Note on the Complexity of the Mixed-Integer Hull of a Polyhedron

We study the complexity of computing the mixed-integer hull $\operatorname{conv}(P\cap\mathbb{Z}^n\times\mathbb{R}^d)$ of a polyhedron $P$. Given an inequality description, with one integer variable, the mixed-integer hull can have exponentially many vertices and facets in $d$. For $n,d$ fixed, we give an algorithm to find the mixed integer hull in polynomial time. Given $P=\operatorname{conv}(V)$ and $n$ fixed, we compute a vertex description of the mixed-integer hull in polynomial time and give bounds on the number of vertices of the mixed integer hull

Equivariant Perturbation in Gomory and Johnson's Infinite Group Problem. VII. Inverse semigroup theory, closures, decomposition of perturbations

In this self-contained paper, we present a theory of the piecewise linear minimal valid functions for the 1-row Gomory-Johnson infinite group problem. The non-extreme minimal valid functions are those that admit effective perturbations. We give a precise description of the space of these perturbations as a direct sum of certain finite- and infinite-dimensional subspaces. The infinite-dimensional subspaces have partial symmetries; to describe them, we develop a theory of inverse semigroups of partial bijections, interacting with the functional equations satisfied by the perturbations. Our paper provides the foundation for grid-free algorithms for the Gomory-Johnson model, in particular for testing extremality of piecewise linear functions whose breakpoints are rational numbers with huge denominators.Comment: 67 pages, 21 figures; v2: changes to sections 10.2-10.3, improved figures; v3: additional figures and minor updates, add reference to IPCO abstract. CC-BY-S

The Triangle Closure is a Polyhedron

Recently, cutting planes derived from maximal lattice-free convex sets have been studied intensively by the integer programming community. An important question in this research area has been to decide whether the closures associated with certain families of lattice-free sets are polyhedra. For a long time, the only result known was the celebrated theorem of Cook, Kannan and Schrijver who showed that the split closure is a polyhedron. Although some fairly general results were obtained by Andersen, Louveaux and Weismantel [ An analysis of mixed integer linear sets based on lattice point free convex sets, Math. Oper. Res. 35 (2010), 233--256] and Averkov [On finitely generated closures in the theory of cutting planes, Discrete Optimization 9 (2012), no. 4, 209--215], some basic questions have remained unresolved. For example, maximal lattice-free triangles are the natural family to study beyond the family of splits and it has been a standing open problem to decide whether the triangle closure is a polyhedron. In this paper, we show that when the number of integer variables $m=2$ the triangle closure is indeed a polyhedron and its number of facets can be bounded by a polynomial in the size of the input data. The techniques of this proof are also used to give a refinement of necessary conditions for valid inequalities being facet-defining due to Cornu\'ejols and Margot [On the facets of mixed integer programs with two integer variables and two constraints, Mathematical Programming 120 (2009), 429--456] and obtain polynomial complexity results about the mixed integer hull.Comment: 39 pages; made self-contained by merging material from arXiv:1107.5068v

Minimizing Cubic and Homogeneous Polynomials over Integers in the Plane

We complete the complexity classification by degree of minimizing a polynomial over the integer points in a polyhedron in $\mathbb{R}^2$. Previous work shows that optimizing a quadratic polynomial over the integer points in a polyhedral region in $\mathbb{R}^2$ can be done in polynomial time, while optimizing a quartic polynomial in the same type of region is NP-hard. We close the gap by showing that this problem can be solved in polynomial time for cubic polynomials. Furthermore, we show that the problem of minimizing a homogeneous polynomial of any fixed degree over the integer points in a bounded polyhedron in $\mathbb{R}^2$ is solvable in polynomial time. We show that this holds for polynomials that can be translated into homogeneous polynomials, even when the translation vector is unknown. We demonstrate that such problems in the unbounded case can have smallest optimal solutions of exponential size in the size of the input, thus requiring a compact representation of solutions for a general polynomial time algorithm for the unbounded case

Finite element analysis solution applications to photoreceptor modules

One of the primary components in a Xerox copier is the print engine. The center of this engine is comprised of a photoreceptor, which is a roller/belt module mounted to a frame. The belt revolves around the module acquiring and transposing toner to sheets of paper as they come into contact with the module. The initial design of these modules can often lead to registration and print quality problems later in the assembly and application phases of design. The current analysis procedure includes lengthy commercial FEA packages that require high designer investment. For this reason, many new ideas are never given the opportunity to develop. The implementation of a low investment analysis step which is designed to reveal problems with a design\u27s general formulation could save the corporation both time and money. The means of statically approximating designs before they are modeled in commercial FEA packages could allow for more module configurations to be analyzed and considered. This low investment means of approximation has been developed here. A user friendly Excel spreadsheet based generic photoreceptor module analyzer is derived, explained, and correlated in the ensuing analysis. Although approximate, the ability to compare designs and choose the best one for the application makes this analysis successful. The generic modeling capability is automated such that user interaction is minimal and navigation is relatively simple. Also included in this thesis is a step by step instruction set for inputting module parameters and running the program. A Nastran FEA model was constructed and correlated to this solver, which was shown to retain the correct order of magnitude (micron level) and overall deformation shape. Future adjustments and other software capabilities are also discussed

Geology, Mantle Tomography, and Inclination Corrected Paleogeographic Trajectories Support Westward Subduction During Cretaceous Orogenesis in the North American Cordillera

Geological evidence, including the presence of two passive margin platforms, juxtaposed and mismatched deformation between North America and more outboard terranes, as well as the lack of rift deposits, suggest that North America was the lower plate during both the Sevier and Laramide events and that subduction dipped westward beneath the Cordilleran Ribbon Continent (Rubia). Terranes within the composite ribbon continent, now present in the Canadian Cordillera, collided with western North America during the 125–105 Ma Sevier event and were transported northward during the ~80–58 Ma Laramide event, which affected the Cordillera from South America to Alaska. New high-resolution mantle tomography beneath North America reveals a huge slab wall that extends vertically for over 1000 km, marks the site of long-lived subduction, and provides independent verification of the westward-dipping subduction model. Other workers analyzed paleogeographic trajectories and concluded that the initial collision took place in Canada at about 160 Ma – a time and place for which there is no deformational thickening on the North American platform – and later farther west where subduction was not likely westward, but eastward. However, by utilizing a meridionally corrected North American paleogeographic trajectory, coupled with the geologically most reasonable location for the initial deformation, the position of western North America with respect to the relict superslab parsimoniously accounts for the timing and extents of both the Sevier and Laramide events. SOMMAIRELes indications géologiques, en particulier la présence de deux marges de plateforme passives, de déformations adjacentes non-conformes entre l’Amérique du Nord et les terranes extérieurs, ainsi que l’absence de gisements de rift, permet de croire que l’Amérique du Nord était la plaque sous-jacente durant les événements de Sevier et de Laramide et que la subduction plongeait vers l’ouest sous le continent rubané de la Cordillères (Rubia).  Les terranes du continent rubané composite, maintenant au sein de la Cordillère canadienne, sont entrés en collision avec l’ouest de l’Amérique du Nord durant l’événement Sevier (125-105 Ma), et ont été transportés vers le nord durant l’événement Laramide (~80–58 Ma), laquelle a affecté la Cordillère, de l’Amérique du Sud à l’Alaska.  Une nouvelle tomographie haute résolution du manteau sous l’Amérique du Nord montre la présence d’un gigantesque mur de plaques vertical qui s’étend sur 1 000 km, marque le site d’une subduction de longue haleine, et offre une validation indépendante du modèle d’une subduction à pendage vers l’ouest.  D’autres chercheurs ont analysé les trajectoires paléogéographiques et conclu que la collision initiale s’est produite au Canada vers 160 Ma – un moment et un endroit sans épaississement par déformation sur la plateforme d’Amérique du Nord – et plus tard plus à l’ouest, là où la subduction n’était vraisemblablement pas vers l’ouest, mais vers l’est.  Cela dit, en considérant une trajectoire paléogéographique de l’Amérique du Nord corrigée longitudinalement, avec la position géologique la plus probable de la déformation initiale, la position de la portion ouest de l’Amérique du Nord par rapport aux restes de la super-plaque explique alors facilement la chronologie et l’étendue des épisodes Sevier et Laramide

Continuous Equality Knapsack with Probit-Style Objectives

We study continuous, equality knapsack problems with uniform separable, non-convex objective functions that are continuous, strictly increasing, antisymmetric about a point, and have concave and convex regions. For example, this model captures a simple allocation problem with the goal of optimizing an expected value where the objective is a sum of cumulative distribution functions of identically distributed normal distributions (i.e., a sum of inverse probit functions). We prove structural results of this model under general assumptions and provide two algorithms for efficient optimization: (1) running in linear time and (2) running in a constant number of operations given preprocessing of the objective function

Arc and Slab-Failure Magmatism in Cordilleran Batholiths I – The Cretaceous Coastal Batholith of Peru and its Role in South American Orogenesis and Hemispheric Subduction Flip

We examined the temporal and spatial relations of rock units within the Western Cordillera of Peru where two Cretaceous basins, the Huarmey-Cañete and the West Peruvian Trough, were considered by previous workers to represent western and eastern parts respectively of the same marginal basin. The Huarmey-Cañete Trough, which sits on Mesoproterozoic basement of the Arequipa block, was filled with up to 9 km of Tithonian to Albian tholeiitic–calc-alkaline volcanic and volcaniclastic rocks. It shoaled to subaerial eastward. At 105–101 Ma the rocks were tightly folded and intruded during and just after the deformation by a suite of 103 ± 2 Ma mafic intrusions, and later in the interval 94–82 Ma by probable subduction-related plutons of the Coastal batholith. The West Peruvian Trough, which sits on Paleozoic metamorphic basement, comprised a west-facing siliciclastic-carbonate platform and adjacent basin filled with up to 5 km of sandstone, shale, marl and thinly bedded limestone deposited continuously throughout the Cretaceous. Rocks of the West Peruvian Trough were detached from their basement, folded and thrust eastward during the Late Cretaceous–Early Tertiary. Because the facies and facing directions of the two basins are incompatible, and their development and subjacent basements also distinct, the two basins could not have developed adjacent to one another.     Based on thickness, composition and magmatic style, we interpret the magmatism of the Huarmey-Cañete Trough to represent a magmatic arc that shut down at about 105 Ma when the arc collided with an unknown terrane. We relate subsequent magmatism of the early 103 ± 2 Ma syntectonic mafic intrusions and dyke swarms to slab failure. The Huarmey-Cañete-Coastal batholithic block and its Mesoproterozoic basement remained offshore until 77 ± 5 Ma when it collided with, and was emplaced upon, the partially subducted western margin of South America to form the east-vergent Marañon fold–thrust belt. A major pulse of 73–62 Ma plutonism and dyke emplacement followed terminal collision and is interpreted to have been related to slab failure of the west-dipping South American lithosphere. Magmatism, 53 Ma and younger, followed terminal collision and was generated by eastward subduction of Pacific oceanic lithosphere beneath South America.    Similar spatial and temporal relations exist over the length of both Americas and represent the terminal collision of an arc-bearing ribbon continent with the Americas during the Late Cretaceous–Early Tertiary Laramide event. It thus separated long-standing westward subduction from the younger period of eastward subduction characteristic of today. We speculate that the Cordilleran Ribbon Continent formed during the Mesozoic over a major zone of downwelling between Tuzo and Jason along the boundary of Panthalassic and Pacific oceanic plates.SOMMAIRENous avons étudié les relations spatiales et temporales des unités de roches dans la portion ouest de la Cordillère du Pérou, où deux bassins crétacés, la fosse d’accumulation de Huarmey-Cañete et la fosse d’accumulation péruvienne de l’ouest, ont été perçues par des auteurs précédents comme les portions ouest et est d’un même bassin de marge.  La fosse de Huarmey-Cañete, qui repose sur le socle mésoprotérozoïque du bloc d’Arequipa, a été comblée par des couches de roches volcaniques tholéitiques – calco-alcalines de l’Albien au Thithonien atteignant 9 km d’épaisseur.  Vers l’est, l’ensemble a fini par former des hauts fonds.  Vers 105 à 101 Ma, les roches ont été plissées fortement puis recoupées par une suite d’intrusions vers 103 ± 2 Ma, durant et juste après la déformation, et plus tard dans l’intervalle 94 – 82 Ma, probablement par des plutons de subduction  du batholite côtier.  Quant à la fosse d’accumulation péruvienne de l’ouest, elle repose sur un socle métamorphique paléozoïque, et elle est constituée d’une plateforme silicoclastique – carbonate à pente ouest et d’un bassin contigu comblé par des grès, des schistes, des marnes et des calcaires finement laminés atteignant 5 km d’épaisseur et qui se sont déposés en continu durant tout le Crétacé.  Les roches de la fosse d’accumulation péruvienne de l’ouest ont été décollées de leur socle, plissées et charriées vers l’est durant la fin du Crétacé et le début du Tertiaire.  Parce que les facies et les profondeurs de sédimentation de ces deux fosses d’accumulation dont incompatibles, et que leur développement et leur socle sont différents, ces deux fosses ne peuvent pas s’être développées côte à côte.     À cause de l’épaisseur accumulée, de sa composition et du style de son magmatisme, nous pensons que la fosse d’accumulation de Huarmey-Cañete représente un arc magmatique qui s’est éteinte vers 105 Ma, lorsque l’arc est entré en collision avec un terrane inconnu.  Nous pensons que le magmatisme subséquent aux premières intrusions mafiques syntectoniques  et aux réseaux de dykes de 103 ± 2 Ma sont à mettre au compte d’une rupture de plaque.  Le bloc Huarmey-Cañete-batholitique côtier et son socle mésoprotérozoïque sont demeurés au large jusqu’à  77 ± 5 Ma, moment où il est entré en collision et a été poussé par-dessus la marge ouest sud-américaine partiellement subduite, pour ainsi former la zone de chevauchement de vergence est de Marañon.  Nous croyons que la séquence majeure de plutonisme et d’intrusion de dykes qui a succédé à la collision finale à 73–62 Ma doit être reliée à une  rupture de la plaque lithosphérique sud-américaine à pendage ouest.  Le magmatisme de 53 Ma et plus récent qui a succédé à la collision finale, a été généré par la subduction vers l’est de la lithosphère océanique du Pacifique sous l’Amérique du Sud.     Des relations temporelles et spatiales similaires qui existent tout le long des deux Amériques représentent la collision terminale d’un ruban continental d’arcs avec les Amériques durant la phase tectonique laramienne de la fin du Crétacé–début du Tertiaire.  Elle a donc séparé la subduction vers l’ouest de longue date de la période de subduction vers l’est plus jeune caractérisant la situation actuelle.  Nous considérons que le ruban continental de la Cordillère s’est constitué durant le Mésozoïque au-dessus d’une zone majeure de convection descendante entre Tuzo et Jason, le long de la limite entre les plaques océaniques Panthalassique et Pacifique