The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the two-dimensional Quickhull algorithm with the general-dimension Beneath-Beyond Algorithm. It is similar to the randomized, incremental algorithms for convex hull and delaunay triangulation. We provide empirical evidence that the algorithm runs faster when the input contains nonextreme points and that it used less memory. computational geometry algorithms have traditionally assumed that input sets are well behaved. When an algorithm is implemented with floating-point arithmetic, this assumption can lead to serous errors. We briefly describe a solution to this problem when computing the convex hull in two, three, or four dimensions. The output is a set of “thick” facets that contain all possible exact convex hulls of the input. A variation is effective in five or more dimensions.

/pdf/the-quickhull-algorithm-for-convex-hulls-ug134bwaff.pdf

The quickhull algorithm for convex hulls

The explicit linear quadratic regulator for constrained systems

Introduction to linear and nonlinear programming

A new era of theoretical computer science addresses fundamental problems about auctions, networks, and human behavior.

http://www.maths.lse.ac.uk/Personal/stengel/TEXTE/agt-stengel.pdf

Algorithmic Game Theory

Direct search methods are best known as unconstrained optimization techniques that do not explicitly use derivatives. Direct search methods were formally proposed and widely applied in the 1960s but fell out of favor with the mathematical optimization community by the early 1970s because they lacked coherent mathematical analysis. Nonetheless, users remained loyal to these methods, most of which were easy to program, some of which were reliable. In the past fifteen years, these methods have seen a revival due, in part, to the appearance of mathematical analysis, as well as to interest in parallel and distributed com- puting. This review begins by briefly summarizing the history of direct search methods and considering the special properties of problems for which they are well suited. Our focus then turns to a broad class of methods for which we provide a unifying framework that lends itself to a variety of convergence results. The underlying principles allow general- ization to handle bound constraints and linear constraints. We also discuss extensions to problems with nonlinear constraints.

https://www.cs.wm.edu/~va/research/sirev.pdf

Optimization by Direct Search: New Perspectives on Some Classical and Modern Methods ∗

Reverse search for enumeration

We present a new pivot-based algorithm which can be used with minor modification for the enumeration of the facets of the convex hull of a set of points, or for the enumeration of the vertices of an arrangement or of a convex polyhedron, in arbitrary dimension. The algorithm has the following properties:(a)Virtually no additional storage is required beyond the input data.(b)The output list produced is free of duplicates.(c)The algorithm is extremely simple, requires no data structures, and handles all degenerate cases.(d)The running time is output sensitive for nondegenerate inputs.(e)The algorithm is easy to parallelize efficiently.

For example, the algorithm finds thev vertices of a polyhedron inRd defined by a nondegenerate system ofn inequalities (or, dually, thev facets of the convex hull ofn points inRd, where each facet contains exactlyd given points) in timeO(ndv) andO(nd) space. Thev vertices in a simple arrangement ofn hyperplanes inRd can be found inO(n2dv) time andO(nd) space complexity. The algorithm is based on inverting finite pivot algorithms for linear programming.

/pdf/a-pivoting-algorithm-for-convex-hulls-and-vertex-enumeration-67riaix85l.pdf

A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra

The double description method is a simple and useful algorithm for enumerating all extreme rays of a general polyhedral cone in ℝd, despite the fact that we can hardly state any interesting theorems on its time and space complexities. In this paper, we reinvestigate this method, introduce some new ideas for efficient implementations, and show some empirical results indicating its practicality in solving highly degenerate problems.

Double Description Method Revisited

We study several known volume computation algorithms for convex d-polytopes by classifying them into two classes, triangulation methods and signed-decomposition methods. By incorporating the detection of simplicial faces and a storing/reusing scheme for face volumes we propose practical and theoretical improvements for two of the algorithms. Finally we present a hybrid method combining advantages from the two algorithmic classes. The behaviour of the algorithms is theoretically analysed for hypercubes and practically tested on a wide range of polytopes, where the new hybrid method proves to be superior.

/pdf/exact-volume-computation-for-polytopes-a-practical-study-1uekjs63l3.pdf

Exact Volume Computation for Polytopes: A Practical Study

https://cs.mcgill.ca/~fukuda/download/paper/minksum031007jsc.pdf

Komei Fukuda

Papers

Reverse search for enumeration

A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra

Double Description Method Revisited

Exact Volume Computation for Polytopes: A Practical Study

From the Zonotope Construction to the Minkowski Addition of Convex Polytopes