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Computing Convex Hulls with a Linear Solver [article]

Florence Benoy and Andy King and Fred Mesnard
2003 arXiv   pre-print
The method enables the computations of convex hulls that are required for polyhedral analysis to be coded with linear constraint solving machinery that is available in many Prolog systems.  ...  A programming tactic involving polyhedra is reported that has been widely applied in the polyhedral analysis of (constraint) logic programs.  ...  We have presented a Prolog program for computing convex hulls using linear solver machinery.  ... 
arXiv:cs/0311002v1 fatcat:mgghnvolhzbmtbmi36zzyk2oqe

Perspective Reformulation and Applications [chapter]

Oktay Günlük, Jeff Linderoth
2011 IMA Volumes in Mathematics and its Applications  
The survey concludes with comments and computations comparing various algorithmic techniques for solving perspective reformulations.  ...  This preprocessing technique is applicable to cases where the MINLP contains binary indicator variables that force continuous decision variables to take the value 0, or to belong to a convex set.  ...  The Convex Hull of a Point and a Convex Set.  ... 
doi:10.1007/978-1-4614-1927-3_3 fatcat:whiomnk4ajahtmjppg4spe3eou

Convex integer optimization with Frank-Wolfe methods [article]

Deborah Hendrych and Hannah Troppens and Mathieu Besançon and Sebastian Pokutta
2022 arXiv   pre-print
In particular, we will study the algorithmic consequences of optimizing with a branch-and-bound approach where the subproblem is solved over the convex hull of the mixed-integer feasible set thanks to  ...  This novel approach computes feasible solutions while working on a single representation of the polyhedral constraints, leveraging the full extent of Mixed-Integer Programming (MIP) solvers without an  ...  Our approach is shown in Figure2cand Figure2d, optimizing over the convex hull with the help of the MIP solver.  ... 
arXiv:2208.11010v4 fatcat:xne4wzi5jjfqhfwpho7f6seyz4

New Insights On Differential And Linear Bounds Using Mixed Integer Linear Programming (Full Version) [article]

Anubhab Baksi
2020 IACR Cryptology ePrint Archive  
In fact, we choose two heuristics from the convex hull modelling. The first uses all the inequalities of a convex hull, while second uses a reduced number of inequalities.  ...  The Convex Hull (CH) modelling, introduced by Sun et al.  ...  [22] , where the convex hull (which is actually a concept in computation geometry [7, 16] ) is used to form the linear constraints.  ... 
dblp:journals/iacr/Baksi20 fatcat:atwzfkghp5aohndozwvexc2zfy

The Bernstein Basis and its Applications in Solving Geometric Constraint Systems

Sebti Foufou, Dominique Michelucci
2012 Reliable Computing  
They are compatible with standard preconditioning methods and fit linear programming techniques. However, current Bernstein-based solvers are limited to small algebraic systems.  ...  We present Bernstein polytopes and show how combining them with linear programming allows us to solve larger systems as well.  ...  Acknowledgements This research work has been funded by NPRP grant number NPRP 09 − 906 − 1 − 137 from the Qatar National Research Fund (a member of the Qatar Foundation).  ... 
dblp:journals/rc/FoufouM12 fatcat:vmw3lj4nkzdftbnptubjijax3m

A greedy algorithm for yield surface approximation

Jérémy Bleyer, Patrick de Buhan
2013 Comptes rendus. Mecanique  
The proposed algorithm constructs an approximation using a convex hull of ellipsoids such that the approximate criterion can be formulated in terms of second-order conic constraints.  ...  This Note presents an approximation method for convex yield surfaces in the framework of yield design theory.  ...  -the resolution of a linear programming problem is the only stage requiring a specific solver.  ... 
doi:10.1016/j.crme.2013.06.003 fatcat:7obfbcfetzdrrih56jhdvl64tu

A Hough Transform Approach to Solving Linear Min-Max Problems [article]

Carmi Grushko
2012 arXiv   pre-print
We also present an algorithm for solving such problems in the 2D case, which is superior to CGAL's linear programming solver, both in performance and in stability.  ...  Several ways to accelerate the solution of 2D/3D linear min-max problems in n constraints are discussed.  ...  A convex hull of a set of points P is unique; one way to see this is to recall one definition of CH (P ) -the intersection of all convex sets that contain P .  ... 
arXiv:1205.5911v1 fatcat:smhc54jcxvai7nistas6ifvx7a

Fully Automatic Trunk Packing with Free Placements [article]

Ernst Althaus, Peter Hachenberger
2010 arXiv   pre-print
We present a new algorithm to compute the volume of a trunk according to the SAE J1100 standard.  ...  Our new algorithm uses state-of-the-art methods from computational geometry and from combinatorial optimization. It finds better solutions than previous approaches for small trunks.  ...  With six machines like ours it is possible to compute the feasible areas and a few sets of their simplified representations on the first half of a day.  ... 
arXiv:1008.3773v1 fatcat:6ybbx3voc5hnflp5vgwy2n6zl4

SAT Modulo Monotonic Theories

Sam Bayless, Noah Bayless, Holger Hoos, Alan Hu
., convex hulls). These constraints arise in problems that are otherwise difficult for SAT solvers to handle, such as procedural content generation.  ...  Boolean satisfiability (SAT) solvers have been successfully applied to a wide variety of difficult combinatorial problems.  ...  Our research has been supported by the use of computing resources provided by WestGrid and Compute/Calcul Canada, and by funding provided by the Natural Sciences and Engineering Research Council of Canada  ... 
doi:10.1609/aaai.v29i1.9755 fatcat:ezcflwu74bhfxoczd6h53itp7q

Overcoming the Convex Barrier for Simplex Inputs

Harkirat Singh Behl, M. Pawan Kumar, Philip H. S. Torr, Krishnamurthy Dvijotham
2021 Neural Information Processing Systems  
We prove a somewhat surprising result that, in this case, not only can one design a tight relaxation that overcomes the convex barrier, but the size of the relaxation remains linear in the number of neurons  ...  Recent progress in neural network verification has challenged the notion of a convex barrier, that is, an inherent weakness in the convex relaxation of the output of a neural network.  ...  Acknowledgements and Disclosure of Funding Harkirat was supported using a Tencent studentship through the University of Oxford. Philip H.S.  ... 
dblp:conf/nips/BehlKTD21 fatcat:ulgofy6bhjfv5hgkrzcseqbd5e

Pseudo basic steps: bound improvement guarantees from Lagrangian decomposition in convex disjunctive programming

Dimitri J. Papageorgiou, Francisco Trespalacios
2017 EURO Journal on Computational Optimization  
provable bounds on this improvement along with techniques to exploit this information when solving a disjunctive program as a convex MINLP.  ...  In this paper, using properties of a convex disjunctive program's hull reformulation and multipliers from Lagrangian decomposition, we introduce an operation called a pseudo basic step and use it to provide  ...  Acknowledgments We wish to thank Ignacio Grossmann, Nick Sawaya, Myun-Seok Cheon, and Ahmet Keha for their feedback on a preliminary manuscript.  ... 
doi:10.1007/s13675-017-0088-0 fatcat:viotmq3cwfclvgkdfoangvissm

The Nearest Polyhedral Convex Conic Regions for High-Dimensional Classification

2020 Turkish Journal of Electrical Engineering and Computer Sciences  
In the nearest-convex-model type classifiers, each class in the training set is approximated with a convex 4 class model, and a test sample is assigned to a class based on the shortest distance from the  ...  To this end, we approximate each class region with a polyhedral convex conic 7 region by utilizing polyhedral conic functions (PCFs) and its extension, Extended PCFs (EPCFs).  ...  A linear subspace class model of a particular class c is the linear subspace spanned by its training samples, i.e., Convex Class Models Model Parameters Solver Type Linear Subspace U c (computed offline  ... 
doi:10.3906/elk-2005-142 fatcat:tv4fwgirmfaupggv2w5ewctsi4

Computational implementation of non-linear convex hull reformulation

Nicolas W. Sawaya, Ignacio E. Grossmann
2007 Computers and Chemical Engineering  
k j J ∈ , each containing a series of equations and/or inequalities representing the constraints of the problem, connected together by the logical OR operator ( ) ∨ that enforces the contents of only one  ...  It is therefore clear that a new method is needed to correctly implement the non-linear convex hull formulation from a computational perspective.  ...  NUMERICAL EXAMPLES All example problems were solved with GAMS [4] v.21.7 on a 2.8 GHz Pentium IV PC (512 MB of RAM) using the SBB solver. Example 1.  ... 
doi:10.1016/j.compchemeng.2006.08.002 fatcat:r5bvmjrkgvc3fhu3dd4hrwo6t4

A Physics-Constrained Data-Driven Approach Based on Locally Convex Reconstruction for Noisy Database [article]

Qizhi He, Jiun-Shyan Chen
2020 arXiv   pre-print
A new data-driven simulation approach coupled with a locally convex reconstruction, termed the local convexity data-driven (LCDD) computing, is proposed to enhance accuracy and robustness against noise  ...  Numerical tests demonstrated that LCDD enhances nearly one order of accuracy compared to the standard distance-minimization data-driven scheme when dealing with noisy database, and a linear exactness is  ...  by the convex hull of the selected k-NN points.  ... 
arXiv:1907.12651v3 fatcat:qmhap7di4vhgthih3gv2qilz3u

Convex power flow models for scalable electricity market modelling

Frederik Geth, Reinhilde D'Hust, Dirk Van Hertem
2018 Zenodo  
Convex relaxation and linear approximation are two such approaches to manage computational tractability.  ...  is linearized with respect to these initial values;  sine function is linearized (sin(θ) = θ);  cosine function, as in the QC formulation, is modelled through a convex hull.  ...  The main relaxation steps are:  convex hulls of sine function and cosine function;  convex hull of quadratic terms;  convex hull of multiplication through McCormick's envelopes.  ... 
doi:10.5281/zenodo.1321189 fatcat:a4nwvo2gmbbsrdyagss22qvaje
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