An important problem in distance geometry is of determining the position of an unknown point in a given convex set such that its longest distance to a set of finite …
An important problem in distance geometry is of determining the position of an unknown point in a given convex set such that its longest distance to a set of finite number of points is shortest. In this paper we present an algorithm based on subgradient method and convex hull computation for solving this problem. A recent improvement of Quickhull algorithm for computing the convex hull of a finite set of planar points is applied to fasten up the computations in our numerical experiments.
In this paper, we introduce a new iteration method for solving a variational inequality over the fixed point set of a firmly nonexpansive mapping in , where the cost function …
In this paper, we introduce a new iteration method for solving a variational inequality over the fixed point set of a firmly nonexpansive mapping in , where the cost function is continuous and monotone, which is called the projection method. The algorithm is a variant of the subgradient method and projection methods.
Abstract: Given a finite set D of n planar discs whose centers are distributed randomly. We are interested in the expected number of extreme discs of the convex hull of …
Abstract: Given a finite set D of n planar discs whose centers are distributed randomly. We are interested in the expected number of extreme discs of the convex hull of D. We show that the expected number of extreme discs is at most O(log2n) for any distribution. This result can be used to derive expected complexity of convex hull algorithms.
 Keywords: Convex hull, computational geometry, expected number.
 Mathematics Subject Classification (2010): 65D18, 52A15, 51N05.
Abstract: Given a finite set D of n planar discs whose centers are distributed randomly. We are interested in the expected number of extreme discs of the convex hull of …
Abstract: Given a finite set D of n planar discs whose centers are distributed randomly. We are interested in the expected number of extreme discs of the convex hull of D. We show that the expected number of extreme discs is at most O(log2n) for any distribution. This result can be used to derive expected complexity of convex hull algorithms.
 Keywords: Convex hull, computational geometry, expected number.
 Mathematics Subject Classification (2010): 65D18, 52A15, 51N05.
In this paper, we introduce a new iteration method for solving a variational inequality over the fixed point set of a firmly nonexpansive mapping in , where the cost function …
In this paper, we introduce a new iteration method for solving a variational inequality over the fixed point set of a firmly nonexpansive mapping in , where the cost function is continuous and monotone, which is called the projection method. The algorithm is a variant of the subgradient method and projection methods.
An important problem in distance geometry is of determining the position of an unknown point in a given convex set such that its longest distance to a set of finite …
An important problem in distance geometry is of determining the position of an unknown point in a given convex set such that its longest distance to a set of finite number of points is shortest. In this paper we present an algorithm based on subgradient method and convex hull computation for solving this problem. A recent improvement of Quickhull algorithm for computing the convex hull of a finite set of planar points is applied to fasten up the computations in our numerical experiments.
article Free Access Share on A New Convex Hull Algorithm for Planar Sets Author: William F. Eddy Department of Statistics, Carnegie-Mellon University, Schenley Park, Pittsburgh, PA Department of Statistics, Carnegie-Mellon …
article Free Access Share on A New Convex Hull Algorithm for Planar Sets Author: William F. Eddy Department of Statistics, Carnegie-Mellon University, Schenley Park, Pittsburgh, PA Department of Statistics, Carnegie-Mellon University, Schenley Park, Pittsburgh, PAView Profile Authors Info & Claims ACM Transactions on Mathematical SoftwareVolume 3Issue 4Dec. 1977 pp 398–403https://doi.org/10.1145/355759.355766Online:01 December 1977Publication History 173citation1,590DownloadsMetricsTotal Citations173Total Downloads1,590Last 12 Months148Last 6 weeks24 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF
In this article, we propose a modified Korpelevich's method for solving variational inequalities. Under some mild assumptions, we show that the suggested method converges strongly to the minimum-norm solution of …
In this article, we propose a modified Korpelevich's method for solving variational inequalities. Under some mild assumptions, we show that the suggested method converges strongly to the minimum-norm solution of some variational inequality in an infinite-dimensional Hilbert space.
The theory of Ky Fan minimax inequalities provides a powerful general framework for the study of convex programming, variational inequalities and economic equilibrium problems. One of the fundamental methods for …
The theory of Ky Fan minimax inequalities provides a powerful general framework for the study of convex programming, variational inequalities and economic equilibrium problems. One of the fundamental methods for finding a solution of Ky Fan minimax inequalities is the proximal point algorithm, where a lot of papers have been dedicated to this subject. In this paper, a general class of two-level hierarchical Ky Fan minimax inequalities is introduced in real Hilbert spaces. For a wide class of Bregman functions, an association of inexact implicit Bregman-penalization proximal and Bregman-splitting proximal algorithms are suggested and analysed. Weak and strong convergences are proved under essentially weaker conditions. We conclude this paper with a hierarchical minimization problem and a numerical example.
We make use of the auxiliary problem principle to develop iterative algorithms for solving equilibrium problems. The first one is an extension of the extragradient algorithm to equilibrium problems. In …
We make use of the auxiliary problem principle to develop iterative algorithms for solving equilibrium problems. The first one is an extension of the extragradient algorithm to equilibrium problems. In this algorithm the equilibrium bifunction is not required to satisfy any monotonicity property, but it must satisfy a certain Lipschitz-type condition. To avoid this requirement we propose linesearch procedures commonly used in variational inequalities to obtain projection-type algorithms for solving equilibrium problems. Applications to mixed variational inequalities are discussed. A special class of equilibrium problems is investigated and some preliminary computational results are reported.
In this article, we present a new hybrid extragradient iteration method for finding a common element of the set of fixed points of a nonexpansive mapping and the set of …
In this article, we present a new hybrid extragradient iteration method for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of equilibrium problems for a pseudomonotone and Lipschitz-type continuous bifunction. We obtain strongly convergent theorems for the sequences generated by these processes in a real Hilbert space.
This paper deals with an iterative method, in a real Hilbert space, for approximating a common element of the set of fixed points of a demicontractive operator (possibly quasi-nonexpansive or …
This paper deals with an iterative method, in a real Hilbert space, for approximating a common element of the set of fixed points of a demicontractive operator (possibly quasi-nonexpansive or strictly pseudocontractive) and the set of solutions of a variational inequality problem for a monotone, Lipschitz continuous mapping. The considered algorithm can be regarded as a combination of a variation of the hybrid steepest descent method and the so-called extragradient method. Under classical conditions, we prove the strong convergence of the sequences of iterates given by the considered scheme.
We propose a method for solving bilevel split variational inequalities involving strongly monotone operators in the leader problems and nonexpansive mappings in the follower ones. The proposed method is a …
We propose a method for solving bilevel split variational inequalities involving strongly monotone operators in the leader problems and nonexpansive mappings in the follower ones. The proposed method is a combination between the projection method for variational inequality and the Krasnoselskii–Mann scheme for fixed points of nonexpansive mappings. Strong convergence of the iterative process is proved. Special cases are considered.
We consider a bilevel problem involving two monotone equilibrium bifunctions and we show that this problem can be solved by a proximal point method with generalized proximal distances. We propose …
We consider a bilevel problem involving two monotone equilibrium bifunctions and we show that this problem can be solved by a proximal point method with generalized proximal distances. We propose a framework for the convergence analysis of the sequence generated by the algorithm. This class of problems is very interesting because it covers mathematical programs and optimization problems under equilibrium constraints. As an application, we consider the problem of the stability and change dynamics of a leader-follower relationship in a hierarchical organization.
The author proposes a generalized proximal method for solving an equilibrium problem which consists in finding x 2 K such that F(x, y) 0, for all y 2 K, where …
The author proposes a generalized proximal method for solving an equilibrium problem which consists in finding x 2 K such that F(x, y) 0, for all y 2 K, where K is a nonempty, convex and closed set of a real Hilbert space X, and F:K ×K !Ris a given bifunction with F(x, x) = for all x 2 K. The weak convergence of the sequence generated by the method is proved under the assumptions of monotonicity and convexity, with respect to the second argument y (for every fixed x 2 K), of the bifunction F. Replacing the assumption of monotonicity with the one of strong monotonicity on F, a strong convergence result is obtained. A second strong convergence theorem is proved under the hypotheses of monotonicity and a new assumption of co-Lipschitz continuity at 0 of the operator F. Applications to convex optimization, to the problem of finding a zero of a maximal monotone operator and to Nash equilibria problems are provided.
The paper introduces and analysizes the convergence of two multi-step proximal-like algorithms for pseudomonotone and Lipschitz-type continuous equilibrium problems in a real Hilbert space. The algorithms are combinations between the …
The paper introduces and analysizes the convergence of two multi-step proximal-like algorithms for pseudomonotone and Lipschitz-type continuous equilibrium problems in a real Hilbert space. The algorithms are combinations between the multi-step proximal-like method and Mann or Halpern iterations. The weakly and strongly convergent theorems are established with the prior knowledge of two Lipschitz-type continuous constants. Moreover, by choosing two sequences of suitable stepsizes, we also show that the multi-step proximal-like algorithm for strongly pseudomonotone and Lipschitz-type continuous equilibrium problems where the construction of solution approximations and the establishing of its convergence do not require the prior knowledge of strongly pseudomonotone and Lipschitz-type continuous constants of bifunctions. Finally, several numerical examples are reported to illustrate the convergence and the performance of the proposed algorithms over classical extragradient-like algorithms.
In this paper, we propose new subgradient extragradient methods for finding a solution of a strongly monotone equilibrium problem over the solution set of another monotone equilibrium problem which usually …
In this paper, we propose new subgradient extragradient methods for finding a solution of a strongly monotone equilibrium problem over the solution set of another monotone equilibrium problem which usually is called monotone bilevel equilibrium problem in Hilbert spaces. The first proposed algorithm is based on the subgradient extragradient method presented by Censor et al. [Censor Y, Gibali A, Reich S. The subgradient extragradient method for solving variational inequalities in Hilbert space. J Optim Theory Appl. 2011;148:318–335]. The strong convergence of the algorithm is established under monotone assumptions of the cost bifunctions with Lipschitz-type continuous conditions recently presented by Mastroeni in the auxiliary problem principle. We also present a modification of the algorithm for solving an equilibrium problem, where the constraint domain is the common solution set of another equilibrium problem and a fixed point problem. Several fundamental experiments are provided to illustrate the numerical behaviour of the algorithms and to compare with others.
In this work, we investigate a new extragradient method for finding an element of the set of solutions of the bilevel pseudomonotone variational inequality problem in real Hilbert spaces. We …
In this work, we investigate a new extragradient method for finding an element of the set of solutions of the bilevel pseudomonotone variational inequality problem in real Hilbert spaces. We only use one projection to construct the algorithm and the strong convergence theorem proved under suitable conditions. Numerical experiments illustrate the performances of our new algorithm and present a comparison with related algorithms.
We generalize the existing formulation and results on linear separability of sets. In order to characterize the solution of the generalized problem, we use the concepts of convex hulls. For …
We generalize the existing formulation and results on linear separability of sets. In order to characterize the solution of the generalized problem, we use the concepts of convex hulls. For finite sets, it is well known the Support Vector Machine technique for finding the optimal separating hyperplane. Here we consider arbitrary sets, allowing infinite, unbounded and nonclosed sets. The problem is formulated as an optimization problem with possibly infinitely many constraints. We prove existence and uniqueness of the solution. Besides, we present some examples and counterexamples to many properties discussed in the text and statements in the literature.
In this paper, we propose an algorithm with two inertial term extrapolation steps for solving bilevel equilibrium problem in a real Hilbert space. The inertial term extrapolation step is introduced …
In this paper, we propose an algorithm with two inertial term extrapolation steps for solving bilevel equilibrium problem in a real Hilbert space. The inertial term extrapolation step is introduced to speed up the rate of convergence of the iteration process. Under some sufficient assumptions on the bifunctions involving pseudomonotone and Lipschitz-type conditions, we obtain the strong convergence of the iterative sequence generated by the proposed algorithm. A numerical experiment is performed to illustrate the numerical behavior of the algorithm and also comparison with some other related algorithms in the literature.
Abstract Research in high energy physics (HEP) requires huge amounts of computing and storage, putting strong constraints on the code speed and resource usage. To meet these requirements, a compiled …
Abstract Research in high energy physics (HEP) requires huge amounts of computing and storage, putting strong constraints on the code speed and resource usage. To meet these requirements, a compiled high-performance language is typically used; while for physicists, who focus on the application when developing the code, better research productivity pleads for a high-level programming language. A popular approach consists of combining Python, used for the high-level interface, and C++, used for the computing intensive part of the code. A more convenient and efficient approach would be to use a language that provides both high-level programming and high-performance. The Julia programming language, developed at MIT especially to allow the use of a single language in research activities, has followed this path. In this paper the applicability of using the Julia language for HEP research is explored, covering the different aspects that are important for HEP code development: runtime performance, handling of large projects, interface with legacy code, distributed computing, training, and ease of programming. The study shows that the HEP community would benefit from a large scale adoption of this programming language. The HEP-specific foundation libraries that would need to be consolidated are identified.