14,790 research outputs found
Best Subset Selection via a Modern Optimization Lens
In the last twenty-five years (1990-2014), algorithmic advances in integer
optimization combined with hardware improvements have resulted in an
astonishing 200 billion factor speedup in solving Mixed Integer Optimization
(MIO) problems. We present a MIO approach for solving the classical best subset
selection problem of choosing out of features in linear regression
given observations. We develop a discrete extension of modern first order
continuous optimization methods to find high quality feasible solutions that we
use as warm starts to a MIO solver that finds provably optimal solutions. The
resulting algorithm (a) provides a solution with a guarantee on its
suboptimality even if we terminate the algorithm early, (b) can accommodate
side constraints on the coefficients of the linear regression and (c) extends
to finding best subset solutions for the least absolute deviation loss
function. Using a wide variety of synthetic and real datasets, we demonstrate
that our approach solves problems with in the 1000s and in the 100s in
minutes to provable optimality, and finds near optimal solutions for in the
100s and in the 1000s in minutes. We also establish via numerical
experiments that the MIO approach performs better than {\texttt {Lasso}} and
other popularly used sparse learning procedures, in terms of achieving sparse
solutions with good predictive power.Comment: This is a revised version (May, 2015) of the first submission in June
201
Microbial metabolism: optimal control of uptake versus synthesis
Microbes require several complex organic molecules for growth. A species may
obtain a required factor by taking up molecules released by other species or by
synthesizing the molecule. The patterns of uptake and synthesis set a flow of
resources through the multiple species that create a microbial community. This
article analyzes a simple mathematical model of the tradeoff between uptake and
synthesis. Key factors include the influx rate from external sources relative
to the outflux rate, the rate of internal decay within cells, and the cost of
synthesis. Aspects of demography also matter, such as cellular birth and death
rates, the expected time course of a local resource flow, and the associated
lifespan of the local population. Spatial patterns of genetic variability and
differentiation between populations may also strongly influence the evolution
of metabolic regulatory controls of individual species and thus the structuring
of microbial communities. The widespread use of optimality approaches in recent
work on microbial metabolism has ignored demography and genetic structure
The Discrete Dantzig Selector: Estimating Sparse Linear Models via Mixed Integer Linear Optimization
We propose a novel high-dimensional linear regression estimator: the Discrete
Dantzig Selector, which minimizes the number of nonzero regression coefficients
subject to a budget on the maximal absolute correlation between the features
and residuals. Motivated by the significant advances in integer optimization
over the past 10-15 years, we present a Mixed Integer Linear Optimization
(MILO) approach to obtain certifiably optimal global solutions to this
nonconvex optimization problem. The current state of algorithmics in integer
optimization makes our proposal substantially more computationally attractive
than the least squares subset selection framework based on integer quadratic
optimization, recently proposed in [8] and the continuous nonconvex quadratic
optimization framework of [33]. We propose new discrete first-order methods,
which when paired with state-of-the-art MILO solvers, lead to good solutions
for the Discrete Dantzig Selector problem for a given computational budget. We
illustrate that our integrated approach provides globally optimal solutions in
significantly shorter computation times, when compared to off-the-shelf MILO
solvers. We demonstrate both theoretically and empirically that in a wide range
of regimes the statistical properties of the Discrete Dantzig Selector are
superior to those of popular -based approaches. We illustrate that
our approach can handle problem instances with p = 10,000 features with
certifiable optimality making it a highly scalable combinatorial variable
selection approach in sparse linear modeling
State-of-the-art in aerodynamic shape optimisation methods
Aerodynamic optimisation has become an indispensable component for any aerodynamic design over the past 60 years, with applications to aircraft, cars, trains, bridges, wind turbines, internal pipe flows, and cavities, among others, and is thus relevant in many facets of technology. With advancements in computational power, automated design optimisation procedures have become more competent, however, there is an ambiguity and bias throughout the literature with regards to relative performance of optimisation architectures and employed algorithms. This paper provides a well-balanced critical review of the dominant optimisation approaches that have been integrated with aerodynamic theory for the purpose of shape optimisation. A total of 229 papers, published in more than 120 journals and conference proceedings, have been classified into 6 different optimisation algorithm approaches. The material cited includes some of the most well-established authors and publications in the field of aerodynamic optimisation. This paper aims to eliminate bias toward certain algorithms by analysing the limitations, drawbacks, and the benefits of the most utilised optimisation approaches. This review provides comprehensive but straightforward insight for non-specialists and reference detailing the current state for specialist practitioners
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