Vector quasi-equilibrium problems: separation,saddle points and error bounds for the solution set

In this paper, we employ the image space analysis (for short, ISA) to investigate vector quasi-equilibrium problems (for short, VQEPs) with a variable ordering relation, the constrained condition of which also consists of a variable ordering relation. The quasi relatively weak VQEP (for short, qr-weak VQEP) are defined by introducing the notion of the quasi relative interior. Linear separation for VQEP (res., qr-weak VQEP) is characterized by utilizing the quasi interior of a regularization of the image and the saddle points of generalized Lagrangian functions. Lagrangian type optimality conditions for VQEP (res., qr-weak VQEP) are then presented. Gap functions for VQEP (res., qr-weak VQEP) are also provided and moreover, it is shown that an error bound holds for the solution set of VQEP (res., qr-weak VQEP) with respect to the gap function under strong monotonicity.     

Error Bounds of Regularized Gap Functions for Nonsmooth Variational Inequality Problems

We study the Clarke–Rockafellar directional derivatives of the regularized gap functions (and of some modified ones) for the variational inequality problem (VIP) defined by a locally Lipschitz but not necessarily differentiable function on a closed convex set in an Euclidean space. As applications we show that, under the strong monotonicity assumption, the regularized gap functions have fractional exponent error bounds and consequently that the sequences provided by an algorithm of Armijo type converge to the solution of the (VIP). The research of this author was supported by an Earmarked Grant from the Research Council of Hong Kong.     

Conditions for Error Bounds and Bounded Level Sets of Some Merit Functions for the Second-Order Cone Complementarity Problem

Recently this author studied several merit functions systematically for the second-order cone complementarity problem. These merit functions were shown to enjoy some favorable properties, to provide error bounds under the condition of strong monotonicity, and to have bounded level sets under the conditions of monotonicity as well as strict feasibility. In this paper, we weaken the condition of strong monotonicity to the so-called uniform P ^*-property, which is a new concept recently developed for linear and nonlinear transformations on Euclidean Jordan algebra. Moreover, we replace the monotonicity and strict feasibility by the so-called R ₀₁ or R ₀₂-functions to keep the property of bounded level sets. This work is partially supported by National Science Council of Taiwan.     

Regularized gap functions for nonsmooth variational inequality problems

Under the condition that the involved function F is locally Lipschitz, but not necessarily differentiable, we investigate the regularized gap function defined by a generalized distance function for the variational inequality problem (VIP). First, we compute exactly the Clarke-Rockafellar directional derivatives of the regularized gap functions (and of some modified ones). Second, using these results, we show that, under the strongly monotonicity assumption, the regularized gap functions have fractional exponent error bounds, and thereby we provide an algorithm of Armijo type to solve the VIP.     

Nonsmooth analysis on smooth manifolds

We study infinitesimal properties of nonsmooth (nondifferentiable) functions on smooth manifolds. The eigenvalue function of a matrix on the manifold of symmetric matrices gives a natural example of such a nonsmooth function.

A subdifferential calculus for lower semicontinuous functions is developed here for studying constrained optimization problems, nonclassical problems of calculus of variations, and generalized solutions of first-order partial differential equations on manifolds. We also establish criteria for monotonicity and invariance of functions and sets with respect to solutions of differential inclusions.

     

Weak sharpness for gap functions in vector variational inequalities

In this paper, characterizations of the set of solutions for VVI are presented by using scalarization approaches. The set of solutions of VVI is shown to be the set of weak sharpness for gap functions of some scalarization of VVI and for gap functions of VVI under semi-strong monotonicity. Some examples are given to illustrate these results.     

无交叉多指标分位数回归模型中的估计与推断

在过去的30年中分位数回归模型的研究已十分深入.然而在实际的应用场景中,由传统估计方法所得到的分位数回归估计量,经常会在不同分位数水平上出现互相交叉的现象,这给分位数回归模型的实际应用造成了解释和预测上的困难.为解决这个问题,本文提出一种带单调约束的半参数多指标分位数回归模型的研究框架.首先将半参数多指标分位数回归模型与充分降维模型相结合,并利用两者间的联系获得指标估计量的相合估计.之后使用张量积样条方法拟合半参数模型在单调约束条件下的非参数结构.通过数值模拟的方式比较所提方法与现有可行方案所得结果在平均预测误差上的差异,实验结果和实际案例的结果都验证了本文所提出模型的可行性.     

Augmented Lagrangian functions for constrained optimization problems

In this paper, in order to obtain some existence results about solutions of the augmented Lagrangian problem for a constrained problem in which the objective function and constraint functions are noncoercive, we construct a new augmented Lagrangian function by using an auxiliary function. We establish a zero duality gap result and a sufficient condition of an exact penalization representation for the constrained problem without the coercive or level-bounded assumption on the objective function and constraint functions. By assuming that the sequence of multipliers is bounded, we obtain the existence of a global minimum and an asymptotically minimizing sequence for the constrained optimization problem.     

Gap functions for constrained vector variational inequalities with applications

In this paper, the image space analysis is applied to investigate scalar-valued gap functions and their applications for a (parametric)-constrained vector variational inequality. Firstly, using a non-linear regular weak separation function in image space, a gap function of a constrained vector variational inequality is obtained without any assumptions. Then, as an application of the gap function, two error bounds for the constrained vector variational inequality are derived by means of the gap function under some mild assumptions. Further, a parametric gap function of a parametric constrained vector variational inequality is presented. As an application of the parametric gap function, a sufficient condition for the continuity of the solution map of the parametric constrained vector variational inequality is established within the continuity and strict convexity of the parametric gap function. These assumptions do not include any information on the solution set of the parametric constrained vector variational inequality.     

Vector Variational Inequalities Involving Set-valued Mappings via Scalarization with Applications to Error Bounds for Gap Functions

In this paper, by using the scalarization approach of Konnov, several kinds of strong and weak scalar variational inequalities (SVI and WVI) are introduced for studying strong and weak vector variational inequalities (SVVI and WVVI) with set-valued mappings, and their gap functions are suggested. The equivalence among SVVI, WVVI, SVI, WVI is then established under suitable conditions and the relations among their gap functions are analyzed. These results are finally applied to the error bounds for gap functions. Some existence theorems of global error bounds for gap functions are obtained under strong monotonicity and several characterizations of global (respectively local) error bounds for the gap functions are derived.     

A unifying theory of exactness of linear penalty functions II: parametric penalty functions

In this article, we develop a general theory of exact parametric penalty functions for constrained optimization problems. The main advantage of the method of parametric penalty functions is the fact that a parametric penalty function can be both smooth and exact unlike the standard (i.e. non-parametric) exact penalty functions that are always nonsmooth. We obtain several necessary and/or sufficient conditions for the exactness of parametric penalty functions, and for the zero duality gap property to hold true for these functions. We also prove some convergence results for the method of parametric penalty functions, and derive necessary and sufficient conditions for a parametric penalty function to not have any stationary points outside the set of feasible points of the constrained optimization problem under consideration. In the second part of the paper, we apply the general theory of exact parametric penalty functions to a class of parametric penalty functions introduced by Huyer and Neumaier, and to smoothing approximations of nonsmooth exact penalty functions. The general approach adopted in this article allowed us to unify and significantly sharpen many existing results on parametric penalty functions.     

The Asymptotic and Numerical Inversion of the Marcum Q‐Function

The generalized Marcum functions appear in problems of technical and scientific areas such as, for example, radar detection and communications. In mathematical statistics and probability theory these functions are called the noncentral gamma or the noncentral chi‐squared cumulative distribution functions. In this paper, we describe a new asymptotic method for inverting the generalized Marcum Q‐function and for the complementary Marcum P‐function. Also, we show how monotonicity and convexity properties of these functions can be used to find initial values for reliable Newton or secant methods to invert the function. We present details of numerical computations that show the reliability of the asymptotic approximations.     

GLOBAL CONVERGENCE OF NONMONOTONIC TRUST REGION ALGORITHM FOR NONLINEAR OPTIMIZATION

§ 1　 IntroductionIn this paper we study the following nonlinear equality constrained optimization prob-lem:minimize f(x) ,subjectto h(x) =0 ,(P)where h(x) =(h1 (x) ,h2 (x) ,...,hm(x) ) T,f and hi(i=1 ,2 ,...,m) are Rn→R twice conti-nously differentiable(m≤n) .Many authors have studied the problem(P) with trustregion method(see,references[1～ 3 ] ) .These methods have the same property:to enforce strict monotonicity for meritfunction at every iteration.Paper[4 ] shows thatstrictmonotonic …     

非线性优化的非单调信赖域算法及全局收敛性

Abstract. A trust region algorithm for equality constrained optimization is given in this paper.The algorithm does not enforce strict monotonicity of the merit function for every iteration.Global convergence of the algorithm is proved under the same conditions of usual trust regionmethod.     

Redheffer type bounds for Bessel and modified Bessel functions of the first kind

In this paper our aim is to show some new inequalities of the Redheffer type for Bessel and modified Bessel functions of the first kind. The key tools in our proofs are some classical results on the monotonicity of quotients of differentiable functions as well as on the monotonicity of quotients of two power series. We also use some known results on the quotients of Bessel and modified Bessel functions of the first kind, and by using the monotonicity of the Dirichlet eta function we prove a sharp inequality for the tangent function. At the end of the paper a conjecture is stated, which may be of interest for further research.     

Functional ANOVA,ultramodularity and monotonicity: Applications in multiattribute utility theory

Utility function properties as monotonicity and concavity play a fundamental role in reflecting a decision-maker’s preference structure. These properties are usually characterized via partial derivatives. However, elicitation methods do not necessarily lead to twice-differentiable utility functions. Furthermore, while in a single-attribute context concavity fully reflects risk aversion, in multiattribute problems such correspondence is not one-to-one. We show that Tsetlin and Winkler’s multivariate risk attitudes imply ultramodularity of the utility function. We demonstrate that geometric properties of a multivariate utility function can be successfully studied by utilizing an integral function expansion (functional ANOVA). The necessary and sufficient conditions under which monotonicity and/or ultramodularity of single-attribute functions imply the monotonicity and/or ultramodularity of the corresponding multiattribute function under additive, preferential and mutual utility independence are then established without reliance on the utility function differentiability. We also investigate the relationship between the presence of interactions among the attributes of a multiattribute utility function and the decision-maker’s multivariate risk attitudes.     

Nonparametric Estimation of Hazard Rate Under the Constraint of Monotonicity

This article shows how to smoothly “monotonize” standard kernel estimators of hazard rate, using bootstrap weights. Our method takes a variety of forms, depending on choice of kernel estimator and on the distance function used to define a certain constrained optimization problem. We confine attention to a particularly simple kernel approach and explore a range of distance functions. It is straightforward to reduce “quadratic” inequality constraints to “linear” equality constraints, and so our method may be implemented using little more than conventional Newton–Raphson iteration. Thus, the necessary computational techniques are very familiar to statisticians. We show both numerically and theoretically that monotonicity, in either direction, can generally be imposed on a kernel hazard rate estimator regardless of the monotonicity or otherwise of the true hazard rate. The case of censored data is easily accommodated. Our methods have straightforward extension to the problem of testing for monotonicity of hazard rate, where the distance function plays the role of a test statistic.     

A branch-reduce-cut algorithm for the global optimization of probabilistically constrained linear programs

We consider probabilistically constrained linear programs with general distributions for the uncertain parameters. These problems involve non-convex feasible sets. We develop a branch-and-bound algorithm that searches for a global optimal solution to this problem by successively partitioning the non-convex feasible region and by using bounds on the objective function to fathom inferior partition elements. This basic algorithm is enhanced by domain reduction and cutting plane strategies to reduce the size of the partition elements and hence tighten bounds. The proposed branch-reduce-cut algorithm exploits the monotonicity properties inherent in the problem, and requires solving linear programming subproblems. We provide convergence proofs for the algorithm. Some illustrative numerical results involving problems with discrete distributions are presented.     

Admissible slopes for monotone and convex interpolation

Summary In many applications, interpolation of experimental data exhibiting some geometric property such as nonnegativity, monotonicity or convexity is unacceptable unless the interpolant reflects these characteristics. This paper identifies admissible slopes at data points of variousC ¹ interpolants which ensure a desirable shape. We discuss this question, in turn for the following function classes commonly used for shape preserving interpolations: monotone polynomials,C ¹ monotone piecewise polynomials, convex polynomials, parametric cubic curves and rational functions.     

Unconstrained minimization approaches to nonlinear complementarity problems

The nonlinear complementarity problem can be reformulated as unconstrained minimization problems by introducing merit functions. Under some assumptions, the solution set of the nonlinear complementarity problem coincides with the set of local minima of the corresponding minimization problem. These results were presented by Mangasarian and Solodov, Yamashita and Fukushima, and Geiger and Kanzow. In this note, we generalize some results of Mangasarian and Solodov, Yamashita and Fukushima, and Geiger and Kanzow to the case where the considered function is only directionally differentiable. Some results are strengthened in the smooth case. For example, it is shown that the strong monotonicity condition can be replaced by the P-uniform property for ensuring a stationary point of the reformulated unconstrained minimization problems to be a solution of the nonlinear complementarity problem. We also present a descent algorithm for solving the nonlinear complementarity problem in the smooth case. Any accumulation point generated by this algorithm is proved to be a solution of the nonlinear complementarity under the monotonicity condition.