use of simplicial facets in the formation of novel convex hull representations for a class of mixed-discrete problems having a subset of their variables taking on discrete, affinely independent realizations. These simplicial facets provide new theoretical machinery necessary to extend the reformulation-linearization technique (RLT) for mixed-binary and mixed-discrete programs. In doing so, new insight is provided which allows for the subsumation of previous mixed-binary and mixed-discrete RLT results. The third contribution presents a novel approach for representing functions of discrete variables and their products using logarithmic numbers of 0-1 variables in order to economize on the number of these binary variables. Here, base-2 expansions are used within linear restrictions to enforce the appropriate behavior of functions of discrete variables. Products amongst functions are handled by scaling these linear restrictions. This approach provides insight into, improves upon, and subsumes recent related linearization methods from the literature
Polyhedral Approximations of Quadratic Semi-Assignment Problems, Disjunctive Programs, and Base-2 Expansions of Integer Variables
This research is concerned with developing improved representations for special families of mixed-discrete programming problems. Such problems can typically be modeled using different mathematical forms, and the representation employed can greatly influence the problem\u27s ability to be solved. Generally speaking, it is desired to obtain mixed 0-1 linear forms whose continuous relaxations provide tight polyhedral outer-approximations to the convex hulls of feasible solutions. This dissertation makes contributions to three distinct problems, providing new forms that improve upon published works. The first emphasis is on devising solution procedures for the classical quadratic semi-assignment problem(QSAP), which is an NP-hard 0-1 quadratic program. The effort begins by using a reformulation-linearization technique to recast the problem as a mixed 0-1 linear program. The resulting form provides insight into identifying special instances that are readily solvable. For the general case, the form is shown to have a tight continuous relaxation, as well as to possess a decomposable structure. Specifically, a Hamiltonian decomposition of a graph interpretation is devised to motivate a Lagrangian dual whose subproblems consist of families of separable acyclic minimum-cost network flows. The result is an efficient approach for computing tight lower bounds on the optimal objective value to the original discrete program. Extensive computational experience is reported to evaluate the tightness of the representation and the expedience of the algorithm. The second contribution uses disjunctive programming arguments to model the convex hull of the union of a finite collection of polytopes. It is well known that the convex hull of the union of n polytopes can be obtained by lifting the problem into a higher-dimensional space using n auxiliary continuous (scaling) variables. When placed within a larger optimization problem, these variables must be restricted to be binary. This work examines an approach that uses fewer binary variables. The same scaling technique is employed, but the variables are treated as continuous by introducing a logarithmic number of new binary variables and constraints. The scaling variables can now be substituted from the problem. Moreover, an emphasis of this work, is that specially structured polytopes lead to well-defined projection operations that yield more concise forms. These special polytopes consist of knapsack problems having SOS-1 and SOS-2 type restrictions. Different projections are defined for the SOS-2 case, leading to forms that serve to both explain and unify alternative representations for piecewise-linear functions, as well as to promote favorable computational experience. The third contribution uses minimal cover and set covering inequalities to define the previously unknown convex hulls of special sets of binary vectors that are lexicographically lower and upper bounded by given vectors. These convex hulls are used to obtain ideal representations for base-2 expansions of bounded integer variables, and also afford a new perspective on, and extend convex hull results for, binary knapsack polytopes having weakly super-decreasing coefficients. Computational experience for base-2 expansions of integer variables exhibits a reduction in effort
Necessary and sufficient conditions for non-perturbative equivalences of large N orbifold gauge theories
Large N coherent state methods are used to study the relation between U(N)
gauge theories containing adjoint representation matter fields and their
orbifold projections. The classical dynamical systems which reproduce the large
N limits of the quantum dynamics in parent and daughter orbifold theories are
compared. We demonstrate that the large N dynamics of the parent theory,
restricted to the subspace invariant under the orbifold projection symmetry,
and the large N dynamics of the daughter theory, restricted to the untwisted
sector invariant under "theory space'' permutations, coincide. This implies
equality, in the large N limit, between appropriately identified connected
correlation functions in parent and daughter theories, provided the orbifold
projection symmetry is not spontaneously broken in the parent theory and the
theory space permutation symmetry is not spontaneously broken in the daughter.
The necessity of these symmetry realization conditions for the validity of the
large N equivalence is unsurprising, but demonstrating the sufficiency of these
conditions is new. This work extends an earlier proof of non-perturbative large
N equivalence which was only valid in the phase of the (lattice regularized)
theories continuously connected to large mass and strong coupling.Comment: 21 page, JHEP styl
Improved Mixed-Integer Models of a Two-Dimensional Cutting Stock Problem
This paper is concerned with a family of two-dimensional cutting stock problems that seeks to cut rectangular regions from a finite collection of sheets in such a manner that the minimum number of sheets is used. A fixed number of rectangles are to be cut, with each rectangle having a known length and width. All sheets are rectangular, and have the same dimension. We review two known mixed-integer mathematical formulations, and then provide new representations that both economize on the number of discrete variables and tighten the continuous relaxations. A key consideration that arises repeatedly in all models is the enforcement of disjunctions that a vector must lie in the union of a finite collection of polytopes. Computational results demonstrate a relative performance of the different formulations
Polynomial Ensembles and Recurrence Coefficients
Polynomial ensembles are determinantal point processes associated with (non
necessarily orthogonal) projections onto polynomial subspaces. The aim of this
survey article is to put forward the use of recurrence coefficients to obtain
the global asymptotic behavior of such ensembles in a rather simple way. We
provide a unified approach to recover well-known convergence results for real
OP ensembles. We study the mutual convergence of the polynomial ensemble and
the zeros of its average characteristic polynomial; we discuss in particular
the complex setting. We also control the variance of linear statistics of
polynomial ensembles and derive comparison results, as well as asymptotic
formulas for real OP ensembles. Finally, we reinterpret the classical algorithm
to sample determinantal point processes so as to cover the setting of
non-orthogonal projection kernels. A few open problems are also suggested.Comment: 23 page
Matrix String Theory and its Moduli Space
The correspondence between Matrix String Theory in the strong coupling limit
and IIA superstring theory can be shown by means of the instanton solutions of
the former. We construct the general instanton solutions of Matrix String
Theory which interpolate between given initial and final string configurations.
Each instanton is characterized by a Riemann surface of genus h with n
punctures, which is realized as a plane curve. We study the moduli space of
such plane curves and find out that, at finite N, it is a discretized version
of the moduli space of Riemann surfaces: instead of 3h-3+n its complex
dimensions are 2h-3+n, the remaining h dimensions being discrete. It turns out
that as tends to infinity, these discrete dimensions become continuous, and
one recovers the full moduli space of string interaction theory.Comment: 30 pages, LaTeX, JHEP.cls class file, minor correction
Higher Spin N=8 Supergravity
The product of two N=8 supersingletons yields an infinite tower of massless
states of higher spin in four dimensional anti de Sitter space. All the states
with spin s > 1/2 correspond to generators of Vasiliev's super higher spin
algebra shs^E (8|4) which contains the D=4, N=8 anti de Sitter superalgebra
OSp(8|4). Gauging the higher spin algebra and introducing a matter multiplet in
a quasi-adjoint representation leads to a consistent and fully nonlinear
equations of motion as shown sometime ago by Vasiliev. We show the embedding of
the N=8 AdS supergravity equations of motion in the full system at the
linearized level and discuss the implications for the embedding of the
interacting theory. We furthermore speculate that the boundary N=8 singleton
field theory yields the dynamics of the N=8 AdS supergravity in the bulk,
including all higher spin massless fields, in an unbroken phase of M-theory.Comment: 64 pages, latex, considerably expanded version, submitted for
publicatio
Theory of weakly nonlinear self sustained detonations
We propose a theory of weakly nonlinear multi-dimensional self sustained
detonations based on asymptotic analysis of the reactive compressible
Navier-Stokes equations. We show that these equations can be reduced to a model
consisting of a forced, unsteady, small disturbance, transonic equation and a
rate equation for the heat release. In one spatial dimension, the model
simplifies to a forced Burgers equation. Through analysis, numerical
calculations and comparison with the reactive Euler equations, the model is
demonstrated to capture such essential dynamical characteristics of detonations
as the steady-state structure, the linear stability spectrum, the
period-doubling sequence of bifurcations and chaos in one-dimensional
detonations and cellular structures in multi- dimensional detonations
Finite volume solution of the compressible boundary-layer equations
A box-type finite volume discretization is applied to the integral form of the compressible boundary layer equations. Boundary layer scaling is introduced through the grid construction: streamwise grid lines follow eta = y/h = const., where y is the normal coordinate and h(x) is a scale factor proportional to the boundary layer thickness. With this grid, similarity can be applied explicity to calculate initial conditions. The finite volume method preserves the physical transparency of the integral equations in the discrete approximation. The resulting scheme is accurate, efficient, and conceptually simple. Computations for similar and non-similar flows show excellent agreement with tabulated results, solutions computed with Keller's Box scheme, and experimental data
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