979 research outputs found
QuickXsort - A Fast Sorting Scheme in Theory and Practice
QuickXsort is a highly efficient in-place sequential sorting scheme that
mixes Hoare's Quicksort algorithm with X, where X can be chosen from a wider
range of other known sorting algorithms, like Heapsort, Insertionsort and
Mergesort. Its major advantage is that QuickXsort can be in-place even if X is
not. In this work we provide general transfer theorems expressing the number of
comparisons of QuickXsort in terms of the number of comparisons of X. More
specifically, if pivots are chosen as medians of (not too fast) growing size
samples, the average number of comparisons of QuickXsort and X differ only by
-terms. For median-of- pivot selection for some constant , the
difference is a linear term whose coefficient we compute precisely. For
instance, median-of-three QuickMergesort uses at most comparisons.
Furthermore, we examine the possibility of sorting base cases with some other
algorithm using even less comparisons. By doing so the average-case number of
comparisons can be reduced down to for a remaining
gap of only comparisons to the known lower bound (while using only
additional space and time overall).
Implementations of these sorting strategies show that the algorithms
challenge well-established library implementations like Musser's Introsort
A Taxonomy of Constraints in Simulation-Based Optimization
The types of constraints encountered in black-box and simulation-based
optimization problems differ significantly from those treated in nonlinear
programming. We introduce a characterization of constraints to address this
situation. We provide formal definitions for several constraint classes and
present illustrative examples in the context of the resulting taxonomy. This
taxonomy, denoted QRAK, is useful for modeling and problem formulation, as well
as optimization software development and deployment. It can also be used as the
basis for a dialog with practitioners in moving problems to increasingly
solvable branches of optimization
Derivative-free optimization methods
In many optimization problems arising from scientific, engineering and
artificial intelligence applications, objective and constraint functions are
available only as the output of a black-box or simulation oracle that does not
provide derivative information. Such settings necessitate the use of methods
for derivative-free, or zeroth-order, optimization. We provide a review and
perspectives on developments in these methods, with an emphasis on highlighting
recent developments and on unifying treatment of such problems in the
non-linear optimization and machine learning literature. We categorize methods
based on assumed properties of the black-box functions, as well as features of
the methods. We first overview the primary setting of deterministic methods
applied to unconstrained, non-convex optimization problems where the objective
function is defined by a deterministic black-box oracle. We then discuss
developments in randomized methods, methods that assume some additional
structure about the objective (including convexity, separability and general
non-smooth compositions), methods for problems where the output of the
black-box oracle is stochastic, and methods for handling different types of
constraints
Bayesian optimization under mixed constraints with a slack-variable augmented Lagrangian
An augmented Lagrangian (AL) can convert a constrained optimization problem
into a sequence of simpler (e.g., unconstrained) problems, which are then
usually solved with local solvers. Recently, surrogate-based Bayesian
optimization (BO) sub-solvers have been successfully deployed in the AL
framework for a more global search in the presence of inequality constraints;
however, a drawback was that expected improvement (EI) evaluations relied on
Monte Carlo. Here we introduce an alternative slack variable AL, and show that
in this formulation the EI may be evaluated with library routines. The slack
variables furthermore facilitate equality as well as inequality constraints,
and mixtures thereof. We show how our new slack "ALBO" compares favorably to
the original. Its superiority over conventional alternatives is reinforced on
several mixed constraint examples.Comment: 24 pages, 5 figure
Simultaneous Sensing Error Recovery and Tomographic Inversion Using an Optimization-based Approach
Tomography can be used to reveal internal properties of a 3D object using any
penetrating wave. Advanced tomographic imaging techniques, however, are
vulnerable to both systematic and random errors associated with the
experimental conditions, which are often beyond the capabilities of the
state-of-the-art reconstruction techniques such as regularizations. Because
they can lead to reduced spatial resolution and even misinterpretation of the
underlying sample structures, these errors present a fundamental obstacle to
full realization of the capabilities of next-generation physical imaging. In
this work, we develop efficient and explicit recovery schemes of the most
common experimental error: movement of the center of rotation during the
experiment. We formulate new physical models to capture the experimental setup,
and we devise new mathematical optimization formulations for reliable inversion
of complex samples. We demonstrate and validate the efficacy of our approach on
synthetic data under known perturbations of the center of rotation
Error Analysis in Nuclear Density Functional Theory
Nuclear density functional theory (DFT) is the only microscopic, global
approach to the structure of atomic nuclei. It is used in numerous
applications, from determining the limits of stability to gaining a deep
understanding of the formation of elements in the universe or the mechanisms
that power stars and reactors. The predictive power of the theory depends on
the amount of physics embedded in the energy density functional as well as on
efficient ways to determine a small number of free parameters and solve the DFT
equations. In this article, we discuss the various sources of uncertainties and
errors encountered in DFT and possible methods to quantify these uncertainties
in a rigorous manner.Comment: 18 pages, 3 figures, 4 tables; Invited paper for the Journal of
Physics G: Nuclear and Particle Physics focus section entitled "Enhancing the
interaction between nuclear experiment and theory through information and
statistics"; Revised version after comments by the referees: Figure 1 and
Table 4 have been correcte
Exploiting Symmetry Reduces the Cost of Training QAOA
A promising approach to the practical application of the Quantum Approximate
Optimization Algorithm (QAOA) is finding QAOA parameters classically in
simulation and sampling the solutions from QAOA with optimized parameters on a
quantum computer. Doing so requires repeated evaluations of QAOA energy in
simulation. We propose a novel approach for accelerating the evaluation of QAOA
energy by leveraging the symmetry of the problem. We show a connection between
classical symmetries of the objective function and the symmetries of the terms
of the cost Hamiltonian with respect to the QAOA energy. We show how by
considering only the terms that are not connected by symmetry, we can
significantly reduce the cost of evaluating the QAOA energy. Our approach is
general and applies to any known subgroup of symmetries and is not limited to
graph problems. Our results are directly applicable to nonlocal QAOA
generalization RQAOA. We outline how available fast graph automorphism solvers
can be leveraged for computing the symmetries of the problem in practice. We
implement the proposed approach on the MaxCut problem using a state-of-the-art
tensor network simulator and a graph automorphism solver on a benchmark of 48
graphs with up to 10,000 nodes. Our approach provides an improvement for
on of the graphs considered, with a median speedup of , on a
benchmark where of the graphs are known to be hard for automorphism
solvers.Comment: minor revisio
Adaptive Sampling Quasi-Newton Methods for Derivative-Free Stochastic Optimization
We consider stochastic zero-order optimization problems, which arise in
settings from simulation optimization to reinforcement learning. We propose an
adaptive sampling quasi-Newton method where we estimate the gradients of a
stochastic function using finite differences within a common random number
framework. We employ modified versions of a norm test and an inner product
quasi-Newton test to control the sample sizes used in the stochastic
approximations. We provide preliminary numerical experiments to illustrate
potential performance benefits of the proposed method.Comment: 7 pages, NeurIPS worksho
Robust Learning of Trimmed Estimators via Manifold Sampling
We adapt a manifold sampling algorithm for the nonsmooth, nonconvex
formulations of learning that arise when imposing robustness to outliers
present in the training data. We demonstrate the approach on objectives based
on trimmed loss. Empirical results show that the method has favorable scaling
properties. Although savings in time come at the expense of not certifying
optimality, the algorithm consistently returns high-quality solutions on the
trimmed linear regression and multiclass classification problems tested.Comment: In ICML 2018 Workshop on Modern Trends in Nonconvex Optimization for
Machine Learnin
Origins and optimization of entanglement in plasmonically coupled quantum dots
A system of two or more quantum dots interacting with a dissipative plasmonic
nanostructure is investigated in detail by using a cavity quantum
electrodynamics approach with a model Hamiltonian. We focus on determining and
understanding system configurations that generate multiple bipartite quantum
entanglements between the occupation states of the quantum dots. These
configurations include allowing for the quantum dots to be asymmetrically
coupled to the plasmonic system. Analytical solution of a simplified limit for
an arbitrary number of quantum dots and numerical simulations and optimization
for the two- and three-dot cases are used to develop guidelines for maximizing
the bipartite entanglements. For any number of quantum dots, we show that
through simple starting states and parameter guidelines, one quantum dot can be
made to share a strong amount of bipartite entanglement with all other quantum
dots in the system, while entangling all other pairs to a lesser degree.Comment: 34 pages, 7 figure
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