19,387 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
Recommended from our members
Design, implementation and testing of an integrated branch and bound algorithm for piecewise linear and discrete programming problems within an LP framework
A number of discrete variable representations are well accepted and find regular use within LP systems. These are Binary variables, General Integer variables, Variable Upper Bounds or Semi Continuous variables, Special Ordered Sets of type One and type Two. The FortLP system has been extended to include these representations. A Branch and Bound algorithm is designed in which the choice of sub-problems and branching variables are kept general. This provides considerable scope of experimentation with tree development heuristics and the tree search can then be guided by search parameters specified by user subroutines. The data structures for representing the variables and the definition of the branch and bound tree are described. The results of experimental investigation for a few test problems are reported
Nonlinear Integer Programming
Research efforts of the past fifty years have led to a development of linear
integer programming as a mature discipline of mathematical optimization. Such a
level of maturity has not been reached when one considers nonlinear systems
subject to integrality requirements for the variables. This chapter is
dedicated to this topic.
The primary goal is a study of a simple version of general nonlinear integer
problems, where all constraints are still linear. Our focus is on the
computational complexity of the problem, which varies significantly with the
type of nonlinear objective function in combination with the underlying
combinatorial structure. Numerous boundary cases of complexity emerge, which
sometimes surprisingly lead even to polynomial time algorithms.
We also cover recent successful approaches for more general classes of
problems. Though no positive theoretical efficiency results are available, nor
are they likely to ever be available, these seem to be the currently most
successful and interesting approaches for solving practical problems.
It is our belief that the study of algorithms motivated by theoretical
considerations and those motivated by our desire to solve practical instances
should and do inform one another. So it is with this viewpoint that we present
the subject, and it is in this direction that we hope to spark further
research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G.
Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50
Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art
Surveys, Springer-Verlag, 2009, ISBN 354068274
Optimising a nonlinear utility function in multi-objective integer programming
In this paper we develop an algorithm to optimise a nonlinear utility
function of multiple objectives over the integer efficient set. Our approach is
based on identifying and updating bounds on the individual objectives as well
as the optimal utility value. This is done using already known solutions,
linear programming relaxations, utility function inversion, and integer
programming. We develop a general optimisation algorithm for use with k
objectives, and we illustrate our approach using a tri-objective integer
programming problem.Comment: 11 pages, 2 tables; v3: minor revisions, to appear in Journal of
Global Optimizatio
- …