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

Lexicographic optimization for the multi-container loading problem with open dimensions for a shoe manufacturer

Motivated by a real-world application, we present a multi-container loading problem with 3-open dimensions. We formulate it as a biobjective mixed-integer nonlinear program with lexicographic objectives in order to reflect the decision maker’s optimization priorities. The first objective is to minimize the number of containers, while the second objective is to minimize the volume of those containers. Besides showing the NP-hardness of this sequential optimization problem, we provide bounds for it which are used in the three proposed algorithms, as well as, on their evaluation when a certificate of optimality is not available. The first is an exact parametric-based approach to tackle the lexicographic optimization through the second objective of the problem. Nevertheless, given that the parametric programs correspond to large nonlinear mixed-integer optimizations, we present a heuristic that is entirely mathematical-programming based. The third algorithm enhances the solution quality of the heuristic. These algorithms are specifically tailored for the real-world application. The effectiveness and efficiency of the devised heuristics is demonstrated with numerical experiments

Designing Volumetric Truss Structures

We present the first algorithm for designing volumetric Michell Trusses. Our method uses a parametrization approach to generate trusses made of structural elements aligned with the primary direction of an object's stress field. Such trusses exhibit high strength-to-weight ratios. We demonstrate the structural robustness of our designs via a posteriori physical simulation. We believe our algorithm serves as an important complement to existing structural optimization tools and as a novel standalone design tool itself

Optimisation of postbuckling stiffened composite structures

The thesis starts off with an introductory chapter on composite materials. This includes a definition of composites, a brief history of composite materials, their use in aerostructures (primarily as stiffened structures), and also optimization of composite structures. A literature review is then presented on postbuckling stiffened structures. This includes both experimental investigations on stiffened composite panels and investigations into secondary instabilities and mode jumping as well as their numerical modelling. Next, the Finite Element (FE) modelling of posthuckling stiffened structures is discussed, relating how ABAQUS models are set up in order to trace stiffened composite panels' buckling and postbuckling responses. An experimental programme conducted on an I-stiffened panel is described, where the panel was tested in compression until collapse. The buckling and postbuckling characteristics of the panel are presented, and then an FE model is described together with its predicted numerical behaviour of the panel's buckling and postbuckling characteristics. Focus then shifts to the modelling of failure in composites, in particular delamination failure. A literature review is conducted, looking at the use of both the Virtual Crack Closure Technique (VCCT) and interface elements in delamination modelling. Two stiffener runout models, representing two specimens previously tested experimentally, are then developed to illustrate how interface elements may be used to model mixed mode delamination. The previously discussed panel is revisited, and a global-local modelling approach used to model the skin-stiffener interface. FE models of a stiffened cylindrical shell are also considered, and again the postbuckling characteristics of the shell are compared with experimental results. . The thesis then moves on to optimization of composite structures. This starts off with a literature review of existing optimization methodologies. A Genetic Algorithm (GA) is devised to increase the damage resistance of the I-stiffened panel. The global-local ABAQUS model discussed earlier is used in conjunction with the GA in order to find a revised stacking sequence of both the panel flanges and skin so as to minimize skin-stiffener debonding subject to a variety of design constraints. A second optimization is then presented, this time linked to the FE model of the stiffened cylindrical shell. The objective is to increase the collapse load of the shell, again subject to specific design constraints. The thesis concludes by summarising the importance of the work conducted. FE models were created and validated against experimental work in order to model a variety of composite stiffened structures in their buckling and postbuckling regimes. These models were able to capture the failure characteristics of these structures relating to delamination at the skin-stiffener interface, a phenomenon widely observed experimentally. Various optimizations, able to account for failure mechanisms which may occur prior to overall structural collapse, were then conducted on the analysed structures in order to obtain more damage resistant designs.Imperial Users onl

A new mathematical model for a 3D container packing problem

Wir betrachten das Problem der Einzelcontainerpackung eines Unternehmens, das seine Kunden bedienen muss, indem es zuerst die Produkte in Kartons legt und diese dann in einen Container lädt. Für dieses Problem entwickeln und lösen wir ein lineares gemischt-ganzzahliges Modell. Unser Modell berücksichtigt geometrische Randbedingungen, beispielsweise Überlappungsverbote, Orientierungs-Bedingungen und Randbedingungen für die relative Positionierung der Kartons. Wir betrachten auch die Erweiterung des Modells durch die Integration der Schwerpunktsabweichung der Packung vom Containermittelpunkt. Das Modell wurde an einer großen Anzahl von realen Instanzen getestet, die bis zu 41 Kartons enthalten. In den meisten Fällen wurden optimale Lösungen erzielt bzw. nah-optimale Lösungen mit beweisbar kleiner Optimalitätslücke.We address the single container packing problem of a company that has to serve its customers by first placing the products in boxes and then loading the boxes into a container. We approach the problem by developing and solving mixed-integer linear models. Our models consider geometric constraints that feature non-overlapping constraints, box orientation constraints, dimensionality constraints, relative packing position constraints, and linearity constraints. We also develop an extension of the models by integrating load balance and the deviation of the center of gravity. We tested the models on a broad set of real instances involving up to 41 boxes and obtained optimal solutions in most cases and very small gaps when optimality could not be proven