变分不等式的极大熵函数方法

利用极大熵函数方法将不等式组及变分不等式的求解问题转化为近似可微优化问题,给出了不等式组及变分不等式问题近似解的可微优化方法,得到了不等式组和变分不等式问题的解集合的示性函数.     

不等式组与变分不等式的极大熵函数方法

利用极大熵函数方法将不等式组及变分不等式的求解问题转化为近似可微优化问题，给出了不等式组及变分不等式问题近似解的可微优化方法，得到了不等式组和变分不等式问题的解集合的示性函数．     

Regression Density Estimation With Variational Methods and Stochastic Approximation

Regression density estimation is the problem of flexibly estimating a response distribution as a function of covariates. An important approach to regression density estimation uses finite mixture models and our article considers flexible mixtures of heteroscedastic regression (MHR) models where the response distribution is a normal mixture, with the component means, variances, and mixture weights all varying as a function of covariates. Our article develops fast variational approximation (VA) methods for inference. Our motivation is that alternative computationally intensive Markov chain Monte Carlo (MCMC) methods for fitting mixture models are difficult to apply when it is desired to fit models repeatedly in exploratory analysis and model choice. Our article makes three contributions. First, a VA for MHR models is described where the variational lower bound is in closed form. Second, the basic approximation can be improved by using stochastic approximation (SA) methods to perturb the initial solution to attain higher accuracy. Third, the advantages of our approach for model choice and evaluation compared with MCMC-based approaches are illustrated. These advantages are particularly compelling for time series data where repeated refitting for one-step-ahead prediction in model choice and diagnostics and in rolling-window computations is very common. Supplementary materials for the article are available online.     

水平集方法与距离函数

讨论了有关水平集方法的基本问题,如保持为距离函数的方法,水平集方程解的存在性和唯一性。主要贡献是证明了,在距离函数约束下,水平集方程在初始零水平集附近有唯一解,它是关于演化界面的有向距离函数。并且用到了一些处理技巧:如注意到原始方程的任意解都是距离函数,将原始方程变化为另一简单形式。由于新的方程组不是一个经典方程组,则它被变换为一个普通形式,其中隐函数方法被采用。     

Some Methods Based on the D-Gap Function for Solving Monotone Variational Inequalities

The D-gap function has been useful in developing unconstrained descent methods for solving strongly monotone variational inequality problems. We show that the D-gap function has certain properties that are useful also for monotone variational inequality problems with bounded feasible set. Accordingly, we develop two unconstrained methods based on them that are similar in spirit to a feasible method of Zhu and Marcotte based on the regularized-gap function. We further discuss a third method based on applying the D-gap function to a regularized problem. Preliminary numerical experience is also reported.     

Variational Methods for Non-Linear Least-Squares

We consider Newton-like line search descent methods for solving non-linear least-squares problems. The basis of our approach is to choose a method, or parameters within a method, by minimizing a variational measure which estimates the error in an inverse Hessian approximation. In one approach we consider sizing methods and choose sizing parameters in an optimal way. In another approach we consider various possibilities for hybrid Gauss-Newton/BFGS methods. We conclude that a simple Gauss-Newton/BFGS hybrid is both efficient and robust and we illustrate this by a range of comparative tests with other methods. These experiments include not only many well known test problems but also some new classes of large residual problem.     

On a Generalized System for Relaxed Cocoercive Variational Inequalities and Projection Methods

Verma introduced a system of nonlinear variational inequalities and proposed projection methods to solve it. This system reduces to a variational inequality problem under certain conditions. So, at least in form, it can be regarded as a extension of a variational inequality problem. In this note, we show that solving this system coincides exactly with solving a variational inequality problem. Therefore, we conclude that it suffices to study the corresponding variational inequalities.This work was supported by the National Natural Science Foundation of China, Grant 10571134.Communicated by M. J. Balas     

Importance Link Function Estimation for Markov Chain Monte Carlo Methods

Abstract

This article focuses on improving estimation for Markov chain Monte Carlo simulation. The proposed methodology is based upon the use of importance link functions. With the help of appropriate importance sampling weights, effective estimates of functionals are developed. The method is most easily applied to irreducible Markov chains, where application is typically immediate. An important conceptual point is the applicability of the method to reducible Markov chains through the use of many-to-many importance link functions. Applications discussed include estimation of marginal genotypic probabilities for pedigree data, estimation for models with and without influential observations, and importance sampling for a target distribution with thick tails.     

On Some Properties of Generalized Proximal Point Methods for Variational Inequalities

We discuss here generalized proximal point methods applied to variational inequality problems. These methods differ from the classical point method in that a so-called Bregman distance substitutes for the Euclidean distance and forces the sequence generated by the algorithm to remain in the interior of the feasible region, assumed to be nonempty. We consider here the case in which this region is a polyhedron (which includes linear and nonlinear programming, monotone linear complementarity problems, and also certain nonlinear complementarity problems), and present two alternatives to deal with linear equality constraints. We prove that the sequences generated by any of these alternatives, which in general are different, converge to the same point, namely the solution of the problem which is closest, in the sense of the Bregman distance, to the initial iterate, for a certain class of operators. This class consists essentially of point-to-point and differentiable operators such that their Jacobian matrices are positive semidefinite (not necessarily symmetric) and their kernels are constant in the feasible region and invariant through symmetrization. For these operators, the solution set of the problem is also a polyhedron. Thus, we extend a previous similar result which covered only linear operators with symmetric and positive-semidefinite matrices.     

On the Application of Accurate Distance Algorithms to Multibody Simulations

Distance algorithms are most frequently used in robotics to determine the distance between two obstacles in the environment of a robot or between a sensor point and an object. We extend the multibody simulation package MOBILE for an application of accurate algorithms for distance computation between objects represented by convex or non-convex polyhedra. These objects are represented by their vertices and oriented facets. As an application example, a multibody system is discussed where a sensor point moves close to a non-convex obstacle. The computed results show that the algorithms developed are suitable for accurate real-time multibody simulations. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Extragradient Methods for Pseudomonotone Variational Inequalities

We consider and analyze some new extragradient-type methods for solving variational inequalities. The modified methods converge for a pseudomonotone operator, which is a much weaker condition than monotonicity. These new iterative methods include the projection, extragradient, and proximal methods as special cases. Our proof of convergence is very simple as compared with other methods.     

Proximal Methods for Mixed Variational Inequalities

A proximal point method for solving mixed variational inequalities is suggested and analyzed by using the auxiliary principle technique. It is shown that the convergence of the proposed method requires only the pseudomonotonicity of the operator, which is a weaker condition than monotonicity. As special cases, we obtain various known and new results for solving variational inequalities and related problems. Our proof of convergence is very simple as compared with other methods.     

强不定问题的变分方法

简要回顾近年来关于强不定问题的变分方法某些研究方面的发展．首先介绍强不定问题,接着叙述建立强不定问题的变分框架的基本思路,进而给出局部凸拓扑线性空间的形变理论,最后陈述几个基于此形变理论的处理强不定问题的临界点定理．这些理论的应用将在后续文章中介绍．     

On Finite Convergence of Iterative Methods for Variational Inequalities in Hilbert Spaces

In a Hilbert space, we study the finite termination of iterative methods for solving a monotone variational inequality under a weak sharpness assumption. Most results to date require that the sequence generated by the method converges strongly to a solution. In this paper, we show that the proximal point algorithm for solving the variational inequality terminates at a solution in a finite number of iterations if the solution set is weakly sharp. Consequently, we derive finite convergence results for the gradient projection and extragradient methods. Our results show that the assumption of strong convergence of sequences can be removed in the Hilbert space case.     

关于Finsler流形中的距离函数与测地球

利用 Finsler流形中的切曲率和旗曲率 ,研究了距离函数与测地球的凸性 ;指出了在单连通完备 Minkowski空间中测地球正好是平面的一部分     

On the Average Taxicab Distance Function and Its Applications

Generalized conics are subsets in the space all of whose points have the same average distance from a given set of points (focal set). The function measuring the average distance is called the average distance function (or the generalized conic function). In general it is a convex function satisfying a kind of growth condition as the preliminary results of Sect. 2 show. Therefore any sublevel set is convex and compact. We can also conclude that such a function has a global minimizer.

The paper is devoted to the special case of the average taxicab distance function given by integration of the taxicab distance on a compact subset of positive Lebesgue measure in the Euclidean coordinate space.

The first application of the average taxicab distance function is related to its minimizer. It is uniquely determined under some natural conditions such as, for example, the connectedness of the integration domain. Geometrically, the minimizer bisects the measure of the integration domain in the sense that each coordinate hyperplane passing through the minimizer cuts the domain into two parts of equal measure. The convexity and the Lipschitzian gradient property allow us to use the gradient descent algorithm that is formulated in terms of a stochastic algorithm (Sect. 3) to find the bisecting point of a set in \(\mathbb{R}^{n}\).

Example 1 in Sect. 4 shows the special form of the average taxicab distance function of a convex polygon. The level curves (generalized conics) admit semidefinite representations as algebraic curves in the plane because the average taxicab distance function is piecewise polynomial of degree at most three.

Some applications in geometric tomography are summarized as our main motivation to investigate the concept of the average taxicab distance function. Its second order partial derivatives give the coordinate \(X\)-rays of the integration domain almost everywhere and vice versa: the average taxicab distance function can be expressed in terms of the coordinate \(X\)-rays. Therefore the reconstruction of the sets given by their coordinate \(X\)-rays can be based on the average taxicab distance function instead of the direct comparison of the \(X\)-rays. In general (especially, in some classes of non-convex sets), the convergence property of the average taxicab distance function with respect to the Hausdorff convergence of the integration domain is better than the convergence of the \(X\)-rays (see regular and \(X\)-regular convergences in Sect. 4.1) and we can apply a standard approximation paradigm (Footnote 1) to solve the problem.

In the last section we prove that any compact convex body is uniquely determined by the diagonalization of its covariogram function measuring the average taxicab distance of the points from the intersection of the body with the translates of its axis parallel bounding box.

     

A Simple Accurate Method for Solving Fractional Variational and Optimal Control Problems

We develop a simple and accurate method to solve fractional variational and fractional optimal control problems with dependence on Caputo and Riemann–Liouville operators. Using known formulas for computing fractional derivatives of polynomials, we rewrite the fractional functional dynamical optimization problem as a classical static optimization problem. The method for classical optimal control problems is called Ritz’s method. Examples show that the proposed approach is more accurate than recent methods available in the literature.     

The Dual Gap Function for Variational Inequalities

In this paper we further study the dual gap function G, which was introduced by Marcotte and Zhu, for the variational inequality problem (VIP). We characterize the directional derivative and subdifferential of G. Based on these, we get a better understanding of the concepts of a global error bound, weak sharpness, and minimum principle sufficiency property for the pseudo-monotone VIP.     

On the Iteration Complexity of Some Projection Methods for Monotone Linear Variational Inequalities

Projection-type methods are important for solving monotone linear variational inequalities. In this paper, we analyze the iteration complexity of two projection methods and accordingly establish their worst-case sublinear convergence rates measured by the iteration complexity in both the ergodic and nonergodic senses. Our analysis does not require any error bound condition or the boundedness of the feasible set, and it is scalable to other methods of the same kind.     

广义混合变分不等式的近似点-投影算法

引入了求解广义混合变分不等式的近似点-投影算法,证明了由算法所生成迭代序列强收敛于非扩张映射不动点集合与广义混合变分不等式解集合的公共元素.方法和结果是新的,且推广了这一领域内许多已知结果.