20,039 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
On orbital allotments for geostationary satellites
The following satellite synthesis problem is addressed: communication satellites are to be allotted positions on the geostationary arc so that interference does not exceed a given acceptable level by enforcing conservative pairwise satellite separation. A desired location is specified for each satellite, and the objective is to minimize the sum of the deviations between the satellites' prescribed and desired locations. Two mixed integer programming models for the satellite synthesis problem are presented. Four solution strategies, branch-and-bound, Benders' decomposition, linear programming with restricted basis entry, and a switching heuristic, are used to find solutions to example synthesis problems. Computational results indicate the switching algorithm yields solutions of good quality in reasonable execution times when compared to the other solution methods. It is demonstrated that the switching algorithm can be applied to synthesis problems with the objective of minimizing the largest deviation between a prescribed location and the corresponding desired location. Furthermore, it is shown that the switching heuristic can use no conservative, location-dependent satellite separations in order to satisfy interference criteria
The Voice of Optimization
We introduce the idea that using optimal classification trees (OCTs) and
optimal classification trees with-hyperplanes (OCT-Hs), interpretable machine
learning algorithms developed by Bertsimas and Dunn [2017, 2018], we are able
to obtain insight on the strategy behind the optimal solution in continuous and
mixed-integer convex optimization problem as a function of key parameters that
affect the problem. In this way, optimization is not a black box anymore.
Instead, we redefine optimization as a multiclass classification problem where
the predictor gives insights on the logic behind the optimal solution. In other
words, OCTs and OCT-Hs give optimization a voice. We show on several realistic
examples that the accuracy behind our method is in the 90%-100% range, while
even when the predictions are not correct, the degree of suboptimality or
infeasibility is very low. We compare optimal strategy predictions of OCTs and
OCT-Hs and feedforward neural networks (NNs) and conclude that the performance
of OCT-Hs and NNs is comparable. OCTs are somewhat weaker but often
competitive. Therefore, our approach provides a novel insightful understanding
of optimal strategies to solve a broad class of continuous and mixed-integer
optimization problems
Estimating Maximally Probable Constrained Relations by Mathematical Programming
Estimating a constrained relation is a fundamental problem in machine
learning. Special cases are classification (the problem of estimating a map
from a set of to-be-classified elements to a set of labels), clustering (the
problem of estimating an equivalence relation on a set) and ranking (the
problem of estimating a linear order on a set). We contribute a family of
probability measures on the set of all relations between two finite, non-empty
sets, which offers a joint abstraction of multi-label classification,
correlation clustering and ranking by linear ordering. Estimating (learning) a
maximally probable measure, given (a training set of) related and unrelated
pairs, is a convex optimization problem. Estimating (inferring) a maximally
probable relation, given a measure, is a 01-linear program. It is solved in
linear time for maps. It is NP-hard for equivalence relations and linear
orders. Practical solutions for all three cases are shown in experiments with
real data. Finally, estimating a maximally probable measure and relation
jointly is posed as a mixed-integer nonlinear program. This formulation
suggests a mathematical programming approach to semi-supervised learning.Comment: 16 page
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