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Semidefinite representation of convex hulls of rational varieties [article]

Didier Henrion
2011 arXiv   pre-print
Using elementary duality properties of positive semidefinite moment matrices and polynomial sum-of-squares decompositions, we prove that the convex hull of rationally parameterized algebraic varieties  ...  by quadratics; and (c) hypersurfaces parameterized by bivariate quartics; all in an ambient space of arbitrary dimension.  ...  I am grateful to Geert Wouters for mentioning reference [13] on oriented projective geometry, and to Bernd Sturmfels for pointing out an error in the coefficients of my original lifted LMI formulation  ... 
arXiv:0901.1821v3 fatcat:xot7f6pcqrd7xp4dcmll37qzna

On semidefinite representations of plane quartics [article]

Didier Henrion
2008 arXiv   pre-print
Conditions are given for the convex hull of a plane quartic to be exactly semidefinite representable with at most 12 lifting variables.  ...  This note focuses on the problem of representing convex sets as projections of the cone of positive semidefinite matrices, in the particular case of sets generated by bivariate polynomials of degree four  ...  Acknowledgments This work benefited from exchanges with Roland Hildebrand, Bill Helton, Jean-Bernard Lasserre and Jiawang Nie.  ... 
arXiv:0809.1826v2 fatcat:fatxnt6ajnfz5o4egz2v7xcpye

Some thoughts on the development of integer programming during my research career – lecture delivered upon receiving the EURO Gold Medal, July 9, 2001, Rotterdam

Egon Balas
2002 European Journal of Operational Research  
We used MIPO to demonstrate that a particular version of disjunctive cuts, which we called lift-and-project cuts, derived in a subspace, then lifted and strengthened, when judiciously combined with branch  ...  which we called lift and project, the reaction was quite different.  ... 
doi:10.1016/s0377-2217(02)00172-8 fatcat:6qb73ktj6jhd3ewa6texfeadwe

Reachability Analysis Using Polygonal Projections [chapter]

Mark R. Greenstreet, Ian Mitchell
1999 Lecture Notes in Computer Science  
Coho represents high-dimensional objects using projections onto planes corresponding to pairs of variables.  ...  This representation is compact and allows e cient algorithms from computational geometry to be exploited while also capturing dependencies in the behaviour of related variables.  ...  Numerical computation The numeric phase of a time step begins by loading a polygon and its convex hull for each projection of the system.  ... 
doi:10.1007/3-540-48983-5_12 fatcat:2uw3tlopnfg5jmivc3psbviq3q

Kinematic Quantification of Gait Asymmetry Based on Characteristics of Angle-Angle Diagrams

2014 Acta Polytechnica Hungarica  
The novel methods are based on the area of the convex hull of the hip-hip diagram or knee-knee diagram, and can be used as an additional method for the determination of gait symmetry of the bipedal walking  ...  The angle-angle diagrams were created for ten patients with leg length inequality (LLI), to quantify gait symmetry before and after an application of the heel lift.  ...  The tables compare the area of convex hulls and symmetry indexes before the application of the heel lift and after the application of the heel lift.  ... 
doi:10.12700/aph.11.05.2014.05.2 fatcat:xr3p243gdfbzxfvlebr4zpkqdi

Erdős-Szekeres and Testing Weak epsilon-Nets are NP-hard in 3 dimensions - and what now? [article]

Christian Knauer, Daniel Werner
2011 arXiv   pre-print
This answers a question by Dobkin, Edelsbrunner and Overmars from 1990. As a corollary, we derive a similar result for the closely related problem of testing weak epsilon-nets in R^3.  ...  We consider the computational versions of the Erd\H os-Szekeres theorem and related problems in 3 dimensions.  ...  Acknowledgements We thank Günter Rote and Nabil Mustafa for pointing us out to [4] and [5] , respectively.  ... 
arXiv:1111.5979v1 fatcat:yokgypobnbe47oic75mx6c6nge

Spectrahedral Lifts of Convex Sets [article]

Rekha R. Thomas
2018 arXiv   pre-print
The main result is that projection representations of a convex set are controlled by factorizations, through closed convex cones, of an operator that comes from the convex set.  ...  It is well-known that sometimes, a complicated convex set can be expressed as the projection of a much simpler set in higher dimensions called a lift of the original set.  ...  I am indebted to all my collaborators on the projects that contributed to this paper. I thank Pablo Parrilo for the construction in Example 1.5 and for several useful conversations.  ... 
arXiv:1803.08079v1 fatcat:t4zjo5nbpzdw3eldn4d6ldlo2i

Regular triangulations of dynamic sets of points

Marc Vigo, Núria Pla, Josep Cotrina
2002 Computer Aided Geometric Design  
Regular triangulations are a generalization of Delaunay triangulations that maintain both their relationship with convex hulls and with Voronoi diagrams.  ...  The algorithms we propose for insertion and deletion are based on a geometrical interpretation of the history data structure in one more dimension and use lifted flips as the unique topological operation  ...  Acknowledgments Research of the authors has been partially supported by the CICYT under grants TIC-98-0586-603-01 and TIC-99-1230-102-02. References [1] F. Aurenhammer.  ... 
doi:10.1016/s0167-8396(01)00082-6 fatcat:u7rukq77uzegdf6g3xn2btn3qe

Convex Hull Formulations for Mixed-Integer Multilinear Functions [article]

Harsha Nagarajan, Kaarthik Sundar, Hassan Hijazi, Russell Bent
2018 arXiv   pre-print
In this paper, we present convex hull formulations for a mixed-integer, multilinear term/function (MIMF) that features products of multiple continuous and binary variables.  ...  We develop two equivalent convex relaxations of an MIMF and study their polyhedral properties in their corresponding higher-dimensional spaces.  ...  Acknowledgements The work was funded by the Center for Nonlinear Studies (CNLS) at LANL and the LANL's directed research and development project "POD: A Polyhedral Outer-approximation, Dynamic-discretization  ... 
arXiv:1807.11007v1 fatcat:rqtxufyexvagfgsg5wo5eegaqu

Inferring argument size relationships with CLP( $$\mathcal{R}$$ ) [chapter]

Florence Benoy, Andy King
1997 Lecture Notes in Computer Science  
To the best of our knowledge, and despite its simplicity, the analysis derives relationships to an accuracy that is either comparable or better than any existing technique.  ...  The analysis is based on polyhedral approximations and uses a simple relaxation technique to calculate least upper bounds and a delay method to improve the precision of widening.  ...  hence P C can be derived through projection. 2 In terms of implementation, the chief technicality is in constructing the equa- tions A i Y i i B i from A i X i B i .  ... 
doi:10.1007/3-540-62718-9_12 fatcat:czkewlaunbcw7k23bhp7ycesoy

Projection, Lifting and Extended Formulation in Integer and Combinatorial Optimization

Egon Balas
2005 Annals of Operations Research  
It discusses the compact representation of the convex hull of a union of polyhedra through extended formulation, the connection between the projection of the latter and the polar of the convex hull, as  ...  In such a case, if the projection cone W is not pointed, then it is not the convex hull of its extreme rays.  ...  to a subspace defined from the LP optimum, and choose a disjunction; • LIFT the disjunctive set to describe its convex hull in a higher dimensional space; • PROJECT the polyhedron describing the convex  ... 
doi:10.1007/s10479-005-3969-1 fatcat:h77rmrcqkvdh5glpxr6tkgrpfi

Ranks of mapping tori via the curve complex [article]

Ian Biringer, Juan Souto
2017 arXiv   pre-print
The -thickened convex hull of f ( X) with respect to Γ, written hull ( f ( X), Γ), is the convex hull of the union of f ( X) and hull (Γ).  ...  If M = H n /Γ, then core (M ) is exactly the projection to M of the -thickened convex hull of Γ. Proof of Proposition 5.8.  ... 
arXiv:1409.1878v2 fatcat:znxsfefzore2rc3caeppsljl4m

P-split formulations: A class of intermediate formulations between big-M and convex hull for disjunctive constraints [article]

Jan Kronqvist and Ruth Misener and Calvin Tsay
2022 arXiv   pre-print
The "P-split" formulations are based on a lifted transformation that splits convex additively separable constraints into P partitions and forms the convex hull of the linearized and partitioned disjunction  ...  The goal of the P-split formulations is to form a strong approximation of the convex hull through a computationally simpler formulation.  ...  The project was also in part financially sponsored by Digital Futures through JK.  ... 
arXiv:2202.05198v1 fatcat:bcyl7xhoovckditvzpzwseauxm

Geometric Aspects of the Space of Triangulations

Pooran Memari
2013 Actes des rencontres du CIRM  
Motivated by practical numerical issues in a number of modeling and simulation problems, we first introduce the notion of a compatible dual complex (made out of convex cells) to a primal triangulation,  ...  revisit the results presented in [21] and [20] (joint works with Patrick Mullen, Fernando De Goes and Mathieu Desbrun).  ...  Then the orthogonal projection of the lower envelope of this lifting partitions the convex hull of X and produces a (full) triangulation coinciding with the Delaunay triangulation of X.  ... 
doi:10.5802/acirm.63 fatcat:s6ozn4ibfjel3hurjc4levfzvq

Unfolding Smooth Prismatoids [article]

Nadia Benbernou, Patricia Cahn, Joseph O'Rourke
2015 arXiv   pre-print
We define a notion for unfolding smooth, ruled surfaces, and prove that every smooth prismatoid (the convex hull of two smooth curves lying in parallel planes), has a nonoverlapping "volcano unfolding.  ...  These unfoldings keep the base intact, unfold the sides outward, splayed around the base, and attach the top to the tip of some side rib.  ...  We benefitted from discussions with Erik Demaine, Meghan Irving, Molly Miller, and Gail Parsloe.  ... 
arXiv:cs/0407063v3 fatcat:5umjaveq2bbh5jgj2qzqcdxpai
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