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Orthogonal convex hull
About: Orthogonal convex hull is a research topic. Over the lifetime, 1211 publications have been published within this topic receiving 27535 citations.
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TL;DR: P can be chosen to I&E the centroid oC the triangle formed by X, y and z and Express each si E S in polar coordinates th origin P and 8 = 0 in the direction of zu~ arhitnry fixed half-line L from P.
1,741 citations
TL;DR: In this paper, a reformulation technique is presented that takes a given linear zero-one programming problem, converts it into a zero one polynomial programming problem and then relinearizes it into an extended linear program.
Abstract: In this paper a reformulation technique is presented that takes a given linear zero-one programming problem, converts it into a zero-one polynomial programming problem, and then relinearizes it into an extended linear program. It is shown that the strength of the resulting reformulation depends on the degree of the terms used to produce the polynomial program at the intermediate step of this method. In fact, as this degree varies from one up to the number of variables in the problem, a hierarchy of sharper representations is obtained with the final relaxation representing the convex hull of feasible solutions. The reformulation technique readily extends to produce a similar hierarchy of linear relaxations for zero-one polynomial programming problems. A characterization of the convex hull in the original variable space is also available through a projection process. The structure of this convex hull characterization (or its other relaxations) can be exploited to generate strong or facetial valid inequaliti...
926 citations
TL;DR: Various methods of finding points from the intersection of sets, using projection on to a separate set as an elementary operation are considered, and the strong convergence of the sequences obtained is proved.
Abstract: MANY mathematical and applied problems can be reduced to finding some common point of a system (finite or infinite) of convex sets. Usually each of the sets is such that it is not difficult to find the projection of any point on to this set. In this paper we shall consider various methods of finding points from the intersection of sets, using projection on to a separate set as an elementary operation. The strong convergence of the sequences obtained in this way is proved. Applications are given to various problems, including the problem of best approximation and problems of optimal control. Particular attention is paid in the latter case to problems with restrictions on the phase coordinates.
771 citations
TL;DR: The presented algorithms use the “divide and conquer” technique and recursively apply a merge procedure for two nonintersecting convex hulls to ensure optimal time complexity within a multiplicative constant.
Abstract: The convex hulls of sets of n points in two and three dimensions can be determined with O(n log n) operations. The presented algorithms use the “divide and conquer” technique and recursively apply a merge procedure for two nonintersecting convex hulls. Since any convex hull algorithm requires at least O(n log n) operations, the time complexity of the proposed algorithms is optimal within a multiplicative constant.
731 citations
692 citations