Iterative schemes for trifunction hemivariational inequalities

In this paper, we consider and study a new class of hemivariational inequalities, which is called trifunction hemivariational inequality. We suggest and analyze a class of iterative methods for solving trifunction hemivariational inequalities using the auxiliary principle technique. We prove that the convergence of these new methods either requires partially relaxed strongly monotonicity or pseudomonotonicity, which is a weaker condition than monotonicity. Results obtained in this paper include several new and known results as special cases.     

Smoothing Methods for Nonlinear Complementarity Problems

In this paper, we present some new smoothing techniques to solve general nonlinear complementarity problems. Under a weaker condition than monotonicity as on the original problems, we prove convergence of our methods. We also present an error estimate under a general monotonicity condition. Some numerical tests confirm the efficiency of the proposed methods.     

Auxiliary Principle Technique for Equilibrium Problems

In this paper, we use the auxiliary principle technique to suggest and analyze a number of iterative methods for solving mixed quasiequilibrium problems. We prove that the convergence of these new methods requires either partially relaxed strongly monotonicity or peudomonotonicity, which is a weaker condition than monotonicity. Our proof of convergence is very simple as compared with others. These new results include several new and known results as special cases. Our results represent refinement and improvement of the previous known results for equilibrium and variational inequalities problems.     

Projection-proximal methods for general variational inequalities

In this paper, we consider and analyze some new projection-proximal methods for solving general variational inequalities. The modified methods converge for pseudomonotone operators which is a weaker condition than monotonicity. The proposed methods include several new and known methods as special cases. Our results can be considered as a novel and important extension of the previously known results. Since the general variational inequalities include the quasi-variational inequalities and implicit complementarity problems as special cases, results proved in this paper continue to hold for these problems.     

多步Runge-Kutta方法的保单调性

一类重要的常微分方程源自用线方法求解非线性双曲型 偏微分方程,这类常微分方程的解具有单调性, 因此要求数值方法能保持原系统的这种性质.本文研究多步Runge-Kutta方法求解常微分方程初值问题的保单调性.分别获得了多步Runge-Kutta方法是条件单调和无条件单调的充分条件.       

New extragradient-type methods for general variational inequalities

In this paper, we consider and analyze a new class of extragradient-type methods for solving general variational inequalities. The modified methods converge for pseudomonotone operators which is weaker condition than monotonicity. Our proof of convergence is very simple as compared with other methods. The proposed methods include several new and known methods as special cases. Our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems.     

Splitting Algorithms for General Pseudomonotone Mixed Variational Inequalities

In this paper, we suggest and analyze a number of resolvent-splitting algorithms for solving general mixed variational inequalities by using the updating technique of the solution. The convergence of these new methods requires either monotonicity or pseudomonotonicity of the operator. Proof of convergence is very simple. Our new methods differ from the existing splitting methods for solving variational inequalities and complementarity problems. The new results are versatile and are easy to implement.     

On monotonicity and boundedness properties of linear multistep methods

In this paper an analysis is provided of nonlinear monotonicity and boundedness properties for linear multistep methods. Instead of strict monotonicity for arbitrary starting values we shall focus on generalized monotonicity or boundedness with Runge-Kutta starting procedures. This allows many multistep methods of practical interest to be included in the theory. In a related manner, we also consider contractivity and stability in arbitrary norms.

     

Modified resolvent splitting algorithms for general mixed variational inequalities

In this paper, we suggest and analyze a number of four-step resolvent splitting algorithms for solving general mixed variational inequalities by using the updating technique of the solution. The convergence of these new methods requires either monotonicity or pseudomonotonicity of the operator. Proof of convergence is very simple. Our new methods differ from the existing splitting methods for solving variational inequalities and complementarity problems. The new results are versatile and are easy to implement.     

Interacting double monotonicity failure with direction of impact under five voting methods

Monotonicity failure is widely considered a severe pathology in a voting method, and some authors regard a voting method that suffers from this pathology to be totally unacceptable. Of the various voting methods discussed prominently in the literature, five methods are subject to monotonicity failures: Alternative Vote (AV), Plurality Runoff (P-R), and Dodgson’s, Nanson’s, and Coombs’ methods. Two of these methods (AV and P-R) are used in practice. In the 2nd World Congress of the Public Choice Societies conducted in March 2012 in Miami, Fl., Nicholas Miller presented a paper in which he defined, inter alia, a novel type of monotonicity failure which he called ‘double monotonicity failure’ and investigated some of its properties in three-candidate elections under the AV and P-R methods. The present paper extends Miller’s investigation by showing that all the aforementioned voting methods may display double monotonicity failure, as well as by interacting double monotonicity failure with the question of whether the monotonicity failures improve or worsen the apparent welfare of the voters whose votes change. We determine which of the four resulting cases can arise in each of the aforementioned five voting methods.     

A class of new iterative methods for general mixed variational inequalities

In this paper, we use the auxiliary principle technique to suggest a class of predictor-corrector methods for solving general mixed variational inequalities. The convergence of the proposed methods only requires the partially relaxed strongly monotonicity of the operator, which is weaker than co-coercivity. As special cases, we obtain a number of known and new results for solving various classes of variational inequalities and related problems.     

A Nonsmooth Global Optimization Technique Using Slopes: The One-Dimensional Case

In this paper we introduce a pruning technique based on slopes in the context of interval branch-and-bound methods for nonsmooth global optimization. We develop the theory for a slope pruning step which can be utilized as an accelerating device similar to the monotonicity test frequently used in interval methods for smooth problems. This pruning step offers the possibility to cut away a large part of the box currently investigated by the optimization algorithm. We underline the new technique's efficiency by comparing two variants of a global optimization model algorithm: one equipped with the monotonicity test and one equipped with the pruning step. For this reason, we compared the required CPU time, the number of function and derivative or slope evaluations, and the necessary storage space when solving several smooth global optimization problems with the two variants. The paper concludes on the test results for several nonsmooth examples.     

Monotonic core solutions: beyond Young’s theorem

Young’s theorem implies that every core concept violates monotonicity. In this paper, we investigate when such a violation of monotonicity by a given core concept is justified. We introduce a new monotonicity property for core concepts. We pose several open questions for this new property. The open questions arise because the most important core concepts (the nucleolus and the per capita nucleolus) do not satisfy the property even in the class of convex games.     

Some Predictor-Corrector Algorithms for Multivalued Variational Inequalities

In this paper, we use the auxiliary principle technique to suggest a new class of predictor-corrector algorithms for solving multivalued variational inequalities. The convergence of the proposed methods requires only the partially-relaxed strong monotonicity of the operator, which is weaker than cocoercivity. As special cases, we obtain a number of known and new results for solving various classes of variational inequalities.     

Extended Projection Methods for Monotone Variational Inequalities

In this paper, we prove that each monotone variational inequality is equivalent to a two-mapping variational inequality problem. On the basis of this fact, a new class of iterative methods for the solution of nonlinear monotone variational inequality problems is presented. The global convergence of the proposed methods is established under the monotonicity assumption. The conditions concerning the implementability of the algorithms are also discussed. The proposed methods have a close relationship to the Douglas–Rachford operator splitting method for monotone variational inequalities.     

The scorix: A popular representation of votes revisited

In earlier work, some of the present authors have advocated that the search for monotonicity of the votrix, a well-known representation of votes, leads to natural ranking rules. In order to exploit hitherto unconsidered information, we introduced a new representation of votes, the votex, and the search for monotonicity was extended to this representation. Previously, we have focused on representations of votes based on pairwise information and we have left aside another well-known representation of votes based on positional information: the scorix. In this paper, we propose to exploit the monotonicity of the scorix and introduce a new family of ranking rules based on the search for this new type of monotonicity. It is shown that, in case the scorix is monotone, all existing scoring ranking rules give the same ranking as output.     

Monotone Convergence of Iterative Methods for Singular Linear Systems

In this paper, monotonicity of iterative methods for solving general solvable singularly systems is discussed. The monotonicity results given by Berman, Plemmons, and Semal are generalized to singular systems. It is shown that for an iterative method introduced by a nonnegative splitting of the coefficient matrix there exist some initial guesses such that the iterative sequence converges towards a solution of the system from below or from above. The monotonicity of the block Gauss-Seidel method for solving a p-cyclic system and Markov chain is considered.     

Multiscale inference for a multivariate density with applications to X-ray astronomy

In this paper, we propose methods for inference of the geometric features of a multivariate density. Our approach uses multiscale tests for the monotonicity of the density at arbitrary points in arbitrary directions. In particular, a significance test for a mode at a specific point is constructed. Moreover, we develop multiscale methods for identifying regions of monotonicity and a general procedure for detecting the modes of a multivariate density. It is shown that the latter method localizes the modes with an effectively optimal rate. The theoretical results are illustrated by means of a simulation study and a data example. The new method is applied to and motivated by the determination and verification of the position of high-energy sources from X-ray observations by the Swift satellite which is important for a multiwavelength analysis of objects such as Active Galactic Nuclei.     

On nonconvex equilibrium problems

In this paper, we introduce a new class of equilibrium problems, known as mixed quasi nonconvex equilibrium problems. We suggest some iterative schemes for solving nonconvex equilibrium problems by using the auxiliary principle technique. The convergence of the proposed methods either requires partially relaxed strongly monotonicity or pseudomonotonicity. As special cases, we obtain a number of known and new results for solving various classes of equilibrium and variational inequality problems.     

A new predictor-corrector method for noncoercive mixed variational inequalities

In this paper, we use the auxiliary principle technique to suggest and analyze a class of predictor-corrector methods for solving noncoercive mixed variational inequalities. The convergence of the proposed method requires only the partially relaxed strongly monotonicity, which is even weaker than the co-coercivity. As special cases, we obtain a number of new and known results for classical variational inequalities.