406 research outputs found
Mixed-Integer Convex Nonlinear Optimization with Gradient-Boosted Trees Embedded
Decision trees usefully represent sparse, high dimensional and noisy data.
Having learned a function from this data, we may want to thereafter integrate
the function into a larger decision-making problem, e.g., for picking the best
chemical process catalyst. We study a large-scale, industrially-relevant
mixed-integer nonlinear nonconvex optimization problem involving both
gradient-boosted trees and penalty functions mitigating risk. This
mixed-integer optimization problem with convex penalty terms broadly applies to
optimizing pre-trained regression tree models. Decision makers may wish to
optimize discrete models to repurpose legacy predictive models, or they may
wish to optimize a discrete model that particularly well-represents a data set.
We develop several heuristic methods to find feasible solutions, and an exact,
branch-and-bound algorithm leveraging structural properties of the
gradient-boosted trees and penalty functions. We computationally test our
methods on concrete mixture design instance and a chemical catalysis industrial
instance
2D Phase Unwrapping via Graph Cuts
Phase imaging technologies such as interferometric synthetic aperture radar (InSAR),
magnetic resonance imaging (MRI), or optical interferometry, are nowadays widespread
and with an increasing usage. The so-called phase unwrapping, which consists in the in-
ference of the absolute phase from the modulo-2π phase, is a critical step in many of their
processing chains, yet still one of its most challenging problems. We introduce an en-
ergy minimization based approach to 2D phase unwrapping. In this approach we address
the problem by adopting a Bayesian point of view and a Markov random field (MRF)
to model the phase. The maximum a posteriori estimation of the absolute phase gives
rise to an integer optimization problem, for which we introduce a family of efficient algo-
rithms based on existing graph cuts techniques. We term our approach and algorithms
PUMA, for Phase Unwrapping MAx flow. As long as the prior potential of the MRF
is convex, PUMA guarantees an exact global solution. In particular it solves exactly all
the minimum L p norm (p ≥ 1) phase unwrapping problems, unifying in that sense, a set
of existing independent algorithms. For non convex potentials we introduce a version of
PUMA that, while yielding only approximate solutions, gives very useful phase unwrap-
ping results. The main characteristic of the introduced solutions is the ability to blindly
preserve discontinuities. Extending the previous versions of PUMA, we tackle denoising by
exploiting a multi-precision idea, which allows us to use the same rationale both for phase
unwrapping and denoising. Finally, the last presented version of PUMA uses a frequency
diversity concept to unwrap phase images having large phase rates. A representative set
of experiences illustrates the performance of PUMA
Efficient Semidefinite Branch-and-Cut for MAP-MRF Inference
We propose a Branch-and-Cut (B&C) method for solving general MAP-MRF
inference problems. The core of our method is a very efficient bounding
procedure, which combines scalable semidefinite programming (SDP) and a
cutting-plane method for seeking violated constraints. In order to further
speed up the computation, several strategies have been exploited, including
model reduction, warm start and removal of inactive constraints.
We analyze the performance of the proposed method under different settings,
and demonstrate that our method either outperforms or performs on par with
state-of-the-art approaches. Especially when the connectivities are dense or
when the relative magnitudes of the unary costs are low, we achieve the best
reported results. Experiments show that the proposed algorithm achieves better
approximation than the state-of-the-art methods within a variety of time
budgets on challenging non-submodular MAP-MRF inference problems.Comment: 21 page
Grey-box Modelling of a Household Refrigeration Unit Using Time Series Data in Application to Demand Side Management
This paper describes the application of stochastic grey-box modeling to
identify electrical power consumption-to-temperature models of a domestic
freezer using experimental measurements. The models are formulated using
stochastic differential equations (SDEs), estimated by maximum likelihood
estimation (MLE), validated through the model residuals analysis and
cross-validated to detect model over-fitting. A nonlinear model based on the
reversed Carnot cycle is also presented and included in the modeling
performance analysis. As an application of the models, we apply model
predictive control (MPC) to shift the electricity consumption of a freezer in
demand response experiments, thereby addressing the model selection problem
also from the application point of view and showing in an experimental context
the ability of MPC to exploit the freezer as a demand side resource (DSR).Comment: Submitted to Sustainable Energy Grids and Networks (SEGAN). Accepted
for publicatio
From combinatorial optimization to real algebraic geometry and back
In this paper, we explain the relations between combinatorial optimization and real algebraic geometry with a special focus to the quadratic assignment problem. We demonstrate how to write a quadratic optimization problem over discrete feasible set as a linear optimization problem over the cone of completely positive matrices. The latter formulation enables a hierarchy of approximations which rely on results from polynomial optimization, a sub-eld of real algebraic geometry
Local Linear Convergence of Approximate Projections onto Regularized Sets
The numerical properties of algorithms for finding the intersection of sets
depend to some extent on the regularity of the sets, but even more importantly
on the regularity of the intersection. The alternating projection algorithm of
von Neumann has been shown to converge locally at a linear rate dependent on
the regularity modulus of the intersection. In many applications, however, the
sets in question come from inexact measurements that are matched to idealized
models. It is unlikely that any such problems in applications will enjoy
metrically regular intersection, let alone set intersection. We explore a
regularization strategy that generates an intersection with the desired
regularity properties. The regularization, however, can lead to a significant
increase in computational complexity. In a further refinement, we investigate
and prove linear convergence of an approximate alternating projection
algorithm. The analysis provides a regularization strategy that fits naturally
with many ill-posed inverse problems, and a mathematically sound stopping
criterion for extrapolated, approximate algorithms. The theory is demonstrated
on the phase retrieval problem with experimental data. The conventional early
termination applied in practice to unregularized, consistent problems in
diffraction imaging can be justified fully in the framework of this analysis
providing, for the first time, proof of convergence of alternating approximate
projections for finite dimensional, consistent phase retrieval problems.Comment: 23 pages, 5 figure
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